Abstract group 1296.3362: $C_6^2.S_3^2$

• Label: 1296.3362
• Order: \( 2^{4} \cdot 3^{4} \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 3 \)
• $\card{Z(G)}$: \( 2^{2} \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{11} \cdot 3^{4} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{9} \cdot 3 \)
• Perm deg.: $18$
• Trans deg.: $144$
• Rank: $3$

Group information

Description:	$C_6^2.S_3^2$
Order:	\(1296\)\(\medspace = 2^{4} \cdot 3^{4} \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$\PSU(3,2).C_6^2.C_2^6$, of order \(165888\)\(\medspace = 2^{11} \cdot 3^{4} \)
Composition factors:	$C_2$ x 4, $C_3$ x 4
Derived length:	$2$

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

Group statistics

Order	1	2	3	4	6	12
Elements	1	123	80	36	768	288	1296
Conjugacy classes	1	7	29	2	143	16	198
Divisions	1	7	19	2	85	6	120
Autjugacy classes	1	4	7	1	18	3	34

Dimension	1	2	4	8
Irr. complex chars.	24	126	48	0	198
Irr. rational chars.	8	30	64	18	120

Minimal presentations

Permutation degree:	$18$
Transitive degree:	$144$
Rank:	$3$
Inequivalent generating triples:	$2184$

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:	$\langle a, b, c, d \mid a^{6}=b^{12}=c^{3}=d^{6}=[a,c]=[a,d]=[c,d]=1, b^{a}=b^{11}, c^{b}=c^{2}, d^{b}=d^{5} \rangle$
Permutation group:	Degree $18$ $\langle(2,3)(5,6)(15,16,17,18), (11,12)(16,18), (13,14)(15,17)(16,18), (7,8,9), (15,17)(16,18), (1,2,3)(4,5,6), (1,2,3), (10,11,12)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 13 & 12 \\ 24 & 25 \end{array}\right), \left(\begin{array}{rr} 25 & 27 \\ 27 & 34 \end{array}\right), \left(\begin{array}{rr} 25 & 0 \\ 0 & 25 \end{array}\right), \left(\begin{array}{rr} 17 & 1 \\ 34 & 19 \end{array}\right), \left(\begin{array}{rr} 29 & 6 \\ 34 & 7 \end{array}\right), \left(\begin{array}{rr} 35 & 18 \\ 34 & 1 \end{array}\right), \left(\begin{array}{rr} 17 & 0 \\ 0 & 17 \end{array}\right), \left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/36\Z)$
Direct product:	$C_2$ $\, \times\, $ $C_3$ $\, \times\, $ $(C_3^2:D_{12})$
Semidirect product:	$(S_3\times C_6^2)$ $\,\rtimes\,$ $S_3$	$(S_3\times C_6^2)$ $\,\rtimes\,$ $C_6$	$(C_6^2:S_3)$ $\,\rtimes\,$ $C_6$	$(C_6^2.C_6)$ $\,\rtimes\,$ $S_3$	all 43
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_6^2$ . $S_3^2$	$C_6^2$ . $(C_6:S_3)$	$C_6^2$ . $(C_6\times S_3)$ (2)	$C_6$ . $(C_6:S_3^2)$	all 21

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:	$C_{2}^{2} \times C_{6}$
Commutator length:	$1$

Subgroups

There are 7616 subgroups in 1348 conjugacy classes, 184 normal (36 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2\times C_6$	$G/Z \simeq$ $C_3:S_3^2$
Commutator:	$G' \simeq$ $C_3^2\times C_6$	$G/G' \simeq$ $C_2^2\times C_6$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_3^4:C_2^3$
Fitting:	$\operatorname{Fit} \simeq$ $C_3^2\times C_6^2$	$G/\operatorname{Fit} \simeq$ $C_2^2$
Radical:	$R \simeq$ $C_6^2.S_3^2$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3^2\times C_6^2$	$G/\operatorname{soc} \simeq$ $C_2^2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^4$

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: $C_3^3:D_{12}$ x 4, $C_3^4:C_2^3$, $C_3:C_6^3$, $C_6\times C_3^2:C_{12}$

Order 432: $C_6^2.D_6$ x 4, $C_6^3:C_2$, $C_6^2.D_6$

Order 324: $C_3^3:D_6$ x 4, $D_6\times C_3^3$ x 4, $C_3^3:C_{12}$ x 2, $C_3^2\times C_6^2$

Order 216: $C_3^2:D_{12}$ x 16, $C_6^2:C_6$ x 9, $S_3\times C_6^2$ x 9, $C_6^2:C_6$ x 4, $C_6.C_6^2$ x 4, $C_3^2:D_{12}$ x 4, $C_6^2.C_6$ x 3, $C_6^3$, $C_6^2:S_3$

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

Order 144: $C_6^2:C_2^2$ x 4, $C_6:D_{12}$ x 4, $C_6^2:C_2^2$, $C_6\times D_{12}$

Order 108: $C_3^2:D_6$ x 44, $C_3^2\times D_6$ x 44, $C_3\times C_6^2$ x 23, $C_3^2:C_{12}$ x 8, $C_3^2:C_{12}$ x 6, $C_3^2:D_6$ x 4

Order 81: $C_3^4$

Order 72: $C_6\times D_6$ x 18, $C_6\wr C_2$ x 16, $C_3:D_{12}$ x 16, $C_6:C_{12}$ x 12, $C_2\times C_6^2$ x 9, $C_6:D_6$ x 9, $C_6^2:C_2$ x 4, $C_3\times D_{12}$ x 4, $C_6\times C_{12}$, $C_6.D_6$

Order 54: $C_3^2\times C_6$ x 59, $C_3^2:C_6$ x 26, $S_3\times C_3^2$ x 26, $C_3^2:S_3$ x 2

Order 48: $C_6:D_4$ x 4, $C_6\times D_4$, $C_2\times D_{12}$

Order 36: $C_6^2$ x 94, $C_6\times S_3$ x 88, $C_6:S_3$ x 44, $C_3:C_{12}$ x 24, $C_3\times C_{12}$ x 2, $C_3^2:C_4$ x 2

Order 27: $C_3^3$ x 19

Order 24: $C_3:D_4$ x 16, $C_2^2\times C_6$ x 10, $C_2\times D_6$ x 10, $C_6:C_4$ x 4, $D_{12}$ x 4, $C_3\times D_4$ x 4, $C_2\times C_{12}$ x 3

Order 18: $C_3\times C_6$ x 176, $C_3\times S_3$ x 52, $C_3:S_3$ x 26

Order 16: $C_2\times D_4$

Order 12: $C_2\times C_6$ x 67, $D_6$ x 48, $C_3:C_4$ x 8, $C_{12}$ x 6

Order 9: $C_3^2$ x 50

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

Order 6: $C_6$ x 85, $S_3$ x 28

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

Order 3: $C_3$ x 19

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

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

Order 648: $C_3^4:C_2^3$, $C_3:C_6^3$, $C_6\times C_3^2:C_{12}$, $C_3^3:D_{12}$

Order 432: $C_6^3:C_2$, $C_6^2.D_6$, $C_6^2.D_6$

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

Order 216: $C_6^2:C_6$ x 3, $S_3\times C_6^2$ x 3, $C_6^2.C_6$ x 3, $C_6^3$, $C_6^2:S_3$, $C_6^2:C_6$, $C_6.C_6^2$, $C_3^2:D_{12}$, $C_3^2:D_{12}$

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

Order 144: $C_6^2:C_2^2$, $C_6^2:C_2^2$, $C_6\times D_{12}$, $C_6:D_{12}$

Order 108: $C_3\times C_6^2$ x 9, $C_3^2:D_6$ x 6, $C_3^2\times D_6$ x 6, $C_3^2:C_{12}$ x 3, $C_3^2:D_6$ x 2, $C_3^2:C_{12}$

Order 81: $C_3^4$

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

Order 54: $C_3^2\times C_6$ x 15, $C_3^2:C_6$ x 3, $S_3\times C_3^2$ x 3, $C_3^2:S_3$

Order 48: $C_6\times D_4$, $C_6:D_4$, $C_2\times D_{12}$

Order 36: $C_6^2$ x 18, $C_6\times S_3$ x 12, $C_6:S_3$ x 6, $C_3:C_{12}$ x 3, $C_3\times C_{12}$, $C_3^2:C_4$

Order 27: $C_3^3$ x 7

Order 24: $C_2^2\times C_6$ x 4, $C_2\times D_6$ x 4, $C_2\times C_{12}$ x 3, $C_3:D_4$, $C_6:C_4$, $D_{12}$, $C_3\times D_4$

Order 18: $C_3\times C_6$ x 27, $C_3\times S_3$ x 6, $C_3:S_3$ x 3

Order 16: $C_2\times D_4$

Order 12: $C_2\times C_6$ x 15, $D_6$ x 8, $C_{12}$ x 3, $C_3:C_4$

Order 9: $C_3^2$ x 12

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

Order 6: $C_6$ x 18, $S_3$ x 4

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

Order 3: $C_3$ x 7

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: $C_3^3:D_{12}$ ($C_2$) x 4, $C_3^4:C_2^3$ ($C_2$), $C_3:C_6^3$ ($C_2$), $C_6\times C_3^2:C_{12}$ ($C_2$)

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

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

Order 216: $S_3\times C_6^2$ ($S_3$) x 4, $C_3^2:D_{12}$ ($C_6$) x 4, $C_6^2:S_3$ ($C_6$), $S_3\times C_6^2$ ($C_6$), $C_6^2.C_6$ ($C_6$), $C_6^2.C_6$ ($S_3$)

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

Order 108: $C_3^2\times D_6$ ($D_6$) x 8, $C_3\times C_6^2$ ($D_6$) x 5, $C_3^2:D_6$ ($C_2\times C_6$) x 2, $C_3^2\times D_6$ ($C_2\times C_6$) x 2, $C_3^2:C_{12}$ ($C_2\times C_6$) x 2, $C_3^2:C_{12}$ ($D_6$) x 2, $C_3\times C_6^2$ ($C_2\times C_6$)

Order 81: $C_3^4$ ($C_2\times D_4$)

Order 72: $C_6\times D_6$ ($C_3\times S_3$) x 4, $C_6\times D_6$ ($C_3:S_3$), $C_6.D_6$ ($C_3\times S_3$)

Order 54: $C_3^2\times C_6$ ($C_3:D_4$) x 8, $C_3^2\times C_6$ ($C_2\times D_6$) x 5, $C_3^2\times C_6$ ($D_{12}$) x 2, $C_3^2\times C_6$ ($C_3\times D_4$) x 2, $C_3^2\times C_6$ ($C_2^2\times C_6$)

Order 36: $C_6\times S_3$ ($C_6\times S_3$) x 8, $C_6^2$ ($C_6\times S_3$) x 5, $C_6^2$ ($S_3^2$) x 4, $C_3^2:C_4$ ($C_6\times S_3$) x 2, $C_6\times S_3$ ($C_6:S_3$) x 2, $C_6^2$ ($C_6:S_3$)

Order 27: $C_3^3$ ($C_6:D_4$) x 4, $C_3^3$ ($C_6\times D_4$), $C_3^3$ ($C_2\times D_{12}$)

Order 24: $C_2\times D_6$ ($C_3^2:C_6$)

Order 18: $C_3\times C_6$ ($C_6\wr C_2$) x 8, $C_3\times C_6$ ($C_3:D_{12}$) x 8, $C_3\times C_6$ ($C_6\times D_6$) x 5, $C_3\times C_6$ ($S_3\times D_6$) x 4, $C_3\times C_6$ ($C_6^2:C_2$) x 2, $C_3\times C_6$ ($C_3\times D_{12}$) x 2, $C_3\times C_6$ ($C_6:D_6$)

Order 12: $C_2\times C_6$ ($C_3\times S_3^2$) x 4, $D_6$ ($C_3^2:D_6$) x 2, $C_2\times C_6$ ($C_3^2:D_6$), $C_2\times C_6$ ($C_3:S_3^2$)

Order 9: $C_3^2$ ($C_6^2:C_2^2$) x 4, $C_3^2$ ($C_6:D_{12}$) x 4, $C_3^2$ ($C_6^2:C_2^2$), $C_3^2$ ($C_6\times D_{12}$)

Order 6: $C_6$ ($C_3^2:D_{12}$) x 8, $C_6$ ($C_6\times S_3^2$) x 4, $C_6$ ($C_6^2:C_6$) x 2, $C_6$ ($C_3^2:D_{12}$) x 2, $C_6$ ($C_6^2:C_6$), $C_6$ ($C_6:S_3^2$)

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

Order 3: $C_3$ ($C_6^2.D_6$) x 4, $C_3$ ($C_6^3:C_2$), $C_3$ ($C_6^2.D_6$)

Order 2: $C_2$ ($C_3^3:D_{12}$) x 2, $C_2$ ($C_3^4:C_2^3$)

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: $C_3^4:C_2^3$ ($C_2$), $C_3:C_6^3$ ($C_2$), $C_6\times C_3^2:C_{12}$ ($C_2$), $C_3^3:D_{12}$ ($C_2$)

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

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

Order 216: $C_6^2:S_3$ ($C_6$), $S_3\times C_6^2$ ($C_6$), $S_3\times C_6^2$ ($S_3$), $C_6^2.C_6$ ($C_6$), $C_6^2.C_6$ ($S_3$), $C_3^2:D_{12}$ ($C_6$)

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

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

Order 81: $C_3^4$ ($C_2\times D_4$)

Order 72: $C_6\times D_6$ ($C_3:S_3$), $C_6\times D_6$ ($C_3\times S_3$), $C_6.D_6$ ($C_3\times S_3$)

Order 54: $C_3^2\times C_6$ ($C_2\times D_6$) x 2, $C_3^2\times C_6$ ($C_3:D_4$), $C_3^2\times C_6$ ($D_{12}$), $C_3^2\times C_6$ ($C_2^2\times C_6$), $C_3^2\times C_6$ ($C_3\times D_4$)

Order 36: $C_6^2$ ($C_6\times S_3$) x 2, $C_3^2:C_4$ ($C_6\times S_3$), $C_6^2$ ($C_6:S_3$), $C_6^2$ ($S_3^2$), $C_6\times S_3$ ($C_6:S_3$), $C_6\times S_3$ ($C_6\times S_3$)

Order 27: $C_3^3$ ($C_6\times D_4$), $C_3^3$ ($C_6:D_4$), $C_3^3$ ($C_2\times D_{12}$)

Order 24: $C_2\times D_6$ ($C_3^2:C_6$)

Order 18: $C_3\times C_6$ ($C_6\times D_6$) x 2, $C_3\times C_6$ ($C_6:D_6$), $C_3\times C_6$ ($S_3\times D_6$), $C_3\times C_6$ ($C_6^2:C_2$), $C_3\times C_6$ ($C_6\wr C_2$), $C_3\times C_6$ ($C_3\times D_{12}$), $C_3\times C_6$ ($C_3:D_{12}$)

Order 12: $C_2\times C_6$ ($C_3^2:D_6$), $C_2\times C_6$ ($C_3:S_3^2$), $C_2\times C_6$ ($C_3\times S_3^2$), $D_6$ ($C_3^2:D_6$)

Order 9: $C_3^2$ ($C_6^2:C_2^2$), $C_3^2$ ($C_6^2:C_2^2$), $C_3^2$ ($C_6\times D_{12}$), $C_3^2$ ($C_6:D_{12}$)

Order 6: $C_6$ ($C_6^2:C_6$), $C_6$ ($C_6:S_3^2$), $C_6$ ($C_6\times S_3^2$), $C_6$ ($C_6^2:C_6$), $C_6$ ($C_3^2:D_{12}$), $C_6$ ($C_3^2:D_{12}$)

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

Order 3: $C_3$ ($C_6^3:C_2$), $C_3$ ($C_6^2.D_6$), $C_3$ ($C_6^2.D_6$)

Order 2: $C_2$ ($C_3^4:C_2^3$), $C_2$ ($C_3^3:D_{12}$)

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

Series

Derived series

$C_6^2.S_3^2$

$\rhd$

$C_3^2\times C_6$

$\rhd$

$C_1$

Chief series

$C_6^2.S_3^2$

$\rhd$

$C_6\times C_3^2:C_{12}$

$\rhd$

$C_6^2.C_6$

$\rhd$

$C_3\times C_6^2$

$\rhd$

$C_3^2\times C_6$

$\rhd$

$C_3\times C_6$

$\rhd$

$C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_6^2.S_3^2$

$\rhd$

$C_3^2\times C_6$

$\rhd$

$C_3^3$

Upper central series

$C_1$

$\lhd$

$C_2\times C_6$

Character theory

Complex character table

See the $198 \times 198$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $120 \times 120$ rational character table.