Abstract group 960.11100: $C_2\times C_{20}.S_4$

• Label: 960.11100
• Order: \( 2^{6} \cdot 3 \cdot 5 \)
• Exponent: \( 2^{3} \cdot 3 \cdot 5 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2^{3} \)
• $\card{\Aut(G)}$: \( 2^{10} \cdot 3 \cdot 5 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{7} \)
• Perm deg.: $23$
• Trans deg.: $160$
• Rank: $3$

Group information

Description:	$C_2\times C_{20}.S_4$
Order:	\(960\)\(\medspace = 2^{6} \cdot 3 \cdot 5 \)
Exponent:	\(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \)
Automorphism group:	$(C_5\times A_4).C_2^4.C_2^4$, of order \(15360\)\(\medspace = 2^{10} \cdot 3 \cdot 5 \)
Composition factors:	$C_2$ x 6, $C_3$, $C_5$
Derived length:	$4$

This group is nonabelian and solvable.

Group statistics

Order	1	2	3	4	5	6	8	10	12	15	20	30	60
Elements	1	135	8	136	4	24	240	60	32	32	64	96	128	960
Conjugacy classes	1	7	1	8	2	3	8	10	4	4	12	12	16	88
Divisions	1	7	1	6	1	3	4	5	2	1	4	3	2	40
Autjugacy classes	1	4	1	4	1	2	2	3	1	1	3	2	1	26

Dimension	1	2	3	4	6	8	12	16	32
Irr. complex chars.	8	36	8	28	8	0	0	0	0	88
Irr. rational chars.	8	4	8	4	0	8	4	2	2	40

Minimal presentations

Permutation degree:	$23$
Transitive degree:	$160$
Rank:	$3$
Inequivalent generating triples:	$15120$

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:	${\langle a, b, c, d, e \mid a^{2}=b^{6}=d^{2}=e^{20}=[c,d]=1, c^{2}=e^{10}, b^{a}= \!\cdots\! \rangle}$
Permutation group:	Degree $23$ $\langle(4,8)(5,7)(9,14)(10,13)(11,12)(15,16)(18,19)(20,21), (1,2)(3,6)(4,7)(5,8) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 36 & 0 \\ 0 & 9 \end{array}\right), \left(\begin{array}{rr} 1 & 17 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 16 & 0 \\ 0 & 16 \end{array}\right), \left(\begin{array}{rr} 16 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 81 & 0 \\ 0 & 81 \end{array}\right), \left(\begin{array}{rr} 51 & 15 \\ 25 & 51 \end{array}\right), \left(\begin{array}{rr} 61 & 5 \\ 5 & 6 \end{array}\right), \left(\begin{array}{rr} 69 & 0 \\ 0 & 69 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/85\Z)$
Direct product:	$C_2$ $\, \times\, $ $(C_{20}.S_4)$
Semidirect product:	$(Q_8.D_{30})$ $\,\rtimes\,$ $C_2$	$(D_4:C_{10})$ $\,\rtimes\,$ $D_6$ (2)	$(D_4:C_2)$ $\,\rtimes\,$ $D_{30}$ (2)	$(C_{10}.S_4)$ $\,\rtimes\,$ $C_2^2$ (2)	all 14
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_2\times C_{20})$ . $S_4$	$C_{20}$ . $(C_2\times S_4)$ (2)	$C_4$ . $(C_{10}:S_4)$ (2)	$(Q_8\times C_{10})$ . $D_6$	all 16

Elements of the group are displayed as matrices in $\GL_{2}(\Z/{85}\Z)$.

Homology

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

Subgroups

There are 2710 subgroups in 306 conjugacy classes, 47 normal (23 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2\times C_4$	$G/Z \simeq$ $C_5:S_4$
Commutator:	$G' \simeq$ $C_5\times \SL(2,3)$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_2^3:D_{30}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{10}.C_2^4$	$G/\operatorname{Fit} \simeq$ $S_3$
Radical:	$R \simeq$ $C_2\times C_{20}.S_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_{10}$	$G/\operatorname{soc} \simeq$ $C_2\times S_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_8:C_2^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5$

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 960: $C_2\times C_{20}.S_4$

Order 480: $C_{20}.S_4$ x 4, $C_2\times \SL(2,3):C_{10}$, $C_{10}:\GL(2,3)$, $Q_8.D_{30}$

Order 320: $C_{20}.C_2^4$

Order 240: $\SL(2,3):C_{10}$ x 2, $C_5:\GL(2,3)$ x 2, $C_{10}.S_4$ x 2, $C_4\times D_{30}$, $C_{10}\times \SL(2,3)$

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

Order 160: $C_{20}.D_4$ x 8, $C_{10}.C_2^4$, $D_{20}:C_2^2$, $C_{10}:Q_{16}$, $Q_8:D_{10}$, $C_{10}:\SD_{16}$, $C_{10}:D_8$, $C_{20}.C_2^3$

Order 120: $C_4\times D_{15}$ x 4, $C_2\times D_{30}$, $C_2\times C_{60}$, $C_{30}:C_4$, $C_5\times \SL(2,3)$

Order 96: $\GL(2,3):C_2$ x 4, $\SL(2,3):C_2^2$, $C_2\times \GL(2,3)$, $C_2^2.S_4$

Order 80: $C_{10}:C_8$ x 6, $D_{20}:C_2$ x 6, $D_4:C_{10}$ x 4, $C_5:Q_{16}$ x 4, $Q_8:D_5$ x 4, $C_5:\SD_{16}$ x 4, $C_5:D_8$ x 4, $Q_8\times C_{10}$, $D_4\times C_{10}$, $C_2^2\times C_{20}$, $C_{10}:D_4$, $C_2\times D_{20}$, $C_4\times D_{10}$, $C_{10}:Q_8$

Order 64: $D_8:C_2^2$

Order 60: $D_{30}$ x 6, $C_{60}$ x 2, $C_{15}:C_4$ x 2, $C_2\times C_{30}$

Order 48: $\SL(2,3):C_2$ x 2, $\GL(2,3)$ x 2, $C_2.S_4$ x 2, $C_4\times D_6$, $C_2\times \SL(2,3)$

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

Order 32: $D_8:C_2$ x 8, $D_4:C_2^2$ x 2, $C_2\times \SD_{16}$ x 2, $C_2\times Q_{16}$, $C_2\times D_8$, $C_2^2\times C_8$

Order 30: $D_{15}$ x 4, $C_{30}$ x 3

Order 24: $C_4\times S_3$ x 4, $C_2\times C_{12}$, $C_6:C_4$, $\SL(2,3)$, $C_2\times D_6$

Order 20: $C_2\times C_{10}$ x 5, $D_{10}$ x 4, $C_{20}$ x 4, $C_5:C_4$ x 2

Order 16: $D_4:C_2$ x 10, $\SD_{16}$ x 8, $C_2\times C_8$ x 6, $Q_{16}$ x 4, $D_8$ x 4, $C_2\times D_4$ x 3, $C_2\times Q_8$ x 2, $C_2^2\times C_4$ x 2

Order 15: $C_{15}$

Order 12: $D_6$ x 6, $C_{12}$ x 2, $C_3:C_4$ x 2, $C_2\times C_6$

Order 10: $C_{10}$ x 5, $D_5$ x 2

Order 8: $C_2\times C_4$ x 11, $D_4$ x 10, $Q_8$ x 5, $C_8$ x 4, $C_2^3$ x 2

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

Order 5: $C_5$

Order 4: $C_2^2$ x 9, $C_4$ x 6

Order 3: $C_3$

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 960: $C_2\times C_{20}.S_4$

Order 480: $C_2\times \SL(2,3):C_{10}$, $C_{20}.S_4$, $C_{10}:\GL(2,3)$, $Q_8.D_{30}$

Order 320: $C_{20}.C_2^4$

Order 240: $C_4\times D_{30}$, $\SL(2,3):C_{10}$, $C_{10}\times \SL(2,3)$, $C_5:\GL(2,3)$, $C_{10}.S_4$

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

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

Order 120: $C_2\times D_{30}$, $C_2\times C_{60}$, $C_{30}:C_4$, $C_4\times D_{15}$, $C_5\times \SL(2,3)$

Order 96: $\SL(2,3):C_2^2$, $\GL(2,3):C_2$, $C_2\times \GL(2,3)$, $C_2^2.S_4$

Order 80: $C_{10}:C_8$ x 3, $D_4:C_{10}$ x 2, $D_{20}:C_2$ x 2, $C_5:Q_{16}$ x 2, $Q_8:D_5$ x 2, $Q_8\times C_{10}$, $D_4\times C_{10}$, $C_2^2\times C_{20}$, $C_{10}:D_4$, $C_2\times D_{20}$, $C_4\times D_{10}$, $C_{10}:Q_8$, $C_5:\SD_{16}$, $C_5:D_8$

Order 64: $D_8:C_2^2$

Order 60: $D_{30}$ x 2, $C_{60}$, $C_{15}:C_4$, $C_2\times C_{30}$

Order 48: $C_4\times D_6$, $\SL(2,3):C_2$, $C_2\times \SL(2,3)$, $\GL(2,3)$, $C_2.S_4$

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

Order 32: $D_4:C_2^2$ x 2, $D_8:C_2$ x 2, $C_2\times \SD_{16}$ x 2, $C_2\times Q_{16}$, $C_2\times D_8$, $C_2^2\times C_8$

Order 30: $C_{30}$ x 2, $D_{15}$

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

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

Order 16: $D_4:C_2$ x 4, $\SD_{16}$ x 3, $C_2\times C_8$ x 3, $C_2\times D_4$ x 3, $Q_{16}$ x 2, $C_2\times Q_8$ x 2, $C_2^2\times C_4$ x 2, $D_8$

Order 15: $C_{15}$

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

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

Order 8: $C_2\times C_4$ x 6, $D_4$ x 5, $Q_8$ x 4, $C_2^3$ x 2, $C_8$ x 2

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

Order 5: $C_5$

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

Order 3: $C_3$

Order 2: $C_2$ x 4

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 960: $C_2\times C_{20}.S_4$ ($C_1$)

Order 480: $C_{20}.S_4$ ($C_2$) x 4, $C_2\times \SL(2,3):C_{10}$ ($C_2$), $C_{10}:\GL(2,3)$ ($C_2$), $Q_8.D_{30}$ ($C_2$)

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

Order 160: $C_{10}.C_2^4$ ($S_3$)

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

Order 96: $\SL(2,3):C_2^2$ ($D_5$)

Order 80: $D_4:C_{10}$ ($D_6$) x 2, $Q_8\times C_{10}$ ($D_6$)

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

Order 40: $C_2\times C_{20}$ ($S_4$), $C_5\times Q_8$ ($C_2\times D_6$)

Order 32: $D_4:C_2^2$ ($D_{15}$)

Order 24: $\SL(2,3)$ ($C_2\times D_{10}$)

Order 20: $C_{20}$ ($C_2\times S_4$) x 2, $C_2\times C_{10}$ ($C_2\times S_4$)

Order 16: $D_4:C_2$ ($D_{30}$) x 2, $C_2\times Q_8$ ($D_{30}$)

Order 10: $C_{10}$ ($\GL(2,3):C_2$) x 2, $C_{10}$ ($C_2^2\times S_4$)

Order 8: $Q_8$ ($C_2\times D_{30}$), $C_2\times C_4$ ($C_5:S_4$)

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

Order 4: $C_4$ ($C_{10}:S_4$) x 2, $C_2^2$ ($C_{10}:S_4$)

Order 2: $C_2$ ($C_{20}.S_4$) x 2, $C_2$ ($C_2^3:D_{30}$)

Order 1: $C_1$ ($C_2\times C_{20}.S_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 960: $C_2\times C_{20}.S_4$ ($C_1$)

Order 480: $C_2\times \SL(2,3):C_{10}$ ($C_2$), $C_{20}.S_4$ ($C_2$), $C_{10}:\GL(2,3)$ ($C_2$), $Q_8.D_{30}$ ($C_2$)

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

Order 160: $C_{10}.C_2^4$ ($S_3$)

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

Order 96: $\SL(2,3):C_2^2$ ($D_5$)

Order 80: $D_4:C_{10}$ ($D_6$), $Q_8\times C_{10}$ ($D_6$)

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

Order 40: $C_2\times C_{20}$ ($S_4$), $C_5\times Q_8$ ($C_2\times D_6$)

Order 32: $D_4:C_2^2$ ($D_{15}$)

Order 24: $\SL(2,3)$ ($C_2\times D_{10}$)

Order 20: $C_2\times C_{10}$ ($C_2\times S_4$), $C_{20}$ ($C_2\times S_4$)

Order 16: $D_4:C_2$ ($D_{30}$), $C_2\times Q_8$ ($D_{30}$)

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

Order 8: $Q_8$ ($C_2\times D_{30}$), $C_2\times C_4$ ($C_5:S_4$)

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

Order 4: $C_2^2$ ($C_{10}:S_4$), $C_4$ ($C_{10}:S_4$)

Order 2: $C_2$ ($C_2^3:D_{30}$), $C_2$ ($C_{20}.S_4$)

Order 1: $C_1$ ($C_2\times C_{20}.S_4$)

Series

Derived series

$C_2\times C_{20}.S_4$

$\rhd$

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

$\rhd$

$Q_8$

$\rhd$

$C_2$

$\rhd$

$C_1$

Chief series

$C_2\times C_{20}.S_4$

$\rhd$

$C_{20}.S_4$

$\rhd$

$\SL(2,3):C_{10}$

$\rhd$

$D_4:C_{10}$

$\rhd$

$C_5\times Q_8$

$\rhd$

$C_{10}$

$\rhd$

$C_5$

$\rhd$

$C_1$

Lower central series

$C_2\times C_{20}.S_4$

$\rhd$

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

Upper central series

$C_1$

$\lhd$

$C_2\times C_4$

Character theory

Complex character table

See the $88 \times 88$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $40 \times 40$ rational character table.