Abstract group 1119744.ej: $C_6^3.C_6^3:S_4$

• Label: 1119744.ej
• Order: \( 2^{9} \cdot 3^{7} \)
• Exponent: \( 2^{2} \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{13} \cdot 3^{7} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \)
• Perm deg.: $30$
• Trans deg.: $36$
• Rank: $2$

Group information

Description:	$C_6^3.C_6^3:S_4$
Order:	\(1119744\)\(\medspace = 2^{9} \cdot 3^{7} \)
Exponent:	\(36\)\(\medspace = 2^{2} \cdot 3^{2} \)
Automorphism group:	$C_3^6.C_2^6.C_6.C_2^6$, of order \(17915904\)\(\medspace = 2^{13} \cdot 3^{7} \)
Composition factors:	$C_2$ x 9, $C_3$ x 7
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	9	12	18	36
Elements	1	12991	11096	81216	159944	82944	564192	82944	124416	1119744
Conjugacy classes	1	23	26	20	608	5	140	5	4	832
Divisions	1	23	24	19	465	5	85	5	2	629
Autjugacy classes	1	19	24	13	346	3	54	3	1	464

Minimal presentations

Permutation degree:	$30$
Transitive degree:	$36$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of faithful linear representations

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

Constructions

Presentation:	${\langle a, b, c, d, e, f, g, h, i \mid b^{12}=c^{2}=d^{6}=e^{6}=f^{6}=g^{6}= \!\cdots\! \rangle}$
Permutation group:	Degree $30$ $\langle(1,3,6)(2,4,7)(5,8,9)(10,11,13,16,12,15)(14,17)(18,20,23,24,25,26)(19,21,22) \!\cdots\! \rangle$
Transitive group:	36T36611				more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$C_6^3$ . $(S_3^3:S_4)$	$C_6^3$ . $(C_6^3:S_4)$	$C_3^6$ . $(C_2^6:D_{12})$	$(C_3^6.C_2^5:S_4)$ . $C_2$	all 37

Elements of the group are displayed as permutations of degree 30.

Homology

Abelianization:	$C_{2}^{2} $
Schur multiplier:	$C_{2}^{5}$
Commutator length:	$2$

Subgroups

There are 47 normal subgroups (43 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_3^6.C_2^5:S_4$
Commutator:	$G' \simeq$ $C_3^6.C_2^6.C_6$	$G/G' \simeq$ $C_2^2$
Frattini:	$\Phi \simeq$ $C_2^3$	$G/\Phi \simeq$ $C_3^3.C_6^3:S_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_3\times C_6^5$	$G/\operatorname{Fit} \simeq$ $C_2\times S_4$
Radical:	$R \simeq$ $C_6^3.C_6^3:S_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3^3\times C_6^3$	$G/\operatorname{soc} \simeq$ $C_2^3:S_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^7.C_2^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2\wr C_3$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: inclusions not computed

Classes of subgroups up to automorphism

No diagram available: inclusions not computed

Normal subgroups

No diagram available: inclusions not computed

Normal subgroups up to automorphism

No diagram available: inclusions not computed

Classes of subgroups up to conjugation

All subgroups of index up to 9 (order at least 124416) are shown, as well as all normal subgroups of any index.

Order 1119744: $C_6^3.C_6^3:S_4$

Order 559872: $C_3^6.C_2^6.D_6$, $C_3^6.C_2^5:S_4$, $C_3^6.C_2^6:C_{12}$

Order 373248: $C_3^6.C_2^6.C_2^3$

Order 279936: $C_3^6.C_2^6.C_6$, $C_3^6.C_2^6.S_3$, $C_6^3.S_3\wr S_3$, $C_6^3.S_3\wr S_3$, $C_3^6.C_2^4:S_4$, $C_6^3.C_6^3:S_3$

Order 186624: $(C_3\times C_6^4).C_6.C_2^3$, $C_3^6.C_2^6.C_2^2$, $C_3^6.C_2^6.C_2^2$, $C_3^6.C_2^6.C_2^2$, $C_3^6.C_2^6.C_2^2$, $C_3^5.C_2^4.C_6.C_2^3$, $C_3^6.C_2^6.C_2^2$, $C_3^6.C_2^6.C_2^2$, $C_3^6.C_2^5.C_2^3$, $C_3^6.C_2^6.C_2^2$, $C_3^6.C_2^6.C_2^2$, $C_3^6.C_2^6.C_2^2$, $C_3^6.C_2^6.C_2^2$, $C_3^6.C_2^6.C_2^2$, $C_3^6.C_2^6.C_2^2$

Order 139968: $C_3^6.C_2^4.D_6$ x 4, $C_3^3.C_6^3:S_4$ x 4, $C_6^3.S_3\wr C_3$ x 2, $C_3^6.C_2^6.C_3$, $C_6^4.C_3^3.C_2^2$, $C_6^3.S_3\wr C_3$, $C_2^5.C_3\wr S_3$, $C_6^3.C_6\wr C_3$

Order 124416: $C_3^5.C_2^6.C_2^3$, $C_3^5.C_2^6.C_2^3$

Order 93312: $C_2\times C_3^6.C_2^6$

Order 46656: $(C_3\times C_6^5).C_2$, $C_2\times C_3^6.C_2^5$, $C_3^6.C_2^5.C_2$, $C_3^6.C_2^5.C_2$

Order 41472: $C_{12}:D_6^2.S_4$, $C_2^5.S_3\wr S_3$

Order 23328: $C_3^6.C_2^5$ x 2, $C_3\times C_6^5$

Order 11664: $C_3^6.C_2^4$ x 2, $C_3^6.C_4.C_2^2$, $C_3^2\times C_6^4$

Order 5832: $C_3^6.C_2^3$ x 2, $C_3^3\times C_6^3$

Order 2916: $C_3^5:D_6$ x 2, $C_3^4\times C_6^2$

Order 2187: $C_3^2\wr C_3$

Order 1728: $C_6^3.D_4$

Order 1458: $C_3^5\times C_6$

Order 864: $C_2^2\times C_6^3$ x 2

Order 729: $C_3^6$

Order 512: $C_2^7.C_2^2$

Order 432: $C_2\times C_6^3$ x 2, $C_6^3.C_2$

Order 216: $C_6^3$ x 2

Order 108: $C_3\times C_6^2$ x 2

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

Order 32: $C_2^5$

Order 27: $C_3^3$ x 2

Order 16: $C_2^4$

Order 8: $C_2^3$

Order 4: $C_2^2$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

All subgroups of index up to 9 (order at least 124416) are shown, as well as all normal subgroups of any index.

Order 1119744: $C_6^3.C_6^3:S_4$

Order 559872: $C_3^6.C_2^6.D_6$, $C_3^6.C_2^5:S_4$, $C_3^6.C_2^6:C_{12}$

Order 373248: $C_3^6.C_2^6.C_2^3$

Order 279936: $C_3^6.C_2^6.C_6$, $C_3^6.C_2^6.S_3$, $C_6^3.S_3\wr S_3$, $C_6^3.S_3\wr S_3$, $C_3^6.C_2^4:S_4$, $C_6^3.C_6^3:S_3$

Order 186624: $(C_3\times C_6^4).C_6.C_2^3$, $C_3^6.C_2^6.C_2^2$, $C_3^6.C_2^6.C_2^2$, $C_3^6.C_2^6.C_2^2$, $C_3^6.C_2^6.C_2^2$, $C_3^5.C_2^4.C_6.C_2^3$, $C_3^6.C_2^6.C_2^2$, $C_3^6.C_2^6.C_2^2$, $C_3^6.C_2^5.C_2^3$, $C_3^6.C_2^6.C_2^2$, $C_3^6.C_2^6.C_2^2$, $C_3^6.C_2^6.C_2^2$, $C_3^6.C_2^6.C_2^2$, $C_3^6.C_2^6.C_2^2$, $C_3^6.C_2^6.C_2^2$

Order 139968: $C_3^6.C_2^4.D_6$ x 3, $C_3^3.C_6^3:S_4$ x 3, $C_6^3.S_3\wr C_3$ x 2, $C_3^6.C_2^6.C_3$, $C_6^4.C_3^3.C_2^2$, $C_6^3.S_3\wr C_3$, $C_2^5.C_3\wr S_3$, $C_6^3.C_6\wr C_3$

Order 124416: $C_3^5.C_2^6.C_2^3$, $C_3^5.C_2^6.C_2^3$

Order 93312: $C_2\times C_3^6.C_2^6$

Order 46656: $(C_3\times C_6^5).C_2$, $C_2\times C_3^6.C_2^5$, $C_3^6.C_2^5.C_2$, $C_3^6.C_2^5.C_2$

Order 41472: $C_{12}:D_6^2.S_4$, $C_2^5.S_3\wr S_3$

Order 23328: $C_3^6.C_2^5$ x 2, $C_3\times C_6^5$

Order 11664: $C_3^6.C_2^4$ x 2, $C_3^6.C_4.C_2^2$, $C_3^2\times C_6^4$

Order 5832: $C_3^3\times C_6^3$, $C_3^6.C_2^3$

Order 2916: $C_3^4\times C_6^2$, $C_3^5:D_6$

Order 2187: $C_3^2\wr C_3$

Order 1728: $C_6^3.D_4$

Order 1458: $C_3^5\times C_6$

Order 864: $C_2^2\times C_6^3$ x 2

Order 729: $C_3^6$

Order 512: $C_2^7.C_2^2$

Order 432: $C_2\times C_6^3$ x 2, $C_6^3.C_2$

Order 216: $C_6^3$ x 2

Order 108: $C_3\times C_6^2$ x 2

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

Order 32: $C_2^5$

Order 27: $C_3^3$ x 2

Order 16: $C_2^4$

Order 8: $C_2^3$

Order 4: $C_2^2$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1119744: $C_6^3.C_6^3:S_4$ ($C_1$)

Order 559872: $C_3^6.C_2^6.D_6$ ($C_2$), $C_3^6.C_2^5:S_4$ ($C_2$), $C_3^6.C_2^6:C_{12}$ ($C_2$)

Order 279936: $C_3^6.C_2^6.C_6$ ($C_2^2$)

Order 186624: $C_3^6.C_2^6.C_2^2$ ($S_3$)

Order 139968: $C_3^6.C_2^6.C_3$ ($D_4$)

Order 93312: $C_2\times C_3^6.C_2^6$ ($D_6$)

Order 46656: $(C_3\times C_6^5).C_2$ ($S_4$), $C_2\times C_3^6.C_2^5$ ($D_{12}$), $C_3^6.C_2^5.C_2$ ($S_4$), $C_3^6.C_2^5.C_2$ ($S_4$)

Order 23328: $C_3^6.C_2^5$ ($C_2\times S_4$) x 2, $C_3\times C_6^5$ ($C_2\times S_4$)

Order 11664: $C_3^6.C_2^4$ ($C_4:S_4$) x 2, $C_3^6.C_4.C_2^2$ ($C_2^2:S_4$), $C_3^2\times C_6^4$ ($C_4:S_4$)

Order 5832: $C_3^6.C_2^3$ ($C_2^3:S_4$) x 2, $C_3^3\times C_6^3$ ($C_2^3:S_4$)

Order 2916: $C_3^5:D_6$ ($C_2^4:D_{12}$) x 2, $C_3^4\times C_6^2$ ($C_2^4:D_{12}$)

Order 1728: $C_6^3.D_4$ ($C_3^3:S_4$)

Order 1458: $C_3^5\times C_6$ ($C_2^5:S_4$)

Order 864: $C_2^2\times C_6^3$ ($C_2\times C_3^3:S_4$), $C_2^2\times C_6^3$ ($S_3\wr S_3$)

Order 729: $C_3^6$ ($C_2^6:D_{12}$)

Order 432: $C_2\times C_6^3$ ($(C_3^2\times C_{12}):S_4$), $C_2\times C_6^3$ ($C_2.S_3\wr S_3$), $C_6^3.C_2$ ($(C_3\times C_6^2):S_4$)

Order 216: $C_6^3$ ($C_6^3:S_4$), $C_6^3$ ($S_3^3:S_4$)

Order 108: $C_3\times C_6^2$ ($(C_6^2\times C_{12}):S_4$), $C_3\times C_6^2$ ($C_6^3.(C_2\times S_4)$)

Order 54: $C_3^2\times C_6$ ($C_6:D_6^2:S_4$), $C_3^2\times C_6$ ($(S_3\times D_6^2).S_4$)

Order 32: $C_2^5$ ($C_3^6:(C_2\times S_4)$)

Order 27: $C_3^3$ ($C_{12}:D_6^2.S_4$), $C_3^3$ ($C_2^5.S_3\wr S_3$)

Order 16: $C_2^4$ ($C_3^6.C_4:S_4$)

Order 8: $C_2^3$ ($C_3^3.C_6^3:S_4$)

Order 4: $C_2^2$ ($C_6^3.S_3\wr S_3$)

Order 2: $C_2$ ($C_3^6.C_2^5:S_4$)

Order 1: $C_1$ ($C_6^3.C_6^3:S_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1119744: $C_6^3.C_6^3:S_4$ ($C_1$)

Order 559872: $C_3^6.C_2^6.D_6$ ($C_2$), $C_3^6.C_2^5:S_4$ ($C_2$), $C_3^6.C_2^6:C_{12}$ ($C_2$)

Order 279936: $C_3^6.C_2^6.C_6$ ($C_2^2$)

Order 186624: $C_3^6.C_2^6.C_2^2$ ($S_3$)

Order 139968: $C_3^6.C_2^6.C_3$ ($D_4$)

Order 93312: $C_2\times C_3^6.C_2^6$ ($D_6$)

Order 46656: $(C_3\times C_6^5).C_2$ ($S_4$), $C_2\times C_3^6.C_2^5$ ($D_{12}$), $C_3^6.C_2^5.C_2$ ($S_4$), $C_3^6.C_2^5.C_2$ ($S_4$)

Order 23328: $C_3^6.C_2^5$ ($C_2\times S_4$) x 2, $C_3\times C_6^5$ ($C_2\times S_4$)

Order 11664: $C_3^6.C_2^4$ ($C_4:S_4$) x 2, $C_3^6.C_4.C_2^2$ ($C_2^2:S_4$), $C_3^2\times C_6^4$ ($C_4:S_4$)

Order 5832: $C_3^3\times C_6^3$ ($C_2^3:S_4$), $C_3^6.C_2^3$ ($C_2^3:S_4$)

Order 2916: $C_3^4\times C_6^2$ ($C_2^4:D_{12}$), $C_3^5:D_6$ ($C_2^4:D_{12}$)

Order 1728: $C_6^3.D_4$ ($C_3^3:S_4$)

Order 1458: $C_3^5\times C_6$ ($C_2^5:S_4$)

Order 864: $C_2^2\times C_6^3$ ($C_2\times C_3^3:S_4$), $C_2^2\times C_6^3$ ($S_3\wr S_3$)

Order 729: $C_3^6$ ($C_2^6:D_{12}$)

Order 432: $C_2\times C_6^3$ ($(C_3^2\times C_{12}):S_4$), $C_2\times C_6^3$ ($C_2.S_3\wr S_3$), $C_6^3.C_2$ ($(C_3\times C_6^2):S_4$)

Order 216: $C_6^3$ ($C_6^3:S_4$), $C_6^3$ ($S_3^3:S_4$)

Order 108: $C_3\times C_6^2$ ($(C_6^2\times C_{12}):S_4$), $C_3\times C_6^2$ ($C_6^3.(C_2\times S_4)$)

Order 54: $C_3^2\times C_6$ ($C_6:D_6^2:S_4$), $C_3^2\times C_6$ ($(S_3\times D_6^2).S_4$)

Order 32: $C_2^5$ ($C_3^6:(C_2\times S_4)$)

Order 27: $C_3^3$ ($C_{12}:D_6^2.S_4$), $C_3^3$ ($C_2^5.S_3\wr S_3$)

Order 16: $C_2^4$ ($C_3^6.C_4:S_4$)

Order 8: $C_2^3$ ($C_3^3.C_6^3:S_4$)

Order 4: $C_2^2$ ($C_6^3.S_3\wr S_3$)

Order 2: $C_2$ ($C_3^6.C_2^5:S_4$)

Order 1: $C_1$ ($C_6^3.C_6^3:S_4$)

Series

Derived series

$C_6^3.C_6^3:S_4$

$\rhd$

$C_3^6.C_2^6.C_6$

$\rhd$

$C_2\times C_3^6.C_2^5$

$\rhd$

$C_3^6$

$\rhd$

$C_1$

Chief series

$C_6^3.C_6^3:S_4$

$\rhd$

$C_3^6.C_2^5:S_4$

$\rhd$

$C_3^6.C_2^6.C_6$

$\rhd$

$C_3^6.C_2^6.C_3$

$\rhd$

$C_2\times C_3^6.C_2^5$

$\rhd$

$C_3^6.C_2^4$

$\rhd$

$C_3^4\times C_6^2$

$\rhd$

$C_3^6$

$\rhd$

$C_3^3$

$\rhd$

$C_1$

Lower central series

$C_6^3.C_6^3:S_4$

$\rhd$

$C_3^6.C_2^6.C_6$

$\rhd$

$C_3^6.C_2^6.C_3$

Upper central series

$C_1$

$\lhd$

$C_2$

Supergroups

This group is a maximal subgroup of 7 larger groups in the database.

This group is a maximal quotient of 2 larger groups in the database.

Character theory

Complex character table

The $832 \times 832$ character table is not available for this group.

Rational character table

The $629 \times 629$ rational character table is not available for this group.