Abstract group 1296.1862: $C_6^2.S_3^2$

• Label: 1296.1862
• Order: \( 2^{4} \cdot 3^{4} \)
• Exponent: \( 2 \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{7} \cdot 3^{6} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \cdot 3^{2} \)
• Perm deg.: $31$
• Trans deg.: $108$
• Rank: $4$

Group information

Description:	$C_6^2.S_3^2$
Order:	\(1296\)\(\medspace = 2^{4} \cdot 3^{4} \)
Exponent:	\(18\)\(\medspace = 2 \cdot 3^{2} \)
Automorphism group:	$C_6^2.C_6^2.C_3^2.C_2^3$, of order \(93312\)\(\medspace = 2^{7} \cdot 3^{6} \)
Composition factors:	$C_2$ x 4, $C_3$ x 4
Derived length:	$3$

This group is nonabelian and supersolvable (hence solvable and monomial).

Group statistics

Order	1	2	3	6	9	18
Elements	1	255	62	690	18	270	1296
Conjugacy classes	1	15	4	24	3	21	68
Divisions	1	15	4	24	1	7	52
Autjugacy classes	1	4	4	7	1	2	19

Dimension	1	2	4	6	18
Irr. complex chars.	16	16	4	32	0	68
Irr. rational chars.	16	16	4	8	8	52

Minimal presentations

Permutation degree:	$31$
Transitive degree:	$108$
Rank:	$4$
Inequivalent generating quadruples:	$8624070$

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:	$\langle a, b, c, d \mid a^{2}=b^{6}=c^{18}=d^{6}=[a,c]=[c,d]=1, b^{a}=b^{5}, d^{a}=d^{5}, c^{b}=c^{5}d^{4}, d^{b}=c^{6}d^{5} \rangle$
Permutation group:	Degree $31$ $\langle(2,5)(3,10)(4,13)(6,9)(7,14)(8,22)(11,24)(12,25)(15,21)(16,20)(17,26)(18,19) \!\cdots\! \rangle$
Direct product:	$C_2$ ${}^2$ $\, \times\, $ $(C_3^2.S_3^2)$
Semidirect product:	$(C_6:D_{18})$ $\,\rtimes\,$ $S_3$	$(C_3:D_{18})$ $\,\rtimes\,$ $D_6$	$(C_6\times C_{18})$ $\,\rtimes\,$ $D_6$	$(C_6^2:S_3)$ $\,\rtimes\,$ $S_3$	all 21
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_6^2$ . $S_3^2$	$C_3^2$ . $D_6^2$	$C_6$ . $(C_6.S_3^2)$	$C_3$ . $(C_6^2:D_6)$	all 9

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

Homology

Abelianization:	$C_{2}^{4} $
Schur multiplier:	$C_{2}^{6}$
Commutator length:	$1$

Subgroups

There are 9784 subgroups in 718 conjugacy classes, 114 normal (18 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2$	$G/Z \simeq$ $C_3^2.S_3^2$
Commutator:	$G' \simeq$ $\He_3:C_3$	$G/G' \simeq$ $C_2^4$
Frattini:	$\Phi \simeq$ $C_3^2$	$G/\Phi \simeq$ $D_6^2$
Fitting:	$\operatorname{Fit} \simeq$ $C_3^2.C_6^2$	$G/\operatorname{Fit} \simeq$ $C_2^2$
Radical:	$R \simeq$ $C_6^2.S_3^2$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_6$	$G/\operatorname{soc} \simeq$ $C_3^2:D_6$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $\He_3:C_3$

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

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Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 1296: $C_6^2.S_3^2$

Order 648: $(C_3\times C_{18}):D_6$ x 12, $(C_6\times C_{18}):S_3$, $(C_6\times C_{18}):C_6$, $(C_6\times C_{18}):S_3$

Order 432: $C_6^2:D_6$, $D_6\times D_{18}$

Order 324: $C_3^2.S_3^2$ x 16, $\He_3:D_6$ x 6, $\He_3.D_6$ x 6, $(C_3\times C_{18}):S_3$ x 6, $C_3^2.C_6^2$

Order 216: $C_6.S_3^2$ x 12, $S_3\times D_{18}$ x 12, $C_6^2:C_6$ x 3, $C_6^2:S_3$, $C_6:D_{18}$, $D_6\times C_{18}$, $C_6\times D_{18}$

Order 162: $\He_3:S_3$ x 4, $\He_3.S_3$ x 4, $\He_3.C_6$ x 4, $\He_3:C_6$ x 3

Order 144: $D_6^2$ x 2, $C_2^2\times D_{18}$

Order 108: $C_3^2:D_6$ x 18, $C_3^2:D_6$ x 16, $S_3\times D_9$ x 16, $C_3^2:D_6$ x 6, $C_3:D_{18}$ x 6, $S_3\times C_{18}$ x 6, $C_3\times D_{18}$ x 6, $C_2^2\times \He_3$ x 2, $C_6\times C_{18}$

Order 81: $\He_3:C_3$

Order 72: $S_3\times D_6$ x 24, $C_2\times D_{18}$ x 14, $C_6\times D_6$ x 4, $C_6:D_6$ x 3, $C_2^2\times C_{18}$

Order 54: $C_3^2:C_6$ x 12, $C_2\times \He_3$ x 6, $C_3^2:S_3$ x 4, $C_3:D_9$ x 4, $S_3\times C_9$ x 4, $C_3\times D_9$ x 4, $C_3\times C_{18}$ x 3

Order 48: $C_2^2\times D_6$ x 3

Order 36: $S_3^2$ x 32, $D_{18}$ x 28, $C_6\times S_3$ x 24, $C_6:S_3$ x 18, $C_2\times C_{18}$ x 7, $C_6^2$ x 3

Order 27: $\He_3$ x 2, $C_3\times C_9$

Order 24: $C_2\times D_6$ x 43, $C_2^2\times C_6$ x 3

Order 18: $C_3\times S_3$ x 16, $C_3:S_3$ x 12, $C_3\times C_6$ x 9, $D_9$ x 8, $C_{18}$ x 7

Order 16: $C_2^4$

Order 12: $D_6$ x 90, $C_2\times C_6$ x 22

Order 9: $C_3^2$ x 3, $C_9$

Order 8: $C_2^3$ x 15

Order 6: $S_3$ x 28, $C_6$ x 24

Order 4: $C_2^2$ x 35

Order 3: $C_3$ x 4

Order 2: $C_2$ x 15

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1296: $C_6^2.S_3^2$

Order 648: $(C_6\times C_{18}):S_3$, $(C_6\times C_{18}):C_6$, $(C_6\times C_{18}):S_3$, $(C_3\times C_{18}):D_6$

Order 432: $C_6^2:D_6$, $D_6\times D_{18}$

Order 324: $C_3^2.C_6^2$, $\He_3:D_6$, $\He_3.D_6$, $(C_3\times C_{18}):S_3$, $C_3^2.S_3^2$

Order 216: $C_6^2:C_6$ x 3, $C_6^2:S_3$, $C_6:D_{18}$, $D_6\times C_{18}$, $C_6\times D_{18}$, $C_6.S_3^2$, $S_3\times D_{18}$

Order 162: $\He_3:C_6$, $\He_3:S_3$, $\He_3.S_3$, $\He_3.C_6$

Order 144: $D_6^2$ x 2, $C_2^2\times D_{18}$

Order 108: $C_3^2:D_6$ x 3, $C_2^2\times \He_3$ x 2, $C_6\times C_{18}$, $C_3^2:D_6$, $C_3:D_{18}$, $S_3\times C_{18}$, $C_3\times D_{18}$, $C_3^2:D_6$, $S_3\times D_9$

Order 81: $\He_3:C_3$

Order 72: $C_6\times D_6$ x 4, $C_6:D_6$ x 3, $C_2\times D_{18}$ x 3, $S_3\times D_6$ x 2, $C_2^2\times C_{18}$

Order 54: $C_3^2:C_6$ x 3, $C_2\times \He_3$ x 2, $C_3\times C_{18}$, $C_3^2:S_3$, $C_3:D_9$, $S_3\times C_9$, $C_3\times D_9$

Order 48: $C_2^2\times D_6$ x 3

Order 36: $C_6\times S_3$ x 4, $D_{18}$ x 3, $C_6^2$ x 3, $C_6:S_3$ x 3, $C_2\times C_{18}$ x 2, $S_3^2$ x 2

Order 27: $\He_3$ x 2, $C_3\times C_9$

Order 24: $C_2\times D_6$ x 10, $C_2^2\times C_6$ x 3

Order 18: $C_3\times S_3$ x 4, $C_3\times C_6$ x 3, $C_3:S_3$ x 3, $C_{18}$ x 2, $D_9$ x 2

Order 16: $C_2^4$

Order 12: $D_6$ x 10, $C_2\times C_6$ x 7

Order 9: $C_3^2$ x 3, $C_9$

Order 8: $C_2^3$ x 4

Order 6: $C_6$ x 7, $S_3$ x 7

Order 4: $C_2^2$ x 5

Order 3: $C_3$ x 4

Order 2: $C_2$ x 4

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1296: $C_6^2.S_3^2$ ($C_1$)

Order 648: $(C_3\times C_{18}):D_6$ ($C_2$) x 12, $(C_6\times C_{18}):S_3$ ($C_2$), $(C_6\times C_{18}):C_6$ ($C_2$), $(C_6\times C_{18}):S_3$ ($C_2$)

Order 324: $C_3^2.S_3^2$ ($C_2^2$) x 16, $\He_3:D_6$ ($C_2^2$) x 6, $\He_3.D_6$ ($C_2^2$) x 6, $(C_3\times C_{18}):S_3$ ($C_2^2$) x 6, $C_3^2.C_6^2$ ($C_2^2$)

Order 216: $C_6^2:S_3$ ($S_3$), $C_6:D_{18}$ ($S_3$)

Order 162: $\He_3:S_3$ ($C_2^3$) x 4, $\He_3.S_3$ ($C_2^3$) x 4, $\He_3.C_6$ ($C_2^3$) x 4, $\He_3:C_6$ ($C_2^3$) x 3

Order 108: $C_3^2:D_6$ ($D_6$) x 6, $C_3:D_{18}$ ($D_6$) x 6, $C_2^2\times \He_3$ ($D_6$), $C_6\times C_{18}$ ($D_6$)

Order 81: $\He_3:C_3$ ($C_2^4$)

Order 54: $C_3^2:S_3$ ($C_2\times D_6$) x 4, $C_3:D_9$ ($C_2\times D_6$) x 4, $C_3\times C_{18}$ ($C_2\times D_6$) x 3, $C_2\times \He_3$ ($C_2\times D_6$) x 3

Order 36: $C_6^2$ ($S_3^2$)

Order 27: $\He_3$ ($C_2^2\times D_6$), $C_3\times C_9$ ($C_2^2\times D_6$)

Order 18: $C_3\times C_6$ ($S_3\times D_6$) x 3

Order 12: $C_2\times C_6$ ($C_3^2:D_6$)

Order 9: $C_3^2$ ($D_6^2$)

Order 6: $C_6$ ($C_6.S_3^2$) x 3

Order 4: $C_2^2$ ($C_3^2.S_3^2$)

Order 3: $C_3$ ($C_6^2:D_6$)

Order 2: $C_2$ ($(C_3\times C_{18}):D_6$) x 3

Order 1: $C_1$ ($C_6^2.S_3^2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1296: $C_6^2.S_3^2$ ($C_1$)

Order 648: $(C_6\times C_{18}):S_3$ ($C_2$), $(C_6\times C_{18}):C_6$ ($C_2$), $(C_6\times C_{18}):S_3$ ($C_2$), $(C_3\times C_{18}):D_6$ ($C_2$)

Order 324: $C_3^2.C_6^2$ ($C_2^2$), $\He_3:D_6$ ($C_2^2$), $\He_3.D_6$ ($C_2^2$), $(C_3\times C_{18}):S_3$ ($C_2^2$), $C_3^2.S_3^2$ ($C_2^2$)

Order 216: $C_6^2:S_3$ ($S_3$), $C_6:D_{18}$ ($S_3$)

Order 162: $\He_3:C_6$ ($C_2^3$), $\He_3:S_3$ ($C_2^3$), $\He_3.S_3$ ($C_2^3$), $\He_3.C_6$ ($C_2^3$)

Order 108: $C_2^2\times \He_3$ ($D_6$), $C_6\times C_{18}$ ($D_6$), $C_3^2:D_6$ ($D_6$), $C_3:D_{18}$ ($D_6$)

Order 81: $\He_3:C_3$ ($C_2^4$)

Order 54: $C_3\times C_{18}$ ($C_2\times D_6$), $C_3^2:S_3$ ($C_2\times D_6$), $C_3:D_9$ ($C_2\times D_6$), $C_2\times \He_3$ ($C_2\times D_6$)

Order 36: $C_6^2$ ($S_3^2$)

Order 27: $\He_3$ ($C_2^2\times D_6$), $C_3\times C_9$ ($C_2^2\times D_6$)

Order 18: $C_3\times C_6$ ($S_3\times D_6$)

Order 12: $C_2\times C_6$ ($C_3^2:D_6$)

Order 9: $C_3^2$ ($D_6^2$)

Order 6: $C_6$ ($C_6.S_3^2$)

Order 4: $C_2^2$ ($C_3^2.S_3^2$)

Order 3: $C_3$ ($C_6^2:D_6$)

Order 2: $C_2$ ($(C_3\times C_{18}):D_6$)

Order 1: $C_1$ ($C_6^2.S_3^2$)

Series

Derived series

$C_6^2.S_3^2$

$\rhd$

$\He_3:C_3$

$\rhd$

$C_3^2$

$\rhd$

$C_1$

Chief series

$C_6^2.S_3^2$

$\rhd$

$(C_3\times C_{18}):D_6$

$\rhd$

$C_3^2.S_3^2$

$\rhd$

$\He_3.C_6$

$\rhd$

$\He_3:C_3$

$\rhd$

$\He_3$

$\rhd$

$C_3\times C_9$

$\rhd$

$C_3^2$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_6^2.S_3^2$

$\rhd$

$\He_3:C_3$

Upper central series

$C_1$

$\lhd$

$C_2^2$

Character theory

Complex character table

See the $68 \times 68$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $52 \times 52$ rational character table.