Abstract group 864.4438: $(C_3\times \SL(2,3)):D_6$

• Label: 864.4438
• Order: \( 2^{5} \cdot 3^{3} \)
• Exponent: \( 2^{3} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{10} \cdot 3^{6} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \cdot 3^{3} \)
• Perm deg.: $22$
• Trans deg.: $144$
• Rank: $4$

Group information

Description:	$(C_3\times \SL(2,3)):D_6$
Order:	\(864\)\(\medspace = 2^{5} \cdot 3^{3} \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Automorphism group:	$C_3^4.(Q_8\times A_4).C_6.C_2^4$, of order \(746496\)\(\medspace = 2^{10} \cdot 3^{6} \)
Composition factors:	$C_2$ x 5, $C_3$ x 3
Derived length:	$4$

This group is nonabelian and solvable. Whether it is monomial has not been computed.

Group statistics

Order	1	2	3	4	6	8	12
Elements	1	223	80	8	128	216	208	864
Conjugacy classes	1	4	13	2	17	2	30	69
Divisions	1	4	13	2	17	2	17	56
Autjugacy classes	1	3	2	2	3	1	3	15

Dimension	1	2	3	4	6	8
Irr. complex chars.	4	26	4	27	8	0	69
Irr. rational chars.	4	26	4	1	8	13	56

Minimal presentations

Permutation degree:	$22$
Transitive degree:	$144$
Rank:	$4$
Inequivalent generating quadruples:	$343980$

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:	${\langle a, b, c, d \mid a^{2}=d^{12}=1, b^{6}=d^{6}, c^{6}=d^{6}, b^{a}= \!\cdots\! \rangle}$
Permutation group:	Degree $22$ $\langle(1,2)(3,6)(4,10)(5,9)(7,14)(8,12)(11,16)(13,15)(18,19)(21,22), (1,3,6,2) \!\cdots\! \rangle$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$C_3$ $\,\rtimes\,$ $(C_{12}.S_4)$	$(C_4.C_6^2)$ $\,\rtimes\,$ $S_3$	$(\SL(2,3):C_6)$ $\,\rtimes\,$ $S_3$	$\SL(2,3)$ $\,\rtimes\,$ $(C_6:S_3)$	all 12
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_6$ . $(C_6:S_4)$	$(C_3\times C_{12})$ . $S_4$	$C_{12}$ . $(C_3:S_4)$	$C_4$ . $(C_3^2:S_4)$	all 9

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

Homology

Abelianization:	$C_{2}^{2} $
Schur multiplier:	$C_{3}^{2} \times C_{6}$
Commutator length:	$1$

Subgroups

There are 6414 subgroups in 460 conjugacy classes, 77 normal (13 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_6^2:D_6$
Commutator:	$G' \simeq$ $C_3^2\times \SL(2,3)$	$G/G' \simeq$ $C_2^2$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_6^2:D_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_4.C_6^2$	$G/\operatorname{Fit} \simeq$ $S_3$
Radical:	$R \simeq$ $(C_3\times \SL(2,3)):D_6$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3\times C_6$	$G/\operatorname{soc} \simeq$ $C_2\times S_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_8:C_2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^3$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: subgroups only stored up to automorphism

Classes of subgroups up to automorphism

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

For the default diagram, subgroups are sorted vertically by the number of prime divisors (counted with multiplicity) in their orders.
To see subgroups sorted vertically by order instead, check this box.

Sorry, your browser does not support the subgroup diagram.

See a full page version of the diagram

See a full page version of the diagram

See a full page version of the diagram

See a full page version of the diagram

Subgroup information

Click on a subgroup in the diagram to see information about it.

Classes of subgroups up to conjugation

Order 864: $(C_3\times \SL(2,3)):D_6$

Order 432: $C_3^2:\GL(2,3)$ x 2, $C_3\times \SL(2,3):C_6$

Order 288: $C_{12}.S_4$ x 12, $C_6^2.D_4$

Order 216: $C_3^2:D_{12}$, $C_3^2\times \SL(2,3)$

Order 144: $C_3:\GL(2,3)$ x 24, $\SL(2,3):C_6$ x 12, $C_{12}.D_6$ x 2, $C_3^2:D_8$ x 2, $C_{12}.D_6$, $C_4.C_6^2$, $C_6:D_{12}$

Order 108: $C_3^2:D_6$ x 2, $C_3^2\times C_{12}$

Order 96: $\GL(2,3):C_2$ x 9, $D_4:D_6$ x 4

Order 72: $C_3:D_{12}$ x 15, $C_3\times \SL(2,3)$ x 12, $C_3^2:C_8$ x 2, $C_6:D_6$, $Q_8\times C_3^2$, $D_4\times C_3^2$, $C_6\times C_{12}$

Order 54: $C_3^2:S_3$ x 2, $C_3^2\times C_6$

Order 48: $\GL(2,3)$ x 18, $\SL(2,3):C_2$ x 9, $Q_8:S_3$ x 8, $C_3:D_8$ x 8, $D_4:C_6$ x 4, $C_2\times D_{12}$ x 4, $C_3:\OD_{16}$ x 4

Order 36: $C_6:S_3$ x 28, $C_3\times C_{12}$ x 14, $C_6^2$

Order 32: $D_8:C_2$

Order 27: $C_3^3$

Order 24: $D_{12}$ x 21, $\SL(2,3)$ x 9, $C_3:C_8$ x 8, $C_2\times C_{12}$ x 4, $C_2\times D_6$ x 4, $C_3\times Q_8$ x 4, $C_3\times D_4$ x 4

Order 18: $C_3:S_3$ x 26, $C_3\times C_6$ x 14

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

Order 12: $D_6$ x 34, $C_{12}$ x 17, $C_2\times C_6$ x 4

Order 9: $C_3^2$ x 13

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

Order 6: $S_3$ x 26, $C_6$ x 17

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

Order 3: $C_3$ x 13

Order 2: $C_2$ x 4

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 864: $(C_3\times \SL(2,3)):D_6$

Order 432: $C_3\times \SL(2,3):C_6$, $C_3^2:\GL(2,3)$

Order 288: $C_{12}.S_4$, $C_6^2.D_4$

Order 216: $C_3^2:D_{12}$, $C_3^2\times \SL(2,3)$

Order 144: $C_{12}.D_6$, $C_3^2:D_8$, $C_{12}.D_6$, $C_4.C_6^2$, $C_6:D_{12}$, $\SL(2,3):C_6$, $C_3:\GL(2,3)$

Order 108: $C_3^2:D_6$, $C_3^2\times C_{12}$

Order 96: $\GL(2,3):C_2$, $D_4:D_6$

Order 72: $C_3:D_{12}$ x 3, $C_6:D_6$, $Q_8\times C_3^2$, $D_4\times C_3^2$, $C_6\times C_{12}$, $C_3\times \SL(2,3)$, $C_3^2:C_8$

Order 54: $C_3^2\times C_6$, $C_3^2:S_3$

Order 48: $D_4:C_6$, $C_2\times D_{12}$, $\SL(2,3):C_2$, $\GL(2,3)$, $Q_8:S_3$, $C_3:D_8$, $C_3:\OD_{16}$

Order 36: $C_3\times C_{12}$ x 3, $C_6:S_3$ x 3, $C_6^2$

Order 32: $D_8:C_2$

Order 27: $C_3^3$

Order 24: $D_{12}$ x 3, $C_2\times C_{12}$, $\SL(2,3)$, $C_2\times D_6$, $C_3\times Q_8$, $C_3\times D_4$, $C_3:C_8$

Order 18: $C_3\times C_6$ x 3, $C_3:S_3$ x 2

Order 16: $\SD_{16}$, $D_8$, $\OD_{16}$, $D_4:C_2$, $C_2\times D_4$

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

Order 9: $C_3^2$ x 2

Order 8: $D_4$ x 3, $C_2^3$, $Q_8$, $C_2\times C_4$, $C_8$

Order 6: $C_6$ x 3, $S_3$ x 2

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

Order 3: $C_3$ x 2

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 864: $(C_3\times \SL(2,3)):D_6$ ($C_1$)

Order 432: $C_3^2:\GL(2,3)$ ($C_2$) x 2, $C_3\times \SL(2,3):C_6$ ($C_2$)

Order 216: $C_3^2\times \SL(2,3)$ ($C_2^2$)

Order 144: $\SL(2,3):C_6$ ($S_3$) x 12, $C_4.C_6^2$ ($S_3$)

Order 72: $C_3\times \SL(2,3)$ ($D_6$) x 12, $Q_8\times C_3^2$ ($D_6$)

Order 48: $\SL(2,3):C_2$ ($C_3:S_3$) x 9, $D_4:C_6$ ($C_3:S_3$) x 4

Order 36: $C_3\times C_{12}$ ($S_4$)

Order 24: $\SL(2,3)$ ($C_6:S_3$) x 9, $C_3\times Q_8$ ($C_6:S_3$) x 4

Order 18: $C_3\times C_6$ ($C_2\times S_4$)

Order 16: $D_4:C_2$ ($C_3^2:S_3$)

Order 12: $C_{12}$ ($C_3:S_4$) x 4

Order 9: $C_3^2$ ($\GL(2,3):C_2$)

Order 8: $Q_8$ ($C_3^2:D_6$)

Order 6: $C_6$ ($C_6:S_4$) x 4

Order 4: $C_4$ ($C_3^2:S_4$)

Order 3: $C_3$ ($C_{12}.S_4$) x 4

Order 2: $C_2$ ($C_6^2:D_6$)

Order 1: $C_1$ ($(C_3\times \SL(2,3)):D_6$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 864: $(C_3\times \SL(2,3)):D_6$ ($C_1$)

Order 432: $C_3\times \SL(2,3):C_6$ ($C_2$), $C_3^2:\GL(2,3)$ ($C_2$)

Order 216: $C_3^2\times \SL(2,3)$ ($C_2^2$)

Order 144: $C_4.C_6^2$ ($S_3$), $\SL(2,3):C_6$ ($S_3$)

Order 72: $Q_8\times C_3^2$ ($D_6$), $C_3\times \SL(2,3)$ ($D_6$)

Order 48: $D_4:C_6$ ($C_3:S_3$), $\SL(2,3):C_2$ ($C_3:S_3$)

Order 36: $C_3\times C_{12}$ ($S_4$)

Order 24: $\SL(2,3)$ ($C_6:S_3$), $C_3\times Q_8$ ($C_6:S_3$)

Order 18: $C_3\times C_6$ ($C_2\times S_4$)

Order 16: $D_4:C_2$ ($C_3^2:S_3$)

Order 12: $C_{12}$ ($C_3:S_4$)

Order 9: $C_3^2$ ($\GL(2,3):C_2$)

Order 8: $Q_8$ ($C_3^2:D_6$)

Order 6: $C_6$ ($C_6:S_4$)

Order 4: $C_4$ ($C_3^2:S_4$)

Order 3: $C_3$ ($C_{12}.S_4$)

Order 2: $C_2$ ($C_6^2:D_6$)

Order 1: $C_1$ ($(C_3\times \SL(2,3)):D_6$)

Series

Derived series

$(C_3\times \SL(2,3)):D_6$

$\rhd$

$C_3^2\times \SL(2,3)$

$\rhd$

$Q_8$

$\rhd$

$C_2$

$\rhd$

$C_1$

Chief series

$(C_3\times \SL(2,3)):D_6$

$\rhd$

$C_3^2:\GL(2,3)$

$\rhd$

$C_3^2\times \SL(2,3)$

$\rhd$

$C_3\times \SL(2,3)$

$\rhd$

$\SL(2,3)$

$\rhd$

$Q_8$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$(C_3\times \SL(2,3)):D_6$

$\rhd$

$C_3^2\times \SL(2,3)$

Upper central series

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_4$

Supergroups

This group is a maximal subgroup of 24 larger groups in the database.

This group is a maximal quotient of 6 larger groups in the database.

Character theory

Complex character table

See the $69 \times 69$ 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.