Abstract group 960.10966: $\GL(2,3):D_{10}$

• Label: 960.10966
• Order: \( 2^{6} \cdot 3 \cdot 5 \)
• Exponent: \( 2^{3} \cdot 3 \cdot 5 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{8} \cdot 3 \cdot 5 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \)
• Perm deg.: $21$
• Trans deg.: $80$
• Rank: $3$

Group information

Description:	$\GL(2,3):D_{10}$
Order:	\(960\)\(\medspace = 2^{6} \cdot 3 \cdot 5 \)
Exponent:	\(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \)
Automorphism group:	$C_2^3\times F_5\times S_4$, of order \(3840\)\(\medspace = 2^{8} \cdot 3 \cdot 5 \)
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	40
Elements	1	127	8	112	4	104	144	108	80	32	48	96	96	960
Conjugacy classes	1	7	1	5	2	3	5	10	1	2	4	6	8	55
Divisions	1	7	1	5	1	3	4	4	1	1	2	2	2	34
Autjugacy classes	1	7	1	5	1	3	4	4	1	1	2	2	2	34

Dimension	1	2	3	4	6	8	12	32
Irr. complex chars.	8	12	8	14	8	5	0	0	55
Irr. rational chars.	8	4	8	4	0	4	4	2	34

Minimal presentations

Permutation degree:	$21$
Transitive degree:	$80$
Rank:	$3$
Inequivalent generating triples:	$60480$

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:	${\langle a, b, c, d, e \mid a^{2}=b^{2}=c^{6}=e^{20}=[a,b]=[b,d]=[b,e]=1, d^{2}= \!\cdots\! \rangle}$
Permutation group:	Degree $21$ $\langle(1,2)(3,7)(4,8)(5,6)(9,14)(10,15)(11,13)(12,16)(18,19)(20,21), (3,9)(4,11) \!\cdots\! \rangle$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$\GL(2,3)$ $\,\rtimes\,$ $D_{10}$ (2)	$(Q_8\times D_5)$ $\,\rtimes\,$ $D_6$	$(Q_8.D_{10})$ $\,\rtimes\,$ $S_3$	$(Q_8.D_{30})$ $\,\rtimes\,$ $C_2$	all 22
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_5:D_4)$ . $S_4$	$Q_8$ . $(S_3\times D_{10})$	$D_{10}$ . $(C_2\times S_4)$	$C_2$ . $(D_{10}\times S_4)$	all 11

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

Homology

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

Subgroups

There are 2918 subgroups in 304 conjugacy classes, 37 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $D_{10}\times S_4$
Commutator:	$G' \simeq$ $C_5\times \SL(2,3)$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $D_{10}\times S_4$
Fitting:	$\operatorname{Fit} \simeq$ $Q_8\times C_{10}$	$G/\operatorname{Fit} \simeq$ $D_6$
Radical:	$R \simeq$ $\GL(2,3):D_{10}$	$G/R \simeq$ $C_1$
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: $\GL(2,3):D_{10}$

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

Order 320: $D_{40}:C_2^2$

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

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

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

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

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 4, $D_4:D_5$ x 3, $D_4\times D_5$ x 3, $D_{20}:C_2$ 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, $C_5:Q_{16}$, $D_{40}$, $Q_8\times C_{10}$, $D_4\times C_{10}$, $C_2\times C_{40}$, $C_5:\OD_{16}$

Order 64: $D_8:C_2^2$

Order 60: $S_3\times C_{10}$ x 4, $S_3\times D_5$ x 2, $C_{15}:C_4$, $C_5:C_{12}$, $C_2\times C_{30}$, $D_{30}$, $C_3\times D_{10}$

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

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 3, $C_{30}$ x 2, $D_{15}$, $C_3\times D_5$

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: $D_{10}$ x 5, $C_2\times C_{10}$ x 4, $C_5:C_4$ x 3, $C_{20}$ x 2

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 4, $D_5$ 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: $C_5$

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

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

Order 320: $D_{40}:C_2^2$

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

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

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

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

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 4, $D_4:D_5$ x 3, $D_4\times D_5$ x 3, $D_{20}:C_2$ 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, $C_5:Q_{16}$, $D_{40}$, $Q_8\times C_{10}$, $D_4\times C_{10}$, $C_2\times C_{40}$, $C_5:\OD_{16}$

Order 64: $D_8:C_2^2$

Order 60: $S_3\times C_{10}$ x 3, $S_3\times D_5$, $C_{15}:C_4$, $C_5:C_{12}$, $C_2\times C_{30}$, $D_{30}$, $C_3\times D_{10}$

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

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 2, $C_5\times S_3$ x 2, $D_{15}$, $C_3\times D_5$

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: $D_{10}$ x 5, $C_2\times C_{10}$ x 4, $C_5:C_4$ x 3, $C_{20}$ x 2

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

Order 10: $C_{10}$ x 4, $D_5$ 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: $C_5$

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 960: $\GL(2,3):D_{10}$ ($C_1$)

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

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

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

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

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

Order 80: $Q_8\times C_{10}$ ($D_6$), $D_{20}:C_2$ ($D_6$), $Q_8\times D_5$ ($D_6$)

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

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

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

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

Order 16: $C_2\times Q_8$ ($S_3\times D_5$)

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

Order 8: $Q_8$ ($S_3\times D_{10}$)

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

Order 4: $C_2^2$ ($D_5\times S_4$)

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

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

Normal subgroups up to automorphism (quotient in parentheses)

Order 960: $\GL(2,3):D_{10}$ ($C_1$)

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

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

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

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

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

Order 80: $Q_8\times C_{10}$ ($D_6$), $D_{20}:C_2$ ($D_6$), $Q_8\times D_5$ ($D_6$)

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

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

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

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

Order 16: $C_2\times Q_8$ ($S_3\times D_5$)

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

Order 8: $Q_8$ ($S_3\times D_{10}$)

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

Order 4: $C_2^2$ ($D_5\times S_4$)

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

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

Series

Derived series

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

$\rhd$

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

$\rhd$

$Q_8$

$\rhd$

$C_2$

$\rhd$

$C_1$

Chief series

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

$\rhd$

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

$\rhd$

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

$\rhd$

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

$\rhd$

$C_5\times Q_8$

$\rhd$

$C_{10}$

$\rhd$

$C_5$

$\rhd$

$C_1$

Lower central series

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

$\rhd$

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

Upper central series

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_2^2$

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 $55 \times 55$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $34 \times 34$ rational character table.