Abstract group 960.10958: $\GL(2,3):D_{10}$

• Label: 960.10958
• Order: \( 2^{6} \cdot 3 \cdot 5 \)
• Exponent: \( 2^{3} \cdot 3 \cdot 5 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{10} \cdot 3 \cdot 5 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \)
• Perm deg.: $23$
• Trans deg.: $160$
• 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:	$(C_2^2\times D_5\times A_4).C_2^5$, 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	40
Elements	1	207	8	32	4	24	144	108	160	32	48	96	96	960
Conjugacy classes	1	9	1	4	2	3	4	10	4	2	4	6	8	58
Divisions	1	9	1	4	1	3	4	5	2	1	2	3	2	38
Autjugacy classes	1	5	1	3	1	2	2	3	1	1	2	2	1	25

Dimension	1	2	3	4	6	8	12	16
Irr. complex chars.	8	12	8	18	8	4	0	0	58
Irr. rational chars.	8	4	8	6	0	4	4	4	38

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:	${\langle a, b, c, d, e \mid a^{2}=b^{2}=c^{6}=e^{20}=1, d^{2}=e^{10}, b^{a}= \!\cdots\! \rangle}$
Permutation group:	Degree $23$ $\langle(1,2,5,6)(3,7,11,13)(4,8,9,14)(10,15,12,16)(18,19)(20,21), (2,6)(3,9)(4,11) \!\cdots\! \rangle$
Direct product:	$C_2$ $\, \times\, $ $(\GL(2,3):D_5)$
Semidirect product:	$\GL(2,3)$ $\,\rtimes\,$ $D_{10}$ (2)	$(Q_8:D_{10})$ $\,\rtimes\,$ $S_3$	$(D_{20}:C_2)$ $\,\rtimes\,$ $D_6$ (2)	$(C_2\times \GL(2,3))$ $\,\rtimes\,$ $D_5$	all 16
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$Q_8$ . $(S_3\times D_{10})$	$C_2$ . $(D_{10}\times S_4)$	$(C_{10}:C_4)$ . $S_4$	$(Q_8\times C_{10})$ . $D_6$	all 10

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 3806 subgroups in 338 conjugacy classes, 41 normal (21 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2$	$G/Z \simeq$ $D_5\times S_4$
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:	$\operatorname{soc} \simeq$ $C_2\times C_{10}$	$G/\operatorname{soc} \simeq$ $C_2\times S_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_8:C_2^3$
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: $\GL(2,3):D_{10}$

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

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

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

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

Order 160: $C_8:D_{10}$ x 8, $Q_8:D_{10}$, $D_4\times D_{10}$, $C_{10}\times \SD_{16}$, $Q_8:D_{10}$, $C_{10}:D_8$, $C_2\times D_{40}$, $C_8:D_{10}$

Order 120: $C_5:D_{12}$ x 4, $C_2\times D_{30}$, $C_{10}\times D_6$, $C_{10}:C_{12}$, $C_5\times \SL(2,3)$

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

Order 80: $D_4\times D_5$ x 6, $D_{40}$ x 4, $C_{40}:C_2$ x 4, $D_{20}:C_2$ x 4, $C_5\times \SD_{16}$ x 4, $Q_8:D_5$ x 4, $C_5:D_8$ x 4, $C_2\times D_{20}$ x 2, $C_{10}:C_8$, $C_2^2\times D_{10}$, $Q_8\times C_{10}$, $D_4\times C_{10}$, $C_{10}:D_4$, $C_4\times D_{10}$, $C_2\times C_{40}$

Order 64: $C_8:C_2^3$

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

Order 48: $\GL(2,3)$ x 4, $\SL(2,3):C_2$ x 2, $C_2\times D_{12}$, $C_2\times \SL(2,3)$

Order 40: $C_2\times D_{10}$ x 10, $D_{20}$ x 6, $C_5:D_4$ x 4, $C_4\times D_5$ x 4, $C_5\times D_4$ x 3, $C_{40}$ 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: $D_8:C_2$ x 8, $C_2\times \SD_{16}$ x 2, $C_2\times D_8$ x 2, $D_4:C_2^2$, $C_2^2\times D_4$, $C_2\times \OD_{16}$

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

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

Order 20: $D_{10}$ x 16, $C_2\times C_{10}$ x 5, $C_{20}$ x 2, $C_5:C_4$ x 2

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

Order 15: $C_{15}$

Order 12: $D_6$ x 8, $C_{12}$ x 2, $C_2\times C_6$

Order 10: $C_{10}$ x 5, $D_5$ x 4

Order 8: $D_4$ x 13, $C_2^3$ x 11, $C_2\times C_4$ x 6, $C_8$ x 4, $Q_8$ x 2

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

Order 5: $C_5$

Order 4: $C_2^2$ x 21, $C_4$ x 4

Order 3: $C_3$

Order 2: $C_2$ x 9

Order 1: $C_1$

Classes of subgroups up to automorphism

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

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

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

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

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

Order 160: $C_8:D_{10}$ x 2, $Q_8:D_{10}$, $D_4\times D_{10}$, $C_{10}\times \SD_{16}$, $Q_8:D_{10}$, $C_{10}:D_8$, $C_2\times D_{40}$, $C_8:D_{10}$

Order 120: $C_2\times D_{30}$, $C_{10}\times D_6$, $C_{10}:C_{12}$, $C_5\times \SL(2,3)$, $C_5:D_{12}$

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

Order 80: $D_{20}:C_2$ x 2, $D_4\times D_5$ x 2, $C_2\times D_{20}$ x 2, $C_5\times \SD_{16}$ x 2, $Q_8:D_5$ x 2, $C_{10}:C_8$, $D_{40}$, $C_2^2\times D_{10}$, $C_{40}:C_2$, $Q_8\times C_{10}$, $D_4\times C_{10}$, $C_{10}:D_4$, $C_4\times D_{10}$, $C_2\times C_{40}$, $C_5:D_8$

Order 64: $C_8:C_2^3$

Order 60: $D_{30}$ x 2, $S_3\times C_{10}$ x 2, $C_5:C_{12}$, $C_2\times C_{30}$

Order 48: $\GL(2,3)$ x 2, $C_2\times D_{12}$, $\SL(2,3):C_2$, $C_2\times \SL(2,3)$

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

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

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

Order 24: $C_2\times D_6$ x 2, $C_2\times C_{12}$, $D_{12}$, $\SL(2,3)$

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

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

Order 15: $C_{15}$

Order 12: $D_6$ x 4, $C_2\times C_6$, $C_{12}$

Order 10: $C_{10}$ x 3, $D_5$ x 2

Order 8: $D_4$ x 7, $C_2^3$ x 5, $C_2\times C_4$ x 4, $Q_8$ x 2, $C_8$ x 2

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

Order 5: $C_5$

Order 4: $C_2^2$ x 8, $C_4$ x 3

Order 3: $C_3$

Order 2: $C_2$ x 5

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

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

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

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

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: $D_{20}:C_2$ ($D_6$) x 2, $Q_8\times C_{10}$ ($D_6$)

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

Order 40: $C_{10}:C_4$ ($S_4$), $C_5\times Q_8$ ($C_2\times D_6$)

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

Order 20: $C_5:C_4$ ($C_2\times S_4$) x 2, $C_2\times C_{10}$ ($C_2\times S_4$)

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

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

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

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

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

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

Order 40: $C_{10}:C_4$ ($S_4$), $C_5\times Q_8$ ($C_2\times D_6$)

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

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

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

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

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: $C_2$ ($\GL(2,3):D_5$), $C_2$ ($D_{10}\times S_4$)

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

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

$\rhd$

$\GL(2,3):D_5$

$\rhd$

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

$\rhd$

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

$\rhd$

$C_5\times Q_8$

$\rhd$

$C_{10}$

$\rhd$

$C_5$

$\rhd$

$C_1$

Lower central series

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

$\rhd$

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

Upper central series

$C_1$

$\lhd$

$C_2^2$

Character theory

Complex character table

See the $58 \times 58$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $38 \times 38$ rational character table.