Abstract group 1296.1067: $C_3\wr S_3\times D_4$

• Label: 1296.1067
• Order: \( 2^{4} \cdot 3^{4} \)
• Exponent: \( 2^{2} \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 3 \)
• $\card{Z(G)}$: \( 2 \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{6} \cdot 3^{4} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \cdot 3 \)
• Perm deg.: $13$
• Trans deg.: $36$
• Rank: $3$

Group information

Description:	$C_3\wr S_3\times D_4$
Order:	\(1296\)\(\medspace = 2^{4} \cdot 3^{4} \)
Exponent:	\(36\)\(\medspace = 2^{2} \cdot 3^{2} \)
Automorphism group:	$(C_6\times \He_3).C_2^5$, of order \(5184\)\(\medspace = 2^{6} \cdot 3^{4} \)
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	4	6	9	12	18	36
Elements	1	59	44	20	652	36	232	180	72	1296
Conjugacy classes	1	7	10	2	62	2	18	6	2	110
Divisions	1	7	6	2	34	1	10	3	1	65
Autjugacy classes	1	4	4	2	12	1	6	2	1	33

Dimension	1	2	3	4	6	8	12
Irr. complex chars.	24	18	48	3	16	0	1	110
Irr. rational chars.	8	14	0	7	28	1	7	65

Minimal presentations

Permutation degree:	$13$
Transitive degree:	$36$
Rank:	$3$
Inequivalent generating triples:	$117936$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	6	12	12
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	$\langle a, b, c, d \mid a^{6}=b^{6}=c^{6}=d^{6}=[a,d]=[b,d]=[c,d]=1, b^{a}=b^{5}c^{2}d^{4}, c^{a}=c^{5}d^{4}, c^{b}=cd \rangle$
Permutation group:	Degree $13$ $\langle(2,5)(6,8)(7,9)(10,11)(12,13), (10,12)(11,13), (12,13), (2,6,7)(5,8,9), (10,11)(12,13), (1,2,5)(3,7,9)(4,6,8), (1,3,4)(2,6,7), (1,4,3)(2,6,7)(5,8,9)\rangle$
Transitive group:	36T1982				more information
Direct product:	$D_4$ $\, \times\, $ $(C_3\wr S_3)$
Semidirect product:	$(D_4\times C_3^3)$ $\,\rtimes\,$ $S_3$	$C_3^3$ $\,\rtimes\,$ $(S_3\times D_4)$	$(C_6^2:S_3)$ $\,\rtimes\,$ $C_6$	$(C_6^2:S_3)$ $\,\rtimes\,$ $C_6$	all 28
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_6^2$ . $(C_6\times S_3)$	$C_3$ . $(C_6^2:D_6)$	$C_6$ . $(C_6^2:C_6)$	$C_{12}$ . $(C_3^2:D_6)$	all 16

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

Homology

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

Subgroups

There are 2800 subgroups in 448 conjugacy classes, 62 normal (34 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_6$	$G/Z \simeq$ $C_6^2:C_6$
Commutator:	$G' \simeq$ $C_2\times \He_3$	$G/G' \simeq$ $C_2^2\times C_6$
Frattini:	$\Phi \simeq$ $C_3\times C_6$	$G/\Phi \simeq$ $C_6\times D_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_3\wr C_3\times D_4$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_3\wr S_3\times D_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_6$	$G/\operatorname{soc} \simeq$ $C_6^2:C_6$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3\wr C_3$

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: $C_3\wr S_3\times D_4$

Order 648: $C_2^2\times C_3\wr S_3$ x 2, $(C_3\times C_6^2):S_3$ x 2, $C_3\wr C_3\times D_4$, $C_3^3:D_{12}$, $C_4\times C_3\wr S_3$

Order 432: $C_{12}:C_6^2$, $C_6^2:D_6$

Order 324: $C_3^3:D_6$ x 7, $C_3^2.C_6^2$ x 2, $C_3^3:C_{12}$, $\He_3:C_{12}$

Order 216: $C_6^2:S_3$ x 2, $S_3\times C_6^2$ x 2, $C_6^2:C_6$ x 2, $C_6^2:S_3$ x 2, $C_{36}:C_6$, $D_4\times \He_3$, $C_3^2:D_{12}$, $C_4\times C_3^2:S_3$, $D_4\times C_3^3$, $C_3^2\times D_{12}$, $C_6.C_6^2$

Order 162: $C_3\wr S_3$ x 4, $C_3^3:C_6$ x 3

Order 144: $C_{12}:D_6$ x 5, $C_4:C_6^2$

Order 108: $C_3^2\times D_6$ x 7, $C_3^2:D_6$ x 7, $C_3\times C_6^2$ x 2, $C_{18}:C_6$ x 2, $C_2^2\times \He_3$ x 2, $C_3^2\times C_{12}$, $C_3^2:C_{12}$, $C_9:C_{12}$, $C_4\times \He_3$, $\He_3:C_4$

Order 81: $C_3\wr C_3$

Order 72: $C_6\times D_6$ x 10, $C_6\wr C_2$ x 10, $D_4\times C_3^2$ x 9, $C_3\times D_{12}$ x 5, $S_3\times C_{12}$ x 5, $C_2\times C_6^2$ x 2, $C_6\times C_{12}$, $D_4\times C_9$

Order 54: $C_3^2:S_3$ x 4, $S_3\times C_3^2$ x 4, $C_3^2\times C_6$ x 3, $C_9:C_6$ x 3, $C_2\times \He_3$ x 3

Order 48: $C_6\times D_4$ x 4, $S_3\times D_4$ x 2

Order 36: $C_6\times S_3$ x 35, $C_6^2$ x 19, $C_3\times C_{12}$ x 7, $C_3:C_{12}$ x 5, $C_2\times C_{18}$ x 2, $C_{36}$

Order 27: $C_3^3$, $C_9:C_3$, $\He_3$

Order 24: $C_3\times D_4$ x 18, $C_2^2\times C_6$ x 8, $C_2\times C_{12}$ x 4, $C_3:D_4$ x 4, $C_2\times D_6$ x 4, $D_{12}$ x 2, $C_4\times S_3$ x 2

Order 18: $C_3\times C_6$ x 22, $C_3\times S_3$ x 20, $C_{18}$ x 3

Order 16: $C_2\times D_4$

Order 12: $C_2\times C_6$ x 40, $D_6$ x 14, $C_{12}$ x 10, $C_3:C_4$ x 2

Order 9: $C_3^2$ x 6, $C_9$

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

Order 6: $C_6$ x 34, $S_3$ x 8

Order 4: $C_2^2$ x 9, $C_4$ x 2

Order 3: $C_3$ x 6

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1296: $C_3\wr S_3\times D_4$

Order 648: $C_2^2\times C_3\wr S_3$, $C_3\wr C_3\times D_4$, $(C_3\times C_6^2):S_3$, $C_3^3:D_{12}$, $C_4\times C_3\wr S_3$

Order 432: $C_{12}:C_6^2$, $C_6^2:D_6$

Order 324: $C_3^3:D_6$ x 3, $C_3^2.C_6^2$, $C_3^3:C_{12}$, $\He_3:C_{12}$

Order 216: $C_{36}:C_6$, $D_4\times \He_3$, $C_6^2:S_3$, $C_3^2:D_{12}$, $C_4\times C_3^2:S_3$, $S_3\times C_6^2$, $D_4\times C_3^3$, $C_6^2:C_6$, $C_3^2\times D_{12}$, $C_6.C_6^2$, $C_6^2:S_3$

Order 162: $C_3^3:C_6$ x 2, $C_3\wr S_3$ x 2

Order 144: $C_{12}:D_6$ x 3, $C_4:C_6^2$

Order 108: $C_3^2\times D_6$ x 3, $C_3^2:D_6$ x 3, $C_3\times C_6^2$, $C_3^2\times C_{12}$, $C_3^2:C_{12}$, $C_{18}:C_6$, $C_2^2\times \He_3$, $C_9:C_{12}$, $C_4\times \He_3$, $\He_3:C_4$

Order 81: $C_3\wr C_3$

Order 72: $D_4\times C_3^2$ x 6, $C_6\times D_6$ x 3, $C_6\wr C_2$ x 3, $C_3\times D_{12}$ x 3, $S_3\times C_{12}$ x 3, $C_2\times C_6^2$, $C_6\times C_{12}$, $D_4\times C_9$

Order 54: $C_3^2:S_3$ x 2, $C_3^2\times C_6$ x 2, $S_3\times C_3^2$ x 2, $C_9:C_6$ x 2, $C_2\times \He_3$ x 2

Order 48: $C_6\times D_4$ x 2, $S_3\times D_4$ x 2

Order 36: $C_6\times S_3$ x 9, $C_6^2$ x 7, $C_3\times C_{12}$ x 5, $C_3:C_{12}$ x 3, $C_2\times C_{18}$, $C_{36}$

Order 27: $C_3^3$, $C_9:C_3$, $\He_3$

Order 24: $C_3\times D_4$ x 8, $C_2\times C_{12}$ x 2, $C_3:D_4$ x 2, $D_{12}$ x 2, $C_4\times S_3$ x 2, $C_2^2\times C_6$ x 2, $C_2\times D_6$ x 2

Order 18: $C_3\times C_6$ x 10, $C_3\times S_3$ x 6, $C_{18}$ x 2

Order 16: $C_2\times D_4$

Order 12: $C_2\times C_6$ x 10, $D_6$ x 6, $C_{12}$ x 6, $C_3:C_4$ x 2

Order 9: $C_3^2$ x 4, $C_9$

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

Order 6: $C_6$ x 12, $S_3$ x 4

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

Order 3: $C_3$ x 4

Order 2: $C_2$ x 4

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1296: $C_3\wr S_3\times D_4$ ($C_1$)

Order 648: $C_2^2\times C_3\wr S_3$ ($C_2$) x 2, $(C_3\times C_6^2):S_3$ ($C_2$) x 2, $C_3\wr C_3\times D_4$ ($C_2$), $C_3^3:D_{12}$ ($C_2$), $C_4\times C_3\wr S_3$ ($C_2$)

Order 432: $C_6^2:D_6$ ($C_3$)

Order 324: $C_3^3:D_6$ ($C_2^2$) x 3, $C_3^2.C_6^2$ ($C_2^2$) x 2, $C_3^3:C_{12}$ ($C_2^2$), $\He_3:C_{12}$ ($C_2^2$)

Order 216: $C_6^2:S_3$ ($C_6$) x 2, $C_6^2:S_3$ ($C_6$) x 2, $D_4\times \He_3$ ($C_6$), $C_3^2:D_{12}$ ($C_6$), $C_4\times C_3^2:S_3$ ($C_6$), $D_4\times C_3^3$ ($S_3$)

Order 162: $C_3\wr S_3$ ($D_4$) x 2, $C_3^3:C_6$ ($C_2^3$)

Order 108: $C_3^2:D_6$ ($C_2\times C_6$) x 3, $C_3\times C_6^2$ ($D_6$) x 2, $C_2^2\times \He_3$ ($C_2\times C_6$) x 2, $C_3^2\times C_{12}$ ($D_6$), $C_4\times \He_3$ ($C_2\times C_6$), $\He_3:C_4$ ($C_2\times C_6$)

Order 81: $C_3\wr C_3$ ($C_2\times D_4$)

Order 72: $D_4\times C_3^2$ ($C_3\times S_3$)

Order 54: $C_3^2:S_3$ ($C_3\times D_4$) x 2, $C_3^2\times C_6$ ($C_2\times D_6$), $C_2\times \He_3$ ($C_2^2\times C_6$)

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

Order 27: $C_3^3$ ($S_3\times D_4$), $\He_3$ ($C_6\times D_4$)

Order 24: $C_3\times D_4$ ($C_3^2:C_6$)

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

Order 12: $C_2\times C_6$ ($C_3^2:D_6$) x 2, $C_{12}$ ($C_3^2:D_6$)

Order 9: $C_3^2$ ($C_{12}:D_6$)

Order 8: $D_4$ ($C_3\wr S_3$)

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

Order 4: $C_2^2$ ($C_3^3:D_6$) x 2, $C_4$ ($C_3^3:D_6$)

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

Order 2: $C_2$ ($C_2^2\times C_3\wr S_3$)

Order 1: $C_1$ ($C_3\wr S_3\times D_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1296: $C_3\wr S_3\times D_4$ ($C_1$)

Order 648: $C_2^2\times C_3\wr S_3$ ($C_2$), $C_3\wr C_3\times D_4$ ($C_2$), $(C_3\times C_6^2):S_3$ ($C_2$), $C_3^3:D_{12}$ ($C_2$), $C_4\times C_3\wr S_3$ ($C_2$)

Order 432: $C_6^2:D_6$ ($C_3$)

Order 324: $C_3^3:D_6$ ($C_2^2$) x 2, $C_3^2.C_6^2$ ($C_2^2$), $C_3^3:C_{12}$ ($C_2^2$), $\He_3:C_{12}$ ($C_2^2$)

Order 216: $D_4\times \He_3$ ($C_6$), $C_6^2:S_3$ ($C_6$), $C_3^2:D_{12}$ ($C_6$), $C_4\times C_3^2:S_3$ ($C_6$), $D_4\times C_3^3$ ($S_3$), $C_6^2:S_3$ ($C_6$)

Order 162: $C_3^3:C_6$ ($C_2^3$), $C_3\wr S_3$ ($D_4$)

Order 108: $C_3^2:D_6$ ($C_2\times C_6$) x 2, $C_3\times C_6^2$ ($D_6$), $C_3^2\times C_{12}$ ($D_6$), $C_2^2\times \He_3$ ($C_2\times C_6$), $C_4\times \He_3$ ($C_2\times C_6$), $\He_3:C_4$ ($C_2\times C_6$)

Order 81: $C_3\wr C_3$ ($C_2\times D_4$)

Order 72: $D_4\times C_3^2$ ($C_3\times S_3$)

Order 54: $C_3^2:S_3$ ($C_3\times D_4$), $C_3^2\times C_6$ ($C_2\times D_6$), $C_2\times \He_3$ ($C_2^2\times C_6$)

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

Order 27: $C_3^3$ ($S_3\times D_4$), $\He_3$ ($C_6\times D_4$)

Order 24: $C_3\times D_4$ ($C_3^2:C_6$)

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

Order 12: $C_2\times C_6$ ($C_3^2:D_6$), $C_{12}$ ($C_3^2:D_6$)

Order 9: $C_3^2$ ($C_{12}:D_6$)

Order 8: $D_4$ ($C_3\wr S_3$)

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

Order 4: $C_2^2$ ($C_3^3:D_6$), $C_4$ ($C_3^3:D_6$)

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

Order 2: $C_2$ ($C_2^2\times C_3\wr S_3$)

Order 1: $C_1$ ($C_3\wr S_3\times D_4$)

Series

Derived series

$C_3\wr S_3\times D_4$

$\rhd$

$C_2\times \He_3$

$\rhd$

$C_3$

$\rhd$

$C_1$

Chief series

$C_3\wr S_3\times D_4$

$\rhd$

$C_3\wr C_3\times D_4$

$\rhd$

$D_4\times \He_3$

$\rhd$

$C_2^2\times \He_3$

$\rhd$

$C_6^2$

$\rhd$

$C_3\times C_6$

$\rhd$

$C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_3\wr S_3\times D_4$

$\rhd$

$C_2\times \He_3$

$\rhd$

$\He_3$

Upper central series

$C_1$

$\lhd$

$C_6$

$\lhd$

$C_3\times D_4$

Supergroups

This group is a maximal subgroup of 8 larger groups in the database.

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

Character theory

Complex character table

See the $110 \times 110$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $65 \times 65$ rational character table.