Abstract group 4800.bk: $C_5\times \GL(2,3):D_{10}$

• Label: 4800.bk
• Order: \( 2^{6} \cdot 3 \cdot 5^{2} \)
• Exponent: \( 2^{3} \cdot 3 \cdot 5 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 5 \)
• $\card{Z(G)}$: \( 2 \cdot 5 \)
• $\card{\Aut(G)}$: \( 2^{10} \cdot 3 \cdot 5 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \)
• Perm deg.: $26$
• Trans deg.: $80$
• Rank: $3$

Group information

Description:	$C_5\times \GL(2,3):D_{10}$
Order:	\(4800\)\(\medspace = 2^{6} \cdot 3 \cdot 5^{2} \)
Exponent:	\(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \)
Automorphism group:	$(C_2\times A_4\times F_5).C_2^5$, of order \(15360\)\(\medspace = 2^{10} \cdot 3 \cdot 5 \)
Composition factors:	$C_2$ x 6, $C_3$, $C_5$ x 2
Derived length:	$4$

This group is nonabelian and solvable.

Group statistics

Order	1	2	3	4	5	6	8	10	12	15	20	30	40	60
Elements	1	127	8	112	24	104	144	1048	80	192	688	896	1056	320	4800
Conjugacy classes	1	7	1	5	14	3	5	78	1	14	40	42	60	4	275
Divisions	1	7	1	5	4	3	4	21	1	4	11	11	10	1	84
Autjugacy classes	1	7	1	5	3	3	4	15	1	3	9	7	8	1	68

Dimension	1	2	3	4	6	8	12	16	24	32
Irr. complex chars.	40	60	40	70	40	25	0	0	0	0	275
Irr. rational chars.	8	4	8	12	0	16	12	4	8	12	84

Minimal presentations

Permutation degree:	$26$
Transitive degree:	$80$
Rank:	$3$
Inequivalent generating triples:	$1874880$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	4	8	32
Arbitrary	4	8	16

Constructions

Presentation:	${\langle a, b, c, d, e \mid a^{10}=b^{2}=c^{6}=e^{20}=[a,b]=[a,d]=[a,e]= \!\cdots\! \rangle}$
Permutation group:	Degree $26$ $\langle(17,18,20,22,26)(19,23,24,21,25), (1,2,3,8)(4,7,9,15)(5,6,10,11)(12,16,14,13) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrr} 3 & 0 & 0 & 0 \\ 0 & 3 & 0 & 0 \\ 0 & 0 & 3 & 0 \\ 0 & 0 & 0 & 3 \end{array}\right), \left(\begin{array}{rrrr} 0 & 7 & 0 & 8 \\ 3 & 1 & 5 & 3 \\ 3 & 6 & 5 & 1 \\ 10 & 5 & 7 & 5 \end{array}\right), \left(\begin{array}{rrrr} 7 & 10 & 10 & 9 \\ 6 & 0 & 5 & 10 \\ 4 & 5 & 9 & 1 \\ 8 & 4 & 5 & 2 \end{array}\right), \left(\begin{array}{rrrr} 5 & 10 & 10 & 9 \\ 6 & 9 & 5 & 10 \\ 4 & 5 & 7 & 1 \\ 8 & 4 & 5 & 0 \end{array}\right), \left(\begin{array}{rrrr} 4 & 6 & 3 & 1 \\ 9 & 7 & 6 & 4 \\ 4 & 8 & 8 & 6 \\ 4 & 6 & 2 & 3 \end{array}\right), \left(\begin{array}{rrrr} 3 & 4 & 10 & 0 \\ 1 & 8 & 0 & 1 \\ 1 & 0 & 8 & 4 \\ 0 & 10 & 1 & 3 \end{array}\right), \left(\begin{array}{rrrr} 8 & 0 & 0 & 0 \\ 0 & 8 & 0 & 0 \\ 0 & 0 & 8 & 0 \\ 0 & 0 & 0 & 8 \end{array}\right), \left(\begin{array}{rrrr} 0 & 8 & 4 & 3 \\ 1 & 10 & 4 & 0 \\ 4 & 9 & 6 & 6 \\ 3 & 8 & 2 & 1 \end{array}\right), \left(\begin{array}{rrrr} 2 & 4 & 1 & 3 \\ 5 & 0 & 6 & 1 \\ 7 & 4 & 0 & 7 \\ 3 & 7 & 6 & 9 \end{array}\right) \right\rangle \subseteq \GL_{4}(\F_{11})$
Direct product:	$C_5$ $\, \times\, $ $(\GL(2,3):D_{10})$
Semidirect product:	$(Q_8.D_{30})$ $\,\rtimes\,$ $C_{10}$	$(C_{20}.D_{10})$ $\,\rtimes\,$ $D_6$	$(D_{20}:C_{10})$ $\,\rtimes\,$ $D_6$	$(C_{10}^2.S_4)$ $\,\rtimes\,$ $C_2$	all 44
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$D_{10}$ . $(C_{10}\times S_4)$	$C_{10}$ . $(D_{10}\times S_4)$	$Q_8$ . $(C_{30}:D_{10})$	$(C_{10}\wr C_2)$ . $S_4$	all 22

Elements of the group are displayed as matrices in $\GL_{4}(\F_{11})$.

Homology

Abelianization:	$C_{2}^{2} \times C_{10} \simeq C_{2}^{3} \times C_{5}$
Schur multiplier:	$C_{2}^{3}$
Commutator length:	$1$

Subgroups

There are 6608 subgroups in 710 conjugacy classes, 74 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_{10}$	$G/Z \simeq$ $D_{10}\times S_4$
Commutator:	$G' \simeq$ $C_5\times \SL(2,3)$	$G/G' \simeq$ $C_2^2\times C_{10}$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $(C_5^2\times A_4).C_2^3$
Fitting:	$\operatorname{Fit} \simeq$ $C_4.C_{10}^2$	$G/\operatorname{Fit} \simeq$ $D_6$
Radical:	$R \simeq$ $C_5\times \GL(2,3):D_{10}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_5\times C_{10}$	$G/\operatorname{soc} \simeq$ $C_2^2\times S_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_8:C_2^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5^2$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available

Classes of subgroups up to automorphism

No diagram available

Normal subgroups

No diagram available

Normal subgroups up to automorphism

No diagram available

Classes of subgroups up to conjugation

Order 4800: $C_5\times \GL(2,3):D_{10}$

Order 2400: $C_5\times \GL(2,3):D_5$, $\SL(2,3):C_{10}^2$, $C_5\times \GL(2,3):D_5$, $C_5\times \GL(2,3):D_5$, $(Q_8\times C_{10}):C_{30}$, $C_5\times D_5\times \GL(2,3)$, $C_{10}^2.S_4$

Order 1600: $C_5\times D_{40}:C_2^2$

Order 1200: $C_5^2\times \GL(2,3)$ x 2, $C_{10}^2.A_4$, $C_{10}^2:D_6$, $C_5\times D_5\times \SL(2,3)$, $C_5\times \SL(2,3):D_5$, $C_5^2:\GL(2,3)$, $C_5\times C_{10}.S_4$

Order 960: $C_5\times \GL(2,3):C_2^2$, $\GL(2,3):D_{10}$

Order 800: $C_{40}.D_{10}$ x 2, $C_{40}.D_{10}$ x 2, $C_{40}:D_{10}$ x 2, $C_{40}:D_{10}$ x 2, $C_8:C_{10}^2$, $C_{10}^2.D_4$, $C_{10}^2.D_4$, $D_{40}:C_{10}$, $C_{40}.D_{10}$, $C_{10}^2.C_2^3$, $C_{10}^2.C_2^3$

Order 600: $D_{30}:C_{10}$, $C_{30}.D_{10}$, $C_{30}.D_{10}$, $S_3\times C_{10}^2$, $C_{30}:D_{10}$, $D_{30}:C_{10}$, $D_{10}:C_{30}$, $C_5^2\times \SL(2,3)$

Order 480: $C_{10}\times \GL(2,3)$ x 5, $\GL(2,3):C_{10}$ x 2, $D_5\times \GL(2,3)$, $\GL(2,3):D_5$, $\GL(2,3):D_5$, $\GL(2,3):D_5$, $(C_2\times Q_8):C_{30}$, $\SL(2,3).D_{10}$, $Q_8.D_{30}$, $\GL(2,3):C_{10}$, $\GL(2,3):C_{10}$

Order 400: $\SD_{16}\times C_5^2$ x 4, $C_{20}.D_{10}$ x 3, $C_{20}:D_{10}$ x 3, $D_{20}:C_{10}$ x 3, $C_5^2:Q_{16}$ x 2, $C_{20}.D_{10}$ x 2, $C_{20}.D_{10}$ x 2, $D_{20}:C_{10}$ x 2, $C_{40}:C_{10}$ x 2, $C_{40}:C_{10}$ x 2, $D_5\times C_{40}$ x 2, $C_{10}^2:C_2^2$ x 2, $D_{20}:C_{10}$ x 2, $C_{20}.D_{10}$ x 2, $C_{20}.C_{20}$, $C_5^2:Q_{16}$, $C_5\times D_{40}$, $C_4.C_{10}^2$, $C_4:C_{10}^2$, $C_{10}\times C_{40}$

Order 320: $C_{40}.C_2^3$, $D_{40}:C_2^2$

Order 300: $D_6\times C_5^2$ x 4, $C_{15}:D_{10}$ x 2, $C_{10}\times C_{30}$, $C_5\times D_{30}$, $C_{15}\times D_{10}$, $C_{15}:C_{20}$, $C_5:C_{60}$

Order 240: $C_5\times \GL(2,3)$ x 9, $C_{10}\times \SL(2,3)$ x 5, $C_{20}:D_6$, $\SL(2,3):C_{10}$, $D_6:D_{10}$, $D_5\times \SL(2,3)$, $\SL(2,3):D_5$, $C_5:\GL(2,3)$, $C_{10}.S_4$, $C_{10}.S_4$

Order 200: $C_{10}\wr C_2$ x 7, $D_5\times C_{20}$ x 4, $D_4\times C_5^2$ x 3, $C_5\times D_{20}$ x 3, $C_5^2:Q_8$ x 3, $C_{10}\times D_{10}$ x 2, $Q_8\times C_5^2$ x 2, $C_{10}:C_{20}$ x 2, $C_5\times C_{40}$ x 2, $C_5:C_{40}$ x 2, $C_2\times C_{10}^2$, $C_{10}\times C_{20}$

Order 192: $\GL(2,3):C_2^2$

Order 160: $C_{10}\times \SD_{16}$ x 6, $Q_{16}:C_{10}$ x 3, $D_8:C_{10}$ x 3, $D_8:C_{10}$ x 3, $C_8.D_{10}$ x 2, $C_8.D_{10}$ x 2, $C_8:D_{10}$ x 2, $D_5\times \SD_{16}$ x 2, $C_{10}.C_2^4$, $C_{10}.C_2^4$, $Q_8.D_{10}$, $D_4:D_{10}$, $\OD_{16}:C_{10}$, $C_{20}.D_4$, $D_{20}:C_2^2$, $D_{40}:C_2$, $C_8.D_{10}$

Order 150: $S_3\times C_5^2$ x 3, $C_5\times C_{30}$ x 2, $D_5\times C_{15}$, $C_5\times D_{15}$

Order 120: $C_{10}\times D_6$ x 5, $C_5\times \SL(2,3)$ x 4, $C_{15}:D_4$ x 2, $C_6.D_{10}$, $S_3\times D_{10}$, $D_4\times C_{15}$, $C_{15}:D_4$, $C_5\times D_{12}$, $S_3\times C_{20}$, $C_{15}:D_4$, $C_5:D_{12}$, $C_{15}:D_4$

Order 100: $C_5\times D_{10}$ x 5, $C_{10}^2$ x 4, $C_5:C_{20}$ x 3, $C_5\times C_{20}$ x 2

Order 96: $\GL(2,3):C_2$ x 2, $C_2\times \GL(2,3)$ x 2, $D_4.A_4$, $\GL(2,3):C_2$, $\GL(2,3):C_2$

Order 80: $C_5\times \SD_{16}$ x 22, $D_4\times C_{10}$ x 9, $D_4:C_{10}$ x 8, $Q_8\times C_{10}$ x 6, $C_2\times C_{40}$ x 6, $D_4:D_5$ x 3, $D_4\times D_5$ x 3, $D_{20}:C_2$ x 3, $C_5\times Q_{16}$ x 3, $C_5\times D_8$ x 3, $C_5\times \OD_{16}$ x 3, $C_{40}:C_2$ x 2, $C_{40}:C_2$ x 2, $C_{10}:D_4$ x 2, $D_{20}:C_2$ x 2, $Q_8\times D_5$ x 2, $C_8\times D_5$ x 2, $C_5:Q_{16}$ x 2, $Q_8:D_5$ x 2, $C_5:\SD_{16}$ x 2, $C_5:D_8$ x 2, $C_5:Q_{16}$, $D_{40}$, $C_5:\OD_{16}$

Order 75: $C_5\times C_{15}$

Order 64: $D_8:C_2^2$

Order 60: $S_3\times C_{10}$ x 23, $C_2\times C_{30}$ x 5, $S_3\times D_5$ x 2, $C_{60}$, $C_{15}:C_4$, $C_5:C_{12}$, $D_{30}$, $C_3\times D_{10}$, $C_3:C_{20}$

Order 50: $C_5\times C_{10}$ x 4, $C_5\times D_5$ x 3

Order 48: $\GL(2,3)$ x 3, $C_2\times \SL(2,3)$ x 2, $S_3\times D_4$, $\SL(2,3):C_2$, $C_2.S_4$

Order 40: $C_5\times D_4$ x 22, $C_5\times Q_8$ x 11, $C_2\times C_{20}$ x 10, $C_{40}$ x 10, $C_5:D_4$ x 7, $C_2^2\times C_{10}$ x 6, $C_4\times D_5$ x 4, $D_{20}$ x 3, $C_5:Q_8$ x 3, $C_{10}:C_4$ x 2, $C_2\times D_{10}$ x 2, $C_5:C_8$ x 2

Order 32: $Q_{16}:C_2$ x 3, $D_8:C_2$ x 3, $D_8:C_2$ x 3, $C_2\times \SD_{16}$ x 3, $C_4.C_2^3$, $D_4:C_2^2$, $\OD_{16}:C_2$

Order 30: $C_5\times S_3$ x 15, $C_{30}$ x 11, $D_{15}$, $C_3\times D_5$

Order 25: $C_5^2$

Order 24: $C_3:D_4$ x 2, $C_2\times D_6$ x 2, $D_{12}$, $C_4\times S_3$, $\SL(2,3)$, $C_3\times D_4$

Order 20: $C_2\times C_{10}$ x 23, $C_{20}$ x 11, $D_{10}$ x 5, $C_5:C_4$ x 3

Order 16: $\SD_{16}$ x 10, $D_4:C_2$ x 8, $C_2\times D_4$ x 6, $Q_{16}$ x 3, $D_8$ x 3, $\OD_{16}$ x 3, $C_2\times C_8$ x 3, $C_2\times Q_8$ x 3

Order 15: $C_{15}$ x 4

Order 12: $D_6$ x 7, $C_2\times C_6$ x 2, $C_{12}$, $C_3:C_4$

Order 10: $C_{10}$ x 21, $D_5$ x 3

Order 8: $D_4$ x 13, $C_2\times C_4$ x 7, $Q_8$ x 5, $C_8$ x 4, $C_2^3$ x 3

Order 6: $S_3$ x 4, $C_6$ x 3

Order 5: $C_5$ x 4

Order 4: $C_2^2$ x 9, $C_4$ x 5

Order 3: $C_3$

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 4800: $C_5\times \GL(2,3):D_{10}$

Order 2400: $C_5\times \GL(2,3):D_5$, $\SL(2,3):C_{10}^2$, $C_5\times \GL(2,3):D_5$, $C_5\times \GL(2,3):D_5$, $(Q_8\times C_{10}):C_{30}$, $C_5\times D_5\times \GL(2,3)$, $C_{10}^2.S_4$

Order 1600: $C_5\times D_{40}:C_2^2$

Order 1200: $C_5^2\times \GL(2,3)$ x 2, $C_{10}^2.A_4$, $C_{10}^2:D_6$, $C_5\times D_5\times \SL(2,3)$, $C_5\times \SL(2,3):D_5$, $C_5^2:\GL(2,3)$, $C_5\times C_{10}.S_4$

Order 960: $C_5\times \GL(2,3):C_2^2$, $\GL(2,3):D_{10}$

Order 800: $C_{40}.D_{10}$ x 2, $C_{40}.D_{10}$ x 2, $C_{40}:D_{10}$ x 2, $C_{40}:D_{10}$ x 2, $C_8:C_{10}^2$, $C_{10}^2.D_4$, $C_{10}^2.D_4$, $D_{40}:C_{10}$, $C_{40}.D_{10}$, $C_{10}^2.C_2^3$, $C_{10}^2.C_2^3$

Order 600: $D_{30}:C_{10}$, $C_{30}.D_{10}$, $C_{30}.D_{10}$, $S_3\times C_{10}^2$, $C_{30}:D_{10}$, $D_{30}:C_{10}$, $D_{10}:C_{30}$, $C_5^2\times \SL(2,3)$

Order 480: $C_{10}\times \GL(2,3)$ x 4, $\GL(2,3):C_{10}$ x 2, $D_5\times \GL(2,3)$, $\GL(2,3):D_5$, $\GL(2,3):D_5$, $\GL(2,3):D_5$, $(C_2\times Q_8):C_{30}$, $\SL(2,3).D_{10}$, $Q_8.D_{30}$, $\GL(2,3):C_{10}$, $\GL(2,3):C_{10}$

Order 400: $\SD_{16}\times C_5^2$ x 4, $C_{20}.D_{10}$ x 3, $C_{20}:D_{10}$ x 3, $D_{20}:C_{10}$ x 3, $C_5^2:Q_{16}$ x 2, $C_{20}.D_{10}$ x 2, $C_{20}.D_{10}$ x 2, $D_{20}:C_{10}$ x 2, $C_{40}:C_{10}$ x 2, $C_{40}:C_{10}$ x 2, $D_5\times C_{40}$ x 2, $C_{10}^2:C_2^2$ x 2, $D_{20}:C_{10}$ x 2, $C_{20}.D_{10}$ x 2, $C_{20}.C_{20}$, $C_5^2:Q_{16}$, $C_5\times D_{40}$, $C_4.C_{10}^2$, $C_4:C_{10}^2$, $C_{10}\times C_{40}$

Order 320: $C_{40}.C_2^3$, $D_{40}:C_2^2$

Order 300: $D_6\times C_5^2$ x 3, $C_{10}\times C_{30}$, $C_5\times D_{30}$, $C_{15}\times D_{10}$, $C_{15}:D_{10}$, $C_{15}:C_{20}$, $C_5:C_{60}$

Order 240: $C_5\times \GL(2,3)$ x 7, $C_{10}\times \SL(2,3)$ x 4, $C_{20}:D_6$, $\SL(2,3):C_{10}$, $D_6:D_{10}$, $D_5\times \SL(2,3)$, $\SL(2,3):D_5$, $C_5:\GL(2,3)$, $C_{10}.S_4$, $C_{10}.S_4$

Order 200: $C_{10}\wr C_2$ x 7, $D_5\times C_{20}$ x 4, $D_4\times C_5^2$ x 3, $C_5\times D_{20}$ x 3, $C_5^2:Q_8$ x 3, $C_{10}\times D_{10}$ x 2, $Q_8\times C_5^2$ x 2, $C_{10}:C_{20}$ x 2, $C_5\times C_{40}$ x 2, $C_5:C_{40}$ x 2, $C_2\times C_{10}^2$, $C_{10}\times C_{20}$

Order 192: $\GL(2,3):C_2^2$

Order 160: $C_{10}\times \SD_{16}$ x 5, $Q_{16}:C_{10}$ x 3, $D_8:C_{10}$ x 3, $D_8:C_{10}$ x 3, $C_8.D_{10}$ x 2, $C_8.D_{10}$ x 2, $C_8:D_{10}$ x 2, $D_5\times \SD_{16}$ x 2, $C_{10}.C_2^4$, $C_{10}.C_2^4$, $Q_8.D_{10}$, $D_4:D_{10}$, $\OD_{16}:C_{10}$, $C_{20}.D_4$, $D_{20}:C_2^2$, $D_{40}:C_2$, $C_8.D_{10}$

Order 150: $C_5\times C_{30}$ x 2, $S_3\times C_5^2$ x 2, $D_5\times C_{15}$, $C_5\times D_{15}$

Order 120: $C_{10}\times D_6$ x 4, $C_5\times \SL(2,3)$ x 3, $C_{15}:D_4$ x 2, $C_6.D_{10}$, $S_3\times D_{10}$, $D_4\times C_{15}$, $C_{15}:D_4$, $C_5\times D_{12}$, $S_3\times C_{20}$, $C_{15}:D_4$, $C_5:D_{12}$, $C_{15}:D_4$

Order 100: $C_5\times D_{10}$ x 5, $C_{10}^2$ x 4, $C_5:C_{20}$ x 3, $C_5\times C_{20}$ x 2

Order 96: $\GL(2,3):C_2$ x 2, $C_2\times \GL(2,3)$ x 2, $D_4.A_4$, $\GL(2,3):C_2$, $\GL(2,3):C_2$

Order 80: $C_5\times \SD_{16}$ x 18, $D_4:C_{10}$ x 8, $D_4\times C_{10}$ x 8, $Q_8\times C_{10}$ x 5, $C_2\times C_{40}$ x 5, $D_4:D_5$ x 3, $D_4\times D_5$ x 3, $D_{20}:C_2$ x 3, $C_5\times Q_{16}$ x 3, $C_5\times D_8$ x 3, $C_5\times \OD_{16}$ x 3, $C_{40}:C_2$ x 2, $C_{40}:C_2$ x 2, $C_{10}:D_4$ x 2, $D_{20}:C_2$ x 2, $Q_8\times D_5$ x 2, $C_8\times D_5$ x 2, $C_5:Q_{16}$ x 2, $Q_8:D_5$ x 2, $C_5:\SD_{16}$ x 2, $C_5:D_8$ x 2, $C_5:Q_{16}$, $D_{40}$, $C_5:\OD_{16}$

Order 75: $C_5\times C_{15}$

Order 64: $D_8:C_2^2$

Order 60: $S_3\times C_{10}$ x 11, $C_2\times C_{30}$ x 4, $S_3\times D_5$, $C_{60}$, $C_{15}:C_4$, $C_5:C_{12}$, $D_{30}$, $C_3\times D_{10}$, $C_3:C_{20}$

Order 50: $C_5\times C_{10}$ x 4, $C_5\times D_5$ x 3

Order 48: $\GL(2,3)$ x 3, $C_2\times \SL(2,3)$ x 2, $S_3\times D_4$, $\SL(2,3):C_2$, $C_2.S_4$

Order 40: $C_5\times D_4$ x 19, $C_2\times C_{20}$ x 9, $C_5\times Q_8$ x 9, $C_{40}$ x 8, $C_5:D_4$ x 7, $C_2^2\times C_{10}$ x 5, $C_4\times D_5$ x 4, $D_{20}$ x 3, $C_5:Q_8$ x 3, $C_{10}:C_4$ x 2, $C_2\times D_{10}$ x 2, $C_5:C_8$ x 2

Order 32: $Q_{16}:C_2$ x 3, $D_8:C_2$ x 3, $D_8:C_2$ x 3, $C_2\times \SD_{16}$ x 3, $C_4.C_2^3$, $D_4:C_2^2$, $\OD_{16}:C_2$

Order 30: $C_{30}$ x 7, $C_5\times S_3$ x 7, $D_{15}$, $C_3\times D_5$

Order 25: $C_5^2$

Order 24: $C_3:D_4$ x 2, $C_2\times D_6$ x 2, $D_{12}$, $C_4\times S_3$, $\SL(2,3)$, $C_3\times D_4$

Order 20: $C_2\times C_{10}$ x 17, $C_{20}$ x 9, $D_{10}$ x 5, $C_5:C_4$ x 3

Order 16: $\SD_{16}$ x 10, $D_4:C_2$ x 8, $C_2\times D_4$ x 6, $Q_{16}$ x 3, $D_8$ x 3, $\OD_{16}$ x 3, $C_2\times C_8$ x 3, $C_2\times Q_8$ x 3

Order 15: $C_{15}$ x 3

Order 12: $D_6$ x 5, $C_2\times C_6$ x 2, $C_{12}$, $C_3:C_4$

Order 10: $C_{10}$ x 15, $D_5$ x 3

Order 8: $D_4$ x 13, $C_2\times C_4$ x 7, $Q_8$ x 5, $C_8$ x 4, $C_2^3$ x 3

Order 6: $C_6$ x 3, $S_3$ x 3

Order 5: $C_5$ x 3

Order 4: $C_2^2$ x 9, $C_4$ x 5

Order 3: $C_3$

Order 2: $C_2$ x 7

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 4800: $C_5\times \GL(2,3):D_{10}$ ($C_1$)

Order 2400: $C_5\times \GL(2,3):D_5$ ($C_2$), $\SL(2,3):C_{10}^2$ ($C_2$), $C_5\times \GL(2,3):D_5$ ($C_2$), $C_5\times \GL(2,3):D_5$ ($C_2$), $(Q_8\times C_{10}):C_{30}$ ($C_2$), $C_5\times D_5\times \GL(2,3)$ ($C_2$), $C_{10}^2.S_4$ ($C_2$)

Order 1200: $C_5^2\times \GL(2,3)$ ($C_2^2$) x 2, $C_{10}^2.A_4$ ($C_2^2$), $C_5\times D_5\times \SL(2,3)$ ($C_2^2$), $C_5\times \SL(2,3):D_5$ ($C_2^2$), $C_5^2:\GL(2,3)$ ($C_2^2$), $C_5\times C_{10}.S_4$ ($C_2^2$)

Order 960: $\GL(2,3):D_{10}$ ($C_5$)

Order 800: $C_{10}^2.C_2^3$ ($S_3$)

Order 600: $C_5^2\times \SL(2,3)$ ($C_2^3$)

Order 480: $D_5\times \GL(2,3)$ ($C_{10}$), $\GL(2,3):D_5$ ($C_{10}$), $\GL(2,3):D_5$ ($C_{10}$), $\GL(2,3):D_5$ ($C_{10}$), $\SL(2,3).D_{10}$ ($C_{10}$), $Q_8.D_{30}$ ($C_{10}$), $C_{10}\times \GL(2,3)$ ($C_{10}$), $C_{10}\times \GL(2,3)$ ($D_5$)

Order 400: $C_4.C_{10}^2$ ($D_6$), $D_{20}:C_{10}$ ($D_6$), $C_{20}.D_{10}$ ($D_6$)

Order 240: $C_5\times \GL(2,3)$ ($C_2\times C_{10}$) x 2, $C_5\times \GL(2,3)$ ($D_{10}$) x 2, $C_{10}\times \SL(2,3)$ ($C_2\times C_{10}$), $C_{10}\times \SL(2,3)$ ($D_{10}$), $D_5\times \SL(2,3)$ ($C_2\times C_{10}$), $\SL(2,3):D_5$ ($C_2\times C_{10}$), $C_5:\GL(2,3)$ ($C_2\times C_{10}$), $C_{10}.S_4$ ($C_2\times C_{10}$)

Order 200: $Q_8\times C_5^2$ ($C_2\times D_6$), $C_{10}\wr C_2$ ($S_4$)

Order 160: $Q_8.D_{10}$ ($C_5\times S_3$)

Order 120: $C_5\times \SL(2,3)$ ($C_2^2\times C_{10}$), $C_5\times \SL(2,3)$ ($C_2\times D_{10}$)

Order 100: $C_5:C_{20}$ ($C_2\times S_4$), $C_{10}^2$ ($C_2\times S_4$), $C_5\times D_{10}$ ($C_2\times S_4$)

Order 96: $C_2\times \GL(2,3)$ ($C_5\times D_5$)

Order 80: $Q_8\times C_{10}$ ($S_3\times D_5$), $Q_8\times C_{10}$ ($S_3\times C_{10}$), $D_{20}:C_2$ ($S_3\times C_{10}$), $Q_8\times D_5$ ($S_3\times C_{10}$)

Order 50: $C_5\times C_{10}$ ($C_2^2\times S_4$)

Order 48: $\GL(2,3)$ ($C_5\times D_{10}$) x 2, $C_2\times \SL(2,3)$ ($C_5\times D_{10}$)

Order 40: $C_5:D_4$ ($C_5\times S_4$), $C_5\times Q_8$ ($C_{10}\times D_6$), $C_5\times Q_8$ ($S_3\times D_{10}$)

Order 25: $C_5^2$ ($\GL(2,3):C_2^2$)

Order 24: $\SL(2,3)$ ($C_{10}\times D_{10}$)

Order 20: $C_2\times C_{10}$ ($C_{10}\times S_4$), $C_2\times C_{10}$ ($D_5\times S_4$), $D_{10}$ ($C_{10}\times S_4$), $C_5:C_4$ ($C_{10}\times S_4$)

Order 16: $C_2\times Q_8$ ($C_{15}:D_{10}$)

Order 10: $C_{10}$ ($C_2\times C_{10}\times S_4$), $C_{10}$ ($D_{10}\times S_4$)

Order 8: $Q_8$ ($C_{30}:D_{10}$)

Order 5: $C_5$ ($C_5\times \GL(2,3):C_2^2$), $C_5$ ($\GL(2,3):D_{10}$)

Order 4: $C_2^2$ ($C_{10}^2:D_6$)

Order 2: $C_2$ ($(C_5^2\times A_4).C_2^3$)

Order 1: $C_1$ ($C_5\times \GL(2,3):D_{10}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 4800: $C_5\times \GL(2,3):D_{10}$ ($C_1$)

Order 2400: $C_5\times \GL(2,3):D_5$ ($C_2$), $\SL(2,3):C_{10}^2$ ($C_2$), $C_5\times \GL(2,3):D_5$ ($C_2$), $C_5\times \GL(2,3):D_5$ ($C_2$), $(Q_8\times C_{10}):C_{30}$ ($C_2$), $C_5\times D_5\times \GL(2,3)$ ($C_2$), $C_{10}^2.S_4$ ($C_2$)

Order 1200: $C_5^2\times \GL(2,3)$ ($C_2^2$) x 2, $C_{10}^2.A_4$ ($C_2^2$), $C_5\times D_5\times \SL(2,3)$ ($C_2^2$), $C_5\times \SL(2,3):D_5$ ($C_2^2$), $C_5^2:\GL(2,3)$ ($C_2^2$), $C_5\times C_{10}.S_4$ ($C_2^2$)

Order 960: $\GL(2,3):D_{10}$ ($C_5$)

Order 800: $C_{10}^2.C_2^3$ ($S_3$)

Order 600: $C_5^2\times \SL(2,3)$ ($C_2^3$)

Order 480: $D_5\times \GL(2,3)$ ($C_{10}$), $\GL(2,3):D_5$ ($C_{10}$), $\GL(2,3):D_5$ ($C_{10}$), $\GL(2,3):D_5$ ($C_{10}$), $\SL(2,3).D_{10}$ ($C_{10}$), $Q_8.D_{30}$ ($C_{10}$), $C_{10}\times \GL(2,3)$ ($C_{10}$), $C_{10}\times \GL(2,3)$ ($D_5$)

Order 400: $C_4.C_{10}^2$ ($D_6$), $D_{20}:C_{10}$ ($D_6$), $C_{20}.D_{10}$ ($D_6$)

Order 240: $C_5\times \GL(2,3)$ ($C_2\times C_{10}$) x 2, $C_5\times \GL(2,3)$ ($D_{10}$) x 2, $C_{10}\times \SL(2,3)$ ($C_2\times C_{10}$), $C_{10}\times \SL(2,3)$ ($D_{10}$), $D_5\times \SL(2,3)$ ($C_2\times C_{10}$), $\SL(2,3):D_5$ ($C_2\times C_{10}$), $C_5:\GL(2,3)$ ($C_2\times C_{10}$), $C_{10}.S_4$ ($C_2\times C_{10}$)

Order 200: $Q_8\times C_5^2$ ($C_2\times D_6$), $C_{10}\wr C_2$ ($S_4$)

Order 160: $Q_8.D_{10}$ ($C_5\times S_3$)

Order 120: $C_5\times \SL(2,3)$ ($C_2^2\times C_{10}$), $C_5\times \SL(2,3)$ ($C_2\times D_{10}$)

Order 100: $C_5:C_{20}$ ($C_2\times S_4$), $C_{10}^2$ ($C_2\times S_4$), $C_5\times D_{10}$ ($C_2\times S_4$)

Order 96: $C_2\times \GL(2,3)$ ($C_5\times D_5$)

Order 80: $Q_8\times C_{10}$ ($S_3\times D_5$), $Q_8\times C_{10}$ ($S_3\times C_{10}$), $D_{20}:C_2$ ($S_3\times C_{10}$), $Q_8\times D_5$ ($S_3\times C_{10}$)

Order 50: $C_5\times C_{10}$ ($C_2^2\times S_4$)

Order 48: $\GL(2,3)$ ($C_5\times D_{10}$) x 2, $C_2\times \SL(2,3)$ ($C_5\times D_{10}$)

Order 40: $C_5:D_4$ ($C_5\times S_4$), $C_5\times Q_8$ ($C_{10}\times D_6$), $C_5\times Q_8$ ($S_3\times D_{10}$)

Order 25: $C_5^2$ ($\GL(2,3):C_2^2$)

Order 24: $\SL(2,3)$ ($C_{10}\times D_{10}$)

Order 20: $C_2\times C_{10}$ ($C_{10}\times S_4$), $C_2\times C_{10}$ ($D_5\times S_4$), $D_{10}$ ($C_{10}\times S_4$), $C_5:C_4$ ($C_{10}\times S_4$)

Order 16: $C_2\times Q_8$ ($C_{15}:D_{10}$)

Order 10: $C_{10}$ ($C_2\times C_{10}\times S_4$), $C_{10}$ ($D_{10}\times S_4$)

Order 8: $Q_8$ ($C_{30}:D_{10}$)

Order 5: $C_5$ ($C_5\times \GL(2,3):C_2^2$), $C_5$ ($\GL(2,3):D_{10}$)

Order 4: $C_2^2$ ($C_{10}^2:D_6$)

Order 2: $C_2$ ($(C_5^2\times A_4).C_2^3$)

Order 1: $C_1$ ($C_5\times \GL(2,3):D_{10}$)

Series

Derived series

$C_5\times \GL(2,3):D_{10}$

$\rhd$

$C_5\times \SL(2,3)$

$\rhd$

$Q_8$

$\rhd$

$C_2$

$\rhd$

$C_1$

Chief series

$C_5\times \GL(2,3):D_{10}$

$\rhd$

$C_5\times \GL(2,3):D_5$

$\rhd$

$C_5\times \SL(2,3):D_5$

$\rhd$

$C_5^2\times \SL(2,3)$

$\rhd$

$C_5\times \SL(2,3)$

$\rhd$

$C_5\times Q_8$

$\rhd$

$Q_8$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$C_5\times \GL(2,3):D_{10}$

$\rhd$

$C_5\times \SL(2,3)$

Upper central series

$C_1$

$\lhd$

$C_{10}$

$\lhd$

$C_2\times C_{10}$

Supergroups

This group is a maximal subgroup of 1 larger groups in the database.

This group is a maximal quotient of 0 larger groups in the database.

Character theory

Complex character table

See the $275 \times 275$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $84 \times 84$ rational character table.