Abstract group 1920.240001: $C_2\times F_5\times \GL(2,3)$

• Label: 1920.240001
• Order: \( 2^{7} \cdot 3 \cdot 5 \)
• Exponent: \( 2^{3} \cdot 3 \cdot 5 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{10} \cdot 3 \cdot 5 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \)
• Perm deg.: $15$
• Trans deg.: $80$
• Rank: $3$

Group information

Description:	$C_2\times F_5\times \GL(2,3)$
Order:	\(1920\)\(\medspace = 2^{7} \cdot 3 \cdot 5 \)
Exponent:	\(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \)
Automorphism group:	$C_3^4:S_3^2$, of order \(15360\)\(\medspace = 2^{10} \cdot 3 \cdot 5 \)
Composition factors:	$C_2$ x 7, $C_3$, $C_5$
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
Elements	1	167	8	472	4	184	384	108	320	32	48	96	96	1920
Conjugacy classes	1	11	1	20	1	7	16	5	8	1	2	3	4	80
Divisions	1	11	1	12	1	7	6	5	4	1	2	3	2	56
Autjugacy classes	1	7	1	10	1	5	4	3	2	1	2	2	1	40

Dimension	1	2	3	4	6	8	12	16
Irr. complex chars.	16	24	16	12	0	6	4	2	80
Irr. rational chars.	8	8	8	14	4	6	4	4	56

Minimal presentations

Permutation degree:	$15$
Transitive degree:	$80$
Rank:	$3$
Inequivalent generating triples:	$120960$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e \mid a^{2}=b^{2}=c^{12}=e^{20}=[a,b]=[a,c]=[a,d]= \!\cdots\! \rangle}$
Permutation group:	Degree $15$ $\langle(1,2)(3,5)(4,6)(7,8)(10,11,12,13), (1,2)(3,7)(5,8), (14,15), (10,12)(11,13) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 2 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 5 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 9 & 5 \\ 5 & 4 \end{array}\right), \left(\begin{array}{rr} 1 & 3 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 4 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 4 & 0 \\ 0 & 4 \end{array}\right), \left(\begin{array}{rr} 11 & 5 \\ 5 & 1 \end{array}\right), \left(\begin{array}{rr} 11 & 0 \\ 0 & 11 \end{array}\right), \left(\begin{array}{rr} 1 & 5 \\ 5 & 11 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/15\Z)$
Direct product:	$C_2$ $\, \times\, $ $F_5$ $\, \times\, $ $\GL(2,3)$
Semidirect product:	$(Q_8\times F_5)$ $\,\rtimes\,$ $D_6$ (2)	$Q_8$ $\,\rtimes\,$ $(D_6\times F_5)$	$(D_{10}.C_2^4)$ $\,\rtimes\,$ $S_3$	$(D_5\times \GL(2,3))$ $\,\rtimes\,$ $C_4$ (2)	all 24
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$D_{10}$ . $(C_4\times S_4)$	$(Q_8\times D_{10})$ . $D_6$	$(C_2^2\times F_5)$ . $S_4$	$C_2^2$ . $(F_5\times S_4)$	all 19
Aut. group:	$\Aut(C_{15}:C_{12})$	$\Aut(C_3^2\times D_{10})$	$\Aut(C_{15}:C_{24})$	$\Aut(C_{30}:C_{12})$

Elements of the group are displayed as matrices in $\GL_{2}(\Z/{15}\Z)$.

Homology

Abelianization:	$C_{2}^{2} \times C_{4} $
Schur multiplier:	$C_{2}^{3}$
Commutator length:	$1$

Subgroups

There are 8858 subgroups in 766 conjugacy classes, 74 normal (34 characteristic).

Characteristic subgroups are shown in this color. Normal (but not characteristic) subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_2^2$	$G/Z \simeq$ $F_5\times S_4$
Commutator:	$G' \simeq$ $C_5\times \SL(2,3)$	$G/G' \simeq$ $C_2^2\times C_4$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_2\times F_5\times S_4$
Fitting:	$\operatorname{Fit} \simeq$ $Q_8\times C_{10}$	$G/\operatorname{Fit} \simeq$ $C_4\times S_3$
Radical:	$R \simeq$ $C_2\times F_5\times \GL(2,3)$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_{10}$	$G/\operatorname{soc} \simeq$ $C_4\times S_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4^2.D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5$

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 1920: $C_2\times F_5\times \GL(2,3)$

Order 960: $F_5\times \GL(2,3)$ x 4, $D_{10}\times \GL(2,3)$, $C_2\times F_5\times \SL(2,3)$, $D_{10}.\GL(2,3)$

Order 640: $C_2\times \SD_{16}\times F_5$

Order 480: $D_5\times \GL(2,3)$ x 4, $F_5\times \SL(2,3)$ x 2, $\SL(2,3):F_5$ x 2, $C_2\times D_6\times F_5$, $D_{10}\times \SL(2,3)$, $C_{10}:\GL(2,3)$, $C_{10}\times \GL(2,3)$

Order 384: $C_2\times C_4\times \GL(2,3)$

Order 320: $\SD_{16}\times F_5$ x 8, $D_{10}.C_2^4$, $D_{10}.C_2^4$, $\SD_{16}\times D_{10}$, $D_{10}.Q_{16}$, $D_{10}.D_8$, $D_{10}.\SD_{16}$, $C_{20}.C_4^2$

Order 240: $D_6\times F_5$ x 12, $D_5\times \SL(2,3)$ x 2, $C_5:\GL(2,3)$ x 2, $C_5\times \GL(2,3)$ x 2, $D_6\times D_{10}$, $D_{10}.D_6$, $D_{10}:C_{12}$, $C_{10}\times \SL(2,3)$

Order 192: $C_4\times \GL(2,3)$ x 4, $C_2\times C_4\times \SL(2,3)$, $C_2^3.S_4$, $C_2^2\times \GL(2,3)$

Order 160: $D_5\times \SD_{16}$ x 8, $D_4\times F_5$ x 6, $Q_8:F_5$ x 4, $D_{20}:C_4$ x 4, $C_{40}:C_4$ x 4, $C_8\times F_5$ x 4, $Q_8\times F_5$ x 4, $D_{10}.D_4$ x 2, $C_2^3\times F_5$, $Q_8\times D_{10}$, $D_4\times D_{10}$, $C_2^3:F_5$, $C_{10}:C_4^2$, $D_{10}:C_8$, $C_{10}\times \SD_{16}$, $Q_8:D_{10}$, $C_{10}:\SD_{16}$, $C_8:D_{10}$, $C_8\times D_{10}$

Order 128: $C_4^2.D_4$

Order 120: $S_3\times F_5$ x 16, $S_3\times D_{10}$ x 12, $C_{30}:C_4$ x 6, $C_6\times F_5$ x 6, $C_2\times D_{30}$, $C_{10}\times D_6$, $C_6\times D_{10}$, $C_5\times \SL(2,3)$

Order 96: $C_2\times \GL(2,3)$ x 6, $C_4\times \SL(2,3)$ x 2, $\SL(2,3):C_4$ x 2, $C_{12}:C_2^3$, $C_2^2\times \SL(2,3)$

Order 80: $C_2^2\times F_5$ x 11, $D_4\times D_5$ x 6, $C_{20}:C_4$ x 6, $C_{40}:C_2$ x 4, $Q_8\times D_5$ x 4, $C_8\times D_5$ x 4, $D_{10}:C_4$ x 4, $C_4\times F_5$ x 4, $C_5\times \SD_{16}$ x 4, $Q_8:D_5$ x 4, $C_5:\SD_{16}$ x 4, $D_5:C_8$ x 3, $C_{10}:C_8$, $C_2^2\times D_{10}$, $Q_8\times C_{10}$, $D_4\times C_{10}$, $C_{10}:D_4$, $C_2\times D_{20}$, $C_4\times D_{10}$, $C_{10}:Q_8$, $C_{10}:C_8$, $C_2\times C_{40}$

Order 64: $C_4\times \SD_{16}$ x 8, $C_2^2.Q_{16}$, $C_2^2.D_8$, $C_2\times C_4\times C_8$, $C_2^2\times \SD_{16}$, $C_4^2.C_2^2$, $C_4^2:C_2^2$, $C_2^3.D_4$

Order 60: $S_3\times D_5$ x 16, $D_{30}$ x 6, $S_3\times C_{10}$ x 6, $C_3\times D_{10}$ x 6, $C_{15}:C_4$ x 4, $C_3\times F_5$ x 4, $C_2\times C_{30}$

Order 48: $C_4\times D_6$ x 12, $\GL(2,3)$ x 4, $C_2\times \SL(2,3)$ x 3, $C_2^2\times D_6$, $C_2^2\times C_{12}$, $C_6.C_2^3$

Order 40: $C_2\times F_5$ x 22, $C_2\times D_{10}$ x 10, $C_5:D_4$ x 4, $C_4\times D_5$ x 4, $D_{20}$ x 3, $C_5:Q_8$ x 3, $C_5\times D_4$ x 3, $C_5:C_8$ x 2, $C_{40}$ x 2, $C_5\times Q_8$ x 2, $C_5:C_8$ x 2, $C_2\times C_{20}$, $C_{10}:C_4$, $C_2^2\times C_{10}$

Order 32: $C_2\times \SD_{16}$ x 12, $C_4\times D_4$ x 6, $D_4:C_4$ x 4, $C_4\times C_8$ x 4, $C_4\times Q_8$ x 4, $C_8:C_4$ x 4, $Q_8:C_4$ x 4, $C_2^2\times C_8$ x 2, $C_2^2.D_4$ x 2, $C_2^2\times Q_8$, $C_2^2\times D_4$, $C_2^3\times C_4$, $C_2^3:C_4$, $C_2\times C_4^2$

Order 30: $D_{15}$ x 4, $C_3\times D_5$ x 4, $C_5\times S_3$ x 4, $C_{30}$ x 3

Order 24: $C_4\times S_3$ x 16, $C_2\times D_6$ x 14, $C_2\times C_{12}$ x 6, $C_6:C_4$ x 6, $\SL(2,3)$, $C_2^2\times C_6$

Order 20: $D_{10}$ x 18, $F_5$ x 8, $C_2\times C_{10}$ x 5, $C_{20}$ x 2, $C_5:C_4$ x 2

Order 16: $\SD_{16}$ x 16, $C_2^2\times C_4$ x 12, $C_2\times C_8$ x 10, $C_2\times D_4$ x 9, $C_4:C_4$ x 6, $C_2\times Q_8$ x 6, $C_2^2:C_4$ x 4, $C_4^2$ x 4, $C_2^4$

Order 15: $C_{15}$

Order 12: $D_6$ x 28, $C_2\times C_6$ x 7, $C_{12}$ x 4, $C_3:C_4$ x 4

Order 10: $D_5$ x 6, $C_{10}$ x 5

Order 8: $C_2\times C_4$ x 28, $C_2^3$ x 11, $D_4$ x 10, $C_8$ x 6, $Q_8$ x 5

Order 6: $S_3$ x 8, $C_6$ x 7

Order 5: $C_5$

Order 4: $C_2^2$ x 23, $C_4$ x 12

Order 3: $C_3$

Order 2: $C_2$ x 11

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1920: $C_2\times F_5\times \GL(2,3)$

Order 960: $D_{10}\times \GL(2,3)$, $C_2\times F_5\times \SL(2,3)$, $D_{10}.\GL(2,3)$, $F_5\times \GL(2,3)$

Order 640: $C_2\times \SD_{16}\times F_5$

Order 480: $D_5\times \GL(2,3)$ x 2, $F_5\times \SL(2,3)$, $\SL(2,3):F_5$, $C_2\times D_6\times F_5$, $D_{10}\times \SL(2,3)$, $C_{10}:\GL(2,3)$, $C_{10}\times \GL(2,3)$

Order 384: $C_2\times C_4\times \GL(2,3)$

Order 320: $\SD_{16}\times F_5$ x 2, $D_{10}.C_2^4$, $D_{10}.C_2^4$, $\SD_{16}\times D_{10}$, $D_{10}.Q_{16}$, $D_{10}.D_8$, $D_{10}.\SD_{16}$, $C_{20}.C_4^2$

Order 240: $D_6\times F_5$ x 2, $D_5\times \SL(2,3)$ x 2, $D_6\times D_{10}$, $D_{10}.D_6$, $D_{10}:C_{12}$, $C_{10}\times \SL(2,3)$, $C_5:\GL(2,3)$, $C_5\times \GL(2,3)$

Order 192: $C_2\times C_4\times \SL(2,3)$, $C_2^3.S_4$, $C_4\times \GL(2,3)$, $C_2^2\times \GL(2,3)$

Order 160: $D_5\times \SD_{16}$ x 4, $Q_8:F_5$ x 2, $Q_8\times F_5$ x 2, $D_4\times F_5$ x 2, $D_{10}.D_4$ x 2, $D_{20}:C_4$, $C_{40}:C_4$, $C_8\times F_5$, $C_2^3\times F_5$, $Q_8\times D_{10}$, $D_4\times D_{10}$, $C_2^3:F_5$, $C_{10}:C_4^2$, $D_{10}:C_8$, $C_{10}\times \SD_{16}$, $Q_8:D_{10}$, $C_{10}:\SD_{16}$, $C_8:D_{10}$, $C_8\times D_{10}$

Order 128: $C_4^2.D_4$

Order 120: $S_3\times D_{10}$ x 4, $C_{30}:C_4$ x 2, $C_6\times F_5$ x 2, $C_2\times D_{30}$, $C_{10}\times D_6$, $C_6\times D_{10}$, $S_3\times F_5$, $C_5\times \SL(2,3)$

Order 96: $C_2\times \GL(2,3)$ x 4, $C_4\times \SL(2,3)$, $\SL(2,3):C_4$, $C_{12}:C_2^3$, $C_2^2\times \SL(2,3)$

Order 80: $C_2^2\times F_5$ x 5, $Q_8\times D_5$ x 4, $D_4\times D_5$ x 4, $C_{20}:C_4$ x 4, $C_{40}:C_2$ x 2, $C_8\times D_5$ x 2, $C_4\times F_5$ x 2, $D_5:C_8$ x 2, $C_5\times \SD_{16}$ x 2, $Q_8:D_5$ x 2, $C_5:\SD_{16}$ x 2, $C_{10}:C_8$, $C_2^2\times D_{10}$, $Q_8\times C_{10}$, $D_4\times C_{10}$, $C_{10}:D_4$, $C_2\times D_{20}$, $C_4\times D_{10}$, $C_{10}:Q_8$, $D_{10}:C_4$, $C_{10}:C_8$, $C_2\times C_{40}$

Order 64: $C_4\times \SD_{16}$ x 2, $C_2^2.Q_{16}$, $C_2^2.D_8$, $C_2\times C_4\times C_8$, $C_2^2\times \SD_{16}$, $C_4^2.C_2^2$, $C_4^2:C_2^2$, $C_2^3.D_4$

Order 60: $C_3\times D_{10}$ x 4, $S_3\times D_5$ x 3, $D_{30}$ x 2, $S_3\times C_{10}$ x 2, $C_{15}:C_4$, $C_3\times F_5$, $C_2\times C_{30}$

Order 48: $C_2\times \SL(2,3)$ x 3, $C_4\times D_6$ x 2, $\GL(2,3)$ x 2, $C_2^2\times D_6$, $C_2^2\times C_{12}$, $C_6.C_2^3$

Order 40: $C_2\times F_5$ x 7, $C_2\times D_{10}$ x 6, $C_4\times D_5$ x 4, $C_5:Q_8$ x 3, $C_5:D_4$ x 2, $D_{20}$ x 2, $C_5\times Q_8$ x 2, $C_5\times D_4$ x 2, $C_2\times C_{20}$, $C_{10}:C_4$, $C_5:C_8$, $C_{40}$, $C_2^2\times C_{10}$, $C_5:C_8$

Order 32: $C_2\times \SD_{16}$ x 8, $C_2^2\times C_8$ x 2, $C_4\times Q_8$ x 2, $C_4\times D_4$ x 2, $C_2^2.D_4$ x 2, $Q_8:C_4$ x 2, $D_4:C_4$, $C_2^2\times Q_8$, $C_2^2\times D_4$, $C_2^3\times C_4$, $C_4\times C_8$, $C_2^3:C_4$, $C_2\times C_4^2$, $C_8:C_4$

Order 30: $C_3\times D_5$ x 3, $C_{30}$ x 2, $D_{15}$, $C_5\times S_3$

Order 24: $C_2\times D_6$ x 6, $C_2\times C_{12}$ x 2, $C_6:C_4$ x 2, $C_4\times S_3$, $\SL(2,3)$, $C_2^2\times C_6$

Order 20: $D_{10}$ x 9, $C_2\times C_{10}$ x 3, $F_5$ x 3, $C_{20}$ x 2, $C_5:C_4$ x 2

Order 16: $\SD_{16}$ x 8, $C_2\times C_8$ x 7, $C_2\times D_4$ x 7, $C_2\times Q_8$ x 6, $C_2^2\times C_4$ x 6, $C_4:C_4$ x 4, $C_4^2$ x 2, $C_2^2:C_4$, $C_2^4$

Order 15: $C_{15}$

Order 12: $D_6$ x 7, $C_2\times C_6$ x 5, $C_{12}$, $C_3:C_4$

Order 10: $D_5$ x 4, $C_{10}$ x 3

Order 8: $C_2\times C_4$ x 13, $C_2^3$ x 7, $D_4$ x 6, $Q_8$ x 5, $C_8$ x 3

Order 6: $C_6$ x 5, $S_3$ x 2

Order 5: $C_5$

Order 4: $C_2^2$ x 12, $C_4$ x 7

Order 3: $C_3$

Order 2: $C_2$ x 7

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1920: $C_2\times F_5\times \GL(2,3)$ ($C_1$)

Order 960: $F_5\times \GL(2,3)$ ($C_2$) x 4, $D_{10}\times \GL(2,3)$ ($C_2$), $C_2\times F_5\times \SL(2,3)$ ($C_2$), $D_{10}.\GL(2,3)$ ($C_2$)

Order 480: $D_5\times \GL(2,3)$ ($C_2^2$) x 2, $D_5\times \GL(2,3)$ ($C_4$) x 2, $F_5\times \SL(2,3)$ ($C_2^2$) x 2, $\SL(2,3):F_5$ ($C_2^2$) x 2, $D_{10}\times \SL(2,3)$ ($C_2^2$), $C_{10}:\GL(2,3)$ ($C_4$), $C_{10}\times \GL(2,3)$ ($C_4$)

Order 320: $D_{10}.C_2^4$ ($S_3$)

Order 240: $C_5:\GL(2,3)$ ($C_2\times C_4$) x 2, $C_5\times \GL(2,3)$ ($C_2\times C_4$) x 2, $C_{10}\times \SL(2,3)$ ($C_2\times C_4$), $D_5\times \SL(2,3)$ ($C_2^3$), $D_5\times \SL(2,3)$ ($C_2\times C_4$)

Order 160: $Q_8\times F_5$ ($D_6$) x 2, $Q_8\times D_{10}$ ($D_6$)

Order 120: $C_5\times \SL(2,3)$ ($C_2^2\times C_4$)

Order 96: $C_2\times \GL(2,3)$ ($F_5$)

Order 80: $C_2^2\times F_5$ ($S_4$), $Q_8\times C_{10}$ ($C_4\times S_3$), $Q_8\times D_5$ ($C_4\times S_3$), $Q_8\times D_5$ ($C_2\times D_6$)

Order 48: $\GL(2,3)$ ($C_2\times F_5$) x 2, $C_2\times \SL(2,3)$ ($C_2\times F_5$)

Order 40: $C_2\times F_5$ ($\GL(2,3)$) x 4, $C_2\times F_5$ ($C_2\times S_4$) x 2, $C_2\times D_{10}$ ($C_2\times S_4$), $C_5\times Q_8$ ($C_4\times D_6$)

Order 24: $\SL(2,3)$ ($C_2^2\times F_5$)

Order 20: $F_5$ ($C_2\times \GL(2,3)$) x 4, $D_{10}$ ($\GL(2,3):C_2$) x 2, $D_{10}$ ($C_2\times \GL(2,3)$) x 2, $C_2\times C_{10}$ ($C_4\times S_4$), $D_{10}$ ($C_2^2\times S_4$), $D_{10}$ ($C_4\times S_4$)

Order 16: $C_2\times Q_8$ ($S_3\times F_5$)

Order 10: $C_{10}$ ($C_4\times \GL(2,3)$) x 2, $D_5$ ($C_4\times \GL(2,3)$) x 2, $C_{10}$ ($C_2^4.D_6$), $D_5$ ($\GL(2,3):C_2^2$), $D_5$ ($C_2^2\times \GL(2,3)$)

Order 8: $Q_8$ ($D_6\times F_5$)

Order 5: $C_5$ ($C_2\times C_4\times \GL(2,3)$)

Order 4: $C_2^2$ ($F_5\times S_4$)

Order 2: $C_2$ ($F_5\times \GL(2,3)$) x 2, $C_2$ ($C_2\times F_5\times S_4$)

Order 1: $C_1$ ($C_2\times F_5\times \GL(2,3)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1920: $C_2\times F_5\times \GL(2,3)$ ($C_1$)

Order 960: $D_{10}\times \GL(2,3)$ ($C_2$), $C_2\times F_5\times \SL(2,3)$ ($C_2$), $D_{10}.\GL(2,3)$ ($C_2$), $F_5\times \GL(2,3)$ ($C_2$)

Order 480: $D_5\times \GL(2,3)$ ($C_2^2$), $D_5\times \GL(2,3)$ ($C_4$), $F_5\times \SL(2,3)$ ($C_2^2$), $\SL(2,3):F_5$ ($C_2^2$), $D_{10}\times \SL(2,3)$ ($C_2^2$), $C_{10}:\GL(2,3)$ ($C_4$), $C_{10}\times \GL(2,3)$ ($C_4$)

Order 320: $D_{10}.C_2^4$ ($S_3$)

Order 240: $C_{10}\times \SL(2,3)$ ($C_2\times C_4$), $D_5\times \SL(2,3)$ ($C_2^3$), $D_5\times \SL(2,3)$ ($C_2\times C_4$), $C_5:\GL(2,3)$ ($C_2\times C_4$), $C_5\times \GL(2,3)$ ($C_2\times C_4$)

Order 160: $Q_8\times D_{10}$ ($D_6$), $Q_8\times F_5$ ($D_6$)

Order 120: $C_5\times \SL(2,3)$ ($C_2^2\times C_4$)

Order 96: $C_2\times \GL(2,3)$ ($F_5$)

Order 80: $C_2^2\times F_5$ ($S_4$), $Q_8\times C_{10}$ ($C_4\times S_3$), $Q_8\times D_5$ ($C_4\times S_3$), $Q_8\times D_5$ ($C_2\times D_6$)

Order 48: $C_2\times \SL(2,3)$ ($C_2\times F_5$), $\GL(2,3)$ ($C_2\times F_5$)

Order 40: $C_2\times D_{10}$ ($C_2\times S_4$), $C_2\times F_5$ ($C_2\times S_4$), $C_2\times F_5$ ($\GL(2,3)$), $C_5\times Q_8$ ($C_4\times D_6$)

Order 24: $\SL(2,3)$ ($C_2^2\times F_5$)

Order 20: $C_2\times C_{10}$ ($C_4\times S_4$), $D_{10}$ ($C_2^2\times S_4$), $D_{10}$ ($\GL(2,3):C_2$), $D_{10}$ ($C_2\times \GL(2,3)$), $D_{10}$ ($C_4\times S_4$), $F_5$ ($C_2\times \GL(2,3)$)

Order 16: $C_2\times Q_8$ ($S_3\times F_5$)

Order 10: $C_{10}$ ($C_4\times \GL(2,3)$), $C_{10}$ ($C_2^4.D_6$), $D_5$ ($C_4\times \GL(2,3)$), $D_5$ ($\GL(2,3):C_2^2$), $D_5$ ($C_2^2\times \GL(2,3)$)

Order 8: $Q_8$ ($D_6\times F_5$)

Order 5: $C_5$ ($C_2\times C_4\times \GL(2,3)$)

Order 4: $C_2^2$ ($F_5\times S_4$)

Order 2: $C_2$ ($C_2\times F_5\times S_4$), $C_2$ ($F_5\times \GL(2,3)$)

Order 1: $C_1$ ($C_2\times F_5\times \GL(2,3)$)

Series

Derived series

$C_2\times F_5\times \GL(2,3)$

$\rhd$

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

$\rhd$

$Q_8$

$\rhd$

$C_2$

$\rhd$

$C_1$

Chief series

$C_2\times F_5\times \GL(2,3)$

$\rhd$

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

$\rhd$

$D_{10}\times \SL(2,3)$

$\rhd$

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

$\rhd$

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

$\rhd$

$C_5\times Q_8$

$\rhd$

$C_{10}$

$\rhd$

$C_5$

$\rhd$

$C_1$

Lower central series

$C_2\times F_5\times \GL(2,3)$

$\rhd$

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

Upper central series

$C_1$

$\lhd$

$C_2^2$

Supergroups

This group is a maximal subgroup of 2 larger groups in the database.

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

Character theory

Complex character table

See the $80 \times 80$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $56 \times 56$ rational character table.