Abstract group 960.11084: $C_5\times \GL(2,3):C_2^2$

• Label: 960.11084
• Order: \( 2^{6} \cdot 3 \cdot 5 \)
• Exponent: \( 2^{3} \cdot 3 \cdot 5 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 5 \)
• $\card{Z(G)}$: \( 2 \cdot 5 \)
• $\card{\Aut(G)}$: \( 2^{9} \cdot 3 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \)
• Perm deg.: $21$
• Trans deg.: $80$
• Rank: $3$

Group information

Description:	$C_5\times \GL(2,3):C_2^2$
Order:	\(960\)\(\medspace = 2^{6} \cdot 3 \cdot 5 \)
Exponent:	\(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \)
Automorphism group:	$C_4\times \GL(2,\mathbb{Z}/4):C_2^2$, of order \(1536\)\(\medspace = 2^{9} \cdot 3 \)
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	40	60
Elements	1	47	8	32	4	40	48	188	16	32	128	160	192	64	960
Conjugacy classes	1	7	1	5	4	3	5	28	1	4	20	12	20	4	115
Divisions	1	7	1	5	1	3	4	7	1	1	5	3	4	1	44
Autjugacy classes	1	5	1	4	1	2	3	5	1	1	4	2	3	1	34

Dimension	1	2	3	4	8	12	32
Irr. complex chars.	40	20	40	10	5	0	0	115
Irr. rational chars.	8	4	8	8	6	8	2	44

Minimal presentations

Permutation degree:	$21$
Transitive degree:	$80$
Rank:	$3$
Inequivalent generating triples:	$156240$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	4	8	32
Arbitrary	4	8	12

Constructions

Presentation:	${\langle a, b, c, d, e \mid a^{2}=b^{2}=c^{6}=e^{20}=[a,b]=[a,d]=[a,e]=1, d^{2}= \!\cdots\! \rangle}$
Permutation group:	Degree $21$ $\langle(1,2)(3,6)(4,5)(7,8)(9,10)(13,14), (1,3)(2,6)(4,9)(5,10)(7,13)(8,14)(11,15) \!\cdots\! \rangle$
Direct product:	$C_5$ $\, \times\, $ $(\GL(2,3):C_2^2)$
Semidirect product:	$(D_4.A_4)$ $\,\rtimes\,$ $C_{10}$	$(Q_8\times C_{10})$ $\,\rtimes\,$ $D_6$ (2)	$(D_4:C_{10})$ $\,\rtimes\,$ $D_6$	$(C_{10}.S_4)$ $\,\rtimes\,$ $C_2^2$	all 24
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_5\times D_4)$ . $S_4$	$D_4$ . $(C_5\times S_4)$	$C_{20}$ . $(C_2\times S_4)$	$C_4$ . $(C_{10}\times S_4)$	all 12

Elements of the group are displayed as words in the presentation generators from the presentation above.

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 1230 subgroups in 304 conjugacy classes, 54 normal (30 characteristic).

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

Special subgroups

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

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

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Classes of subgroups up to automorphism

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Normal subgroups

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 960: $C_5\times \GL(2,3):C_2^2$

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

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

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

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

Order 160: $Q_{16}:C_{10}$ x 3, $D_8:C_{10}$ x 3, $D_8:C_{10}$ x 3, $C_{10}\times \SD_{16}$ x 3, $C_{10}.C_2^4$, $C_{10}.C_2^4$, $\OD_{16}:C_{10}$

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

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 10, $D_4:C_{10}$ x 8, $D_4\times C_{10}$ x 6, $Q_8\times C_{10}$ x 3, $C_5\times Q_{16}$ x 3, $C_5\times D_8$ x 3, $C_5\times \OD_{16}$ x 3, $C_2\times C_{40}$ x 3

Order 64: $D_8:C_2^2$

Order 60: $S_3\times C_{10}$ x 7, $C_2\times C_{30}$ x 2, $C_{60}$, $C_3:C_{20}$

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 13, $C_2\times C_{20}$ x 7, $C_5\times Q_8$ x 5, $C_{40}$ x 4, $C_2^2\times C_{10}$ x 3

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 4, $C_{30}$ x 3

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 9, $C_{20}$ x 5

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: $C_{15}$

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

Order 10: $C_{10}$ x 7

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: $C_5$

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 960: $C_5\times \GL(2,3):C_2^2$

Order 480: $(C_2\times Q_8):C_{30}$, $\GL(2,3):C_{10}$, $\GL(2,3):C_{10}$, $\GL(2,3):C_{10}$, $C_{10}\times \GL(2,3)$

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

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

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

Order 160: $Q_{16}:C_{10}$ x 2, $D_8:C_{10}$ x 2, $D_8:C_{10}$ x 2, $C_{10}\times \SD_{16}$ x 2, $C_{10}.C_2^4$, $C_{10}.C_2^4$, $\OD_{16}:C_{10}$

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

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

Order 80: $D_4:C_{10}$ x 6, $C_5\times \SD_{16}$ x 6, $D_4\times C_{10}$ x 4, $Q_8\times C_{10}$ x 2, $C_5\times Q_{16}$ x 2, $C_5\times D_8$ x 2, $C_5\times \OD_{16}$ x 2, $C_2\times C_{40}$ x 2

Order 64: $D_8:C_2^2$

Order 60: $S_3\times C_{10}$ x 3, $C_{60}$, $C_2\times C_{30}$, $C_3:C_{20}$

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

Order 40: $C_5\times D_4$ x 9, $C_2\times C_{20}$ x 5, $C_5\times Q_8$ x 4, $C_{40}$ x 3, $C_2^2\times C_{10}$ x 2

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

Order 30: $C_{30}$ x 2, $C_5\times S_3$ x 2

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

Order 20: $C_2\times C_{10}$ x 6, $C_{20}$ x 4

Order 16: $\SD_{16}$ x 6, $D_4:C_2$ x 6, $C_2\times D_4$ x 4, $Q_{16}$ x 2, $D_8$ x 2, $\OD_{16}$ x 2, $C_2\times C_8$ x 2, $C_2\times Q_8$ x 2

Order 15: $C_{15}$

Order 12: $D_6$ x 3, $C_2\times C_6$, $C_{12}$, $C_3:C_4$

Order 10: $C_{10}$ x 5

Order 8: $D_4$ x 9, $C_2\times C_4$ x 5, $Q_8$ x 4, $C_8$ x 3, $C_2^3$ x 2

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

Order 5: $C_5$

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

Order 3: $C_3$

Order 2: $C_2$ x 5

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 960: $C_5\times \GL(2,3):C_2^2$ ($C_1$)

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

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

Order 192: $\GL(2,3):C_2^2$ ($C_5$)

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

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

Order 96: $\GL(2,3):C_2$ ($C_{10}$) x 2, $C_2\times \GL(2,3)$ ($C_{10}$) x 2, $D_4.A_4$ ($C_{10}$), $\GL(2,3):C_2$ ($C_{10}$), $\GL(2,3):C_2$ ($C_{10}$)

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

Order 48: $\GL(2,3)$ ($C_2\times C_{10}$) x 3, $C_2\times \SL(2,3)$ ($C_2\times C_{10}$) x 2, $\SL(2,3):C_2$ ($C_2\times C_{10}$), $C_2.S_4$ ($C_2\times C_{10}$)

Order 40: $C_5\times Q_8$ ($C_2\times D_6$), $C_5\times D_4$ ($S_4$)

Order 32: $C_4.C_2^3$ ($C_5\times S_3$)

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

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

Order 16: $C_2\times Q_8$ ($S_3\times C_{10}$) x 2, $D_4:C_2$ ($S_3\times C_{10}$)

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

Order 8: $Q_8$ ($C_{10}\times D_6$), $D_4$ ($C_5\times S_4$)

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

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

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

Order 1: $C_1$ ($C_5\times \GL(2,3):C_2^2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 960: $C_5\times \GL(2,3):C_2^2$ ($C_1$)

Order 480: $(C_2\times Q_8):C_{30}$ ($C_2$), $\GL(2,3):C_{10}$ ($C_2$), $\GL(2,3):C_{10}$ ($C_2$), $\GL(2,3):C_{10}$ ($C_2$), $C_{10}\times \GL(2,3)$ ($C_2$)

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

Order 192: $\GL(2,3):C_2^2$ ($C_5$)

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

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

Order 96: $D_4.A_4$ ($C_{10}$), $\GL(2,3):C_2$ ($C_{10}$), $\GL(2,3):C_2$ ($C_{10}$), $\GL(2,3):C_2$ ($C_{10}$), $C_2\times \GL(2,3)$ ($C_{10}$)

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

Order 48: $\GL(2,3)$ ($C_2\times C_{10}$) x 2, $\SL(2,3):C_2$ ($C_2\times C_{10}$), $C_2\times \SL(2,3)$ ($C_2\times C_{10}$), $C_2.S_4$ ($C_2\times C_{10}$)

Order 40: $C_5\times Q_8$ ($C_2\times D_6$), $C_5\times D_4$ ($S_4$)

Order 32: $C_4.C_2^3$ ($C_5\times S_3$)

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

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

Order 16: $D_4:C_2$ ($S_3\times C_{10}$), $C_2\times Q_8$ ($S_3\times C_{10}$)

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

Order 8: $Q_8$ ($C_{10}\times D_6$), $D_4$ ($C_5\times S_4$)

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

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

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

Order 1: $C_1$ ($C_5\times \GL(2,3):C_2^2$)

Series

Derived series

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

$\rhd$

$\SL(2,3)$

$\rhd$

$Q_8$

$\rhd$

$C_2$

$\rhd$

$C_1$

Chief series

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

$\rhd$

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

$\rhd$

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

$\rhd$

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

$\rhd$

$C_5\times Q_8$

$\rhd$

$C_{10}$

$\rhd$

$C_5$

$\rhd$

$C_1$

Lower central series

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

$\rhd$

$\SL(2,3)$

Upper central series

$C_1$

$\lhd$

$C_{10}$

$\lhd$

$C_5\times D_4$

Supergroups

This group is a maximal subgroup of 5 larger groups in the database.

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

Character theory

Complex character table

See the $115 \times 115$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $44 \times 44$ rational character table.