Abstract group 1296.2052: $C_6.S_3^3$

• Label: 1296.2052
• Order: \( 2^{4} \cdot 3^{4} \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{7} \cdot 3^{4} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \)
• Perm deg.: $20$
• Trans deg.: $144$
• Rank: $3$

Group information

Description:	$C_6.S_3^3$
Order:	\(1296\)\(\medspace = 2^{4} \cdot 3^{4} \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$(C_3\times \He_3).C_2^6.C_2$, of order \(10368\)\(\medspace = 2^{7} \cdot 3^{4} \)
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	12
Elements	1	43	80	180	524	468	1296
Conjugacy classes	1	4	9	5	23	9	51
Divisions	1	4	9	4	22	6	46
Autjugacy classes	1	3	7	3	13	5	32

Dimension	1	2	4	6	8	12	24
Irr. complex chars.	8	14	15	8	2	4	0	51
Irr. rational chars.	8	12	16	4	2	2	2	46

Minimal presentations

Permutation degree:	$20$
Transitive degree:	$144$
Rank:	$3$
Inequivalent generating triples:	$48384$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	12	24	48
Arbitrary	8	12	12

Constructions

Presentation:	${\langle a, b, c, d, e \mid a^{2}=b^{6}=c^{3}=d^{6}=e^{6}=[a,c]=[a,e]=[b,e]= \!\cdots\! \rangle}$
Permutation group:	Degree $20$ $\langle(2,4)(3,6)(5,7)(8,9)(10,11)(12,16)(13,14)(15,17), (3,6)(5,9)(7,8)(11,14) \!\cdots\! \rangle$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_3^3:D_4)$ $\,\rtimes\,$ $S_3$ (2)	$C_3^3$ $\,\rtimes\,$ $(D_4:S_3)$ (2)	$(\He_3:D_4)$ $\,\rtimes\,$ $S_3$	$\He_3$ $\,\rtimes\,$ $(D_4:S_3)$	all 12
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_6$ . $S_3^3$	$(C_6\times S_3)$ . $S_3^2$	$(C_6:S_3)$ . $S_3^2$ (2)	$C_6$ . $(C_6.S_3^2)$	all 22

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

Homology

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

Subgroups

There are 3796 subgroups in 390 conjugacy classes, 51 normal (23 characteristic).

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

Special subgroups

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

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

There are too many subgroups to show. See a full page version of the diagram.

Classes of subgroups up to automorphism

There are too many subgroups to show. See a full page version of the diagram.

Normal subgroups

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.

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

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

Classes of subgroups up to conjugation

Order 1296: $C_6.S_3^3$

Order 648: $C_3^3:(C_4\times S_3)$ x 2, $(C_3^2\times D_6):C_6$ x 2, $(C_3\times C_6).S_3^2$, $S_3\times \He_3:C_4$, $C_3\times \He_3:D_4$

Order 432: $C_2.S_3^3$ x 2, $C_6^2.D_6$

Order 324: $C_3^3:C_{12}$ x 2, $C_3^2:C_6^2$ x 2, $\He_3:C_{12}$, $D_6\times \He_3$, $(C_3\times \He_3):C_4$

Order 216: $C_6.S_3^2$ x 4, $C_6.S_3^2$ x 3, $C_6^2:C_6$ x 2, $C_6.S_3^2$ x 2, $C_3^3:Q_8$ x 2, $C_3^3:D_4$ x 2, $C_6.S_3^2$ x 2, $C_3^2:D_{12}$ x 2, $C_3^3:D_4$ x 2, $C_6^2.S_3$, $\He_3:D_4$, $C_6.S_3^2$

Order 162: $C_3^3:C_6$ x 2, $C_6\times \He_3$, $S_3\times \He_3$

Order 144: $D_6.D_6$ x 2, $D_6.D_6$ x 2, $D_{12}:S_3$

Order 108: $C_3^2\times D_6$ x 5, $C_3^2:C_{12}$ x 5, $\He_3:C_4$ x 5, $C_3^2:C_{12}$ x 4, $C_3^2:D_6$ x 4, $C_3^2:C_{12}$ x 3, $C_3^2:D_6$ x 2, $C_3^3:C_4$ x 2, $C_2^2\times \He_3$

Order 81: $C_3\times \He_3$

Order 72: $C_6.D_6$ x 11, $C_6\wr C_2$ x 6, $C_3^2:Q_8$ x 6, $D_6:S_3$ x 4, $C_6:C_{12}$ x 3, $C_6^2:C_2$ x 2, $C_6.D_6$ x 2, $C_3:D_{12}$ x 2, $C_6.D_6$ x 2, $C_3^2:Q_8$, $C_3\times D_{12}$, $S_3\times C_{12}$

Order 54: $C_3^2:C_6$ x 6, $S_3\times C_3^2$ x 6, $C_2\times \He_3$ x 5, $C_3^2\times C_6$ x 3, $C_3^2:C_6$ x 2

Order 48: $D_4:S_3$ x 3, $D_{12}:C_2$

Order 36: $C_3:C_{12}$ x 22, $C_3^2:C_4$ x 13, $C_6\times S_3$ x 12, $C_6^2$ x 5, $C_6:S_3$ x 2, $C_3\times C_{12}$

Order 27: $\He_3$ x 4, $C_3^3$ x 3

Order 24: $C_6:C_4$ x 9, $C_3:D_4$ x 8, $C_4\times S_3$ x 5, $C_3:Q_8$ x 5, $C_3\times D_4$ x 3, $C_2\times C_{12}$, $D_{12}$

Order 18: $C_3\times C_6$ x 19, $C_3\times S_3$ x 15, $C_3:S_3$ x 2

Order 16: $D_4:C_2$

Order 12: $C_3:C_4$ x 20, $C_2\times C_6$ x 10, $C_{12}$ x 6, $D_6$ x 5

Order 9: $C_3^2$ x 13

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

Order 6: $C_6$ x 22, $S_3$ x 5

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

Order 3: $C_3$ x 9

Order 2: $C_2$ x 4

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1296: $C_6.S_3^3$

Order 648: $(C_3\times C_6).S_3^2$, $S_3\times \He_3:C_4$, $C_3^3:(C_4\times S_3)$, $(C_3^2\times D_6):C_6$, $C_3\times \He_3:D_4$

Order 432: $C_2.S_3^3$, $C_6^2.D_6$

Order 324: $\He_3:C_{12}$, $C_3^3:C_{12}$, $D_6\times \He_3$, $C_3^2:C_6^2$, $(C_3\times \He_3):C_4$

Order 216: $C_6.S_3^2$ x 2, $C_6.S_3^2$ x 2, $C_6^2.S_3$, $C_6^2:C_6$, $\He_3:D_4$, $C_6.S_3^2$, $C_6.S_3^2$, $C_3^3:Q_8$, $C_3^3:D_4$, $C_6.S_3^2$, $C_3^2:D_{12}$, $C_3^3:D_4$

Order 162: $C_6\times \He_3$, $S_3\times \He_3$, $C_3^3:C_6$

Order 144: $D_6.D_6$, $D_6.D_6$, $D_{12}:S_3$

Order 108: $\He_3:C_4$ x 4, $C_3^2\times D_6$ x 3, $C_3^2:C_{12}$ x 3, $C_3^2:C_{12}$ x 2, $C_3^2:C_{12}$ x 2, $C_3^2:D_6$ x 2, $C_3^2:D_6$, $C_3^3:C_4$, $C_2^2\times \He_3$

Order 81: $C_3\times \He_3$

Order 72: $C_6.D_6$ x 6, $C_6\wr C_2$ x 3, $C_3^2:Q_8$ x 3, $C_6:C_{12}$ x 2, $D_6:S_3$ x 2, $C_6^2:C_2$, $C_6.D_6$, $C_3^2:Q_8$, $C_3\times D_{12}$, $S_3\times C_{12}$, $C_3:D_{12}$, $C_6.D_6$

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

Order 48: $D_4:S_3$ x 2, $D_{12}:C_2$

Order 36: $C_3:C_{12}$ x 14, $C_3^2:C_4$ x 7, $C_6\times S_3$ x 7, $C_6^2$ x 3, $C_3\times C_{12}$, $C_6:S_3$

Order 27: $\He_3$ x 3, $C_3^3$ x 2

Order 24: $C_6:C_4$ x 5, $C_3:D_4$ x 4, $C_3:Q_8$ x 4, $C_4\times S_3$ x 3, $C_3\times D_4$ x 2, $C_2\times C_{12}$, $D_{12}$

Order 18: $C_3\times C_6$ x 12, $C_3\times S_3$ x 7, $C_3:S_3$

Order 16: $D_4:C_2$

Order 12: $C_3:C_4$ x 12, $C_2\times C_6$ x 6, $C_{12}$ x 5, $D_6$ x 3

Order 9: $C_3^2$ x 9

Order 8: $D_4$ x 2, $C_2\times C_4$ x 2, $Q_8$

Order 6: $C_6$ x 13, $S_3$ x 3

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

Order 3: $C_3$ x 7

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1296: $C_6.S_3^3$ ($C_1$)

Order 648: $C_3^3:(C_4\times S_3)$ ($C_2$) x 2, $(C_3^2\times D_6):C_6$ ($C_2$) x 2, $(C_3\times C_6).S_3^2$ ($C_2$), $S_3\times \He_3:C_4$ ($C_2$), $C_3\times \He_3:D_4$ ($C_2$)

Order 324: $C_3^3:C_{12}$ ($C_2^2$) x 2, $C_3^2:C_6^2$ ($C_2^2$) x 2, $\He_3:C_{12}$ ($C_2^2$), $D_6\times \He_3$ ($C_2^2$), $(C_3\times \He_3):C_4$ ($C_2^2$)

Order 216: $C_3^3:D_4$ ($S_3$) x 2, $\He_3:D_4$ ($S_3$)

Order 162: $C_6\times \He_3$ ($C_2^3$)

Order 108: $C_3^2:D_6$ ($D_6$) x 2, $C_3^2\times D_6$ ($D_6$) x 2, $C_3^3:C_4$ ($D_6$) x 2, $C_3^2:D_6$ ($D_6$) x 2, $\He_3:C_4$ ($D_6$)

Order 81: $C_3\times \He_3$ ($D_4:C_2$)

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

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

Order 27: $C_3^3$ ($D_4:S_3$) x 2, $\He_3$ ($D_4:S_3$)

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

Order 12: $D_6$ ($C_3^2:D_6$)

Order 9: $C_3^2$ ($D_6.D_6$) x 3

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

Order 3: $C_3$ ($C_2.S_3^3$), $C_3$ ($C_6^2.D_6$)

Order 2: $C_2$ ($C_3.S_3^3$)

Order 1: $C_1$ ($C_6.S_3^3$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1296: $C_6.S_3^3$ ($C_1$)

Order 648: $(C_3\times C_6).S_3^2$ ($C_2$), $S_3\times \He_3:C_4$ ($C_2$), $C_3^3:(C_4\times S_3)$ ($C_2$), $(C_3^2\times D_6):C_6$ ($C_2$), $C_3\times \He_3:D_4$ ($C_2$)

Order 324: $\He_3:C_{12}$ ($C_2^2$), $C_3^3:C_{12}$ ($C_2^2$), $D_6\times \He_3$ ($C_2^2$), $C_3^2:C_6^2$ ($C_2^2$), $(C_3\times \He_3):C_4$ ($C_2^2$)

Order 216: $\He_3:D_4$ ($S_3$), $C_3^3:D_4$ ($S_3$)

Order 162: $C_6\times \He_3$ ($C_2^3$)

Order 108: $C_3^2:D_6$ ($D_6$), $C_3^2\times D_6$ ($D_6$), $C_3^3:C_4$ ($D_6$), $C_3^2:D_6$ ($D_6$), $\He_3:C_4$ ($D_6$)

Order 81: $C_3\times \He_3$ ($D_4:C_2$)

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

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

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

Order 18: $C_3\times C_6$ ($S_3\times D_6$) x 2

Order 12: $D_6$ ($C_3^2:D_6$)

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

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

Order 3: $C_3$ ($C_2.S_3^3$), $C_3$ ($C_6^2.D_6$)

Order 2: $C_2$ ($C_3.S_3^3$)

Order 1: $C_1$ ($C_6.S_3^3$)

Series

Derived series

$C_6.S_3^3$

$\rhd$

$C_6\times \He_3$

$\rhd$

$C_3$

$\rhd$

$C_1$

Chief series

$C_6.S_3^3$

$\rhd$

$(C_3^2\times D_6):C_6$

$\rhd$

$D_6\times \He_3$

$\rhd$

$C_3^2\times D_6$

$\rhd$

$C_6\times S_3$

$\rhd$

$C_3\times C_6$

$\rhd$

$C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_6.S_3^3$

$\rhd$

$C_6\times \He_3$

$\rhd$

$C_3\times \He_3$

Upper central series

$C_1$

$\lhd$

$C_2$

Character theory

Complex character table

See the $51 \times 51$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $46 \times 46$ rational character table.