Abstract group 78732.je: $C_3^7.(C_6\times S_3)$

• Label: 78732.je
• Order: \( 2^{2} \cdot 3^{9} \)
• Exponent: \( 2 \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 3 \)
• $\card{Z(G)}$: \( 3 \)
• $\card{\Aut(G)}$: \( 2^{4} \cdot 3^{12} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \cdot 3^{4} \)
• Perm deg.: $36$
• Trans deg.: $36$
• Rank: $2$

Group information

Description:	$C_3^7.(C_6\times S_3)$
Order:	\(78732\)\(\medspace = 2^{2} \cdot 3^{9} \)
Exponent:	\(18\)\(\medspace = 2 \cdot 3^{2} \)
Automorphism group:	$C_{1655}:C_{165}$, of order \(8503056\)\(\medspace = 2^{4} \cdot 3^{12} \)
Composition factors:	$C_2$ x 2, $C_3$ x 9
Derived length:	$3$

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

Group statistics

Order	1	2	3	6	9	18
Elements	1	567	6560	19116	13122	39366	78732
Conjugacy classes	1	3	121	25	108	36	294
Divisions	1	3	76	17	46	8	151
Autjugacy classes	1	2	32	10	8	4	57

Minimal presentations

Permutation degree:	$36$
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:	${\langle a, b, c, d, e, f, g, h, i \mid b^{6}=c^{3}=d^{3}=e^{3}=f^{3}=g^{3}= \!\cdots\! \rangle}$
Permutation group:	Degree $36$ $\langle(1,5,27,29,14,17,3,4,26,28,15,18,2,6,25,30,13,16)(7,10,32,36,19,24,9,11,33,35,21,22,8,12,31,34,20,23) \!\cdots\! \rangle$
Transitive group:	36T18899				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:	$(C_3^7.C_6)$ . $S_3$ (2)	$C_3^7$ . $(C_6\times S_3)$	$(C_3^7.D_6)$ . $C_3$	$(C_3^7.C_3)$ . $D_6$	all 37

Elements of the group are displayed as permutations of degree 36.

Homology

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

Subgroups

There are 65 normal subgroups (31 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_3$	$G/Z \simeq$ $C_3^5.C_3^3.C_2^2$
Commutator:	$G' \simeq$ $C_3\times C_3^6.C_3$	$G/G' \simeq$ $C_2\times C_6$
Frattini:	$\Phi \simeq$ $C_3^5$	$G/\Phi \simeq$ $C_3\wr C_2^2$
Fitting:	$\operatorname{Fit} \simeq$ $C_3^7.C_3^2$	$G/\operatorname{Fit} \simeq$ $C_2^2$
Radical:	$R \simeq$ $C_3^7.(C_6\times S_3)$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3^4$	$G/\operatorname{soc} \simeq$ $C_3^3:S_3^2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^7.C_3^2$

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 81 (order at least 972) are shown, as well as all normal subgroups of any index.

Order 78732: $C_3^7.(C_6\times S_3)$

Order 39366: $C_3^6.C_3^3.C_2$ x 2, $C_3^7.C_3.C_6$

Order 26244: $C_3^6.(C_6\times S_3)$ x 2, $C_3^7.D_6$, $C_3^5.(S_3\times C_{18})$

Order 19683: $C_3^7.C_3^2$

Order 13122: $C_3^6.C_3.C_6$ x 4, $C_3^5.C_3^3.C_2$ x 4, $C_3^7.C_6$ x 3, $C_3^6.C_3.C_6$ x 2, $C_3^6.C_3.C_6$ x 2, $C_3^6.C_3.C_6$, $C_3^7.C_6$, $C_3^7.C_6$

Order 8748: $C_3^6.C_6.C_2$ x 3, $C_3^6:D_6$ x 3, $C_3^6.D_6$ x 2, $C_3^7:C_2^2$, $C_3^5.(C_6\times S_3)$, $C_3^6:D_6$

Order 6561: $C_3^6.C_3^2$ x 8, $C_3\times C_3^5.C_3^2$ x 5, $C_3\times C_3^6.C_3$, $C_3^7.C_3$, $C_3^7.C_3$

Order 4374: $C_3^5.C_3.C_6$ x 12, $C_3.C_3^5.C_6$ x 10, $C_3^6.C_6$ x 9, $C_3^6.C_6$ x 6, $C_3^6.C_6$ x 5, $C_3^4.C_3^3.C_2$ x 4, $C_3^7.C_2$ x 3, $C_3^6.C_6$ x 3, $C_3^6.C_6$ x 3, $C_3^5.C_3.C_6$ x 3, $C_3^6:C_6$ x 3, $C_3^6.S_3$ x 2, $C_3^3.\He_3.C_6$ x 2, $C_3.C_3^5.C_6$ x 2, $C_3^6.C_6$, $C_3^6.C_6$, $C_3^5.C_3.C_6$, $C_3^6.C_6$

Order 2916: $(C_3^2\times \He_3).D_6$ x 6, $C_3^5:D_6$ x 6, $C_3^5.C_6.C_2$ x 3, $C_3^5.C_6.C_2$ x 3, $C_3^5:D_6$ x 3, $C_3^4.(C_6\times S_3)$ x 2, $C_3^5:D_6$ x 2, $C_3^5:D_6$ x 2, $C_3^5.D_6$, $C_3^5:D_6$, $C_3^5:D_6$

Order 2187: $C_3^2\times C_3^4.C_3$ x 66, $C_3^5.C_3^2$ x 42, $C_3^3.C_3^4$ x 25, $C_3^6.C_3$ x 12, $C_3^2\times C_3^4:C_3$ x 10, $C_3\times C_3^5.C_3$ x 10, $C_3^4\times C_9:C_3$ x 2, $C_3^3\times C_3^2:C_9$ x 2, $C_3^7$

Order 1458: $C_3^5:S_3$ x 42, $C_3^5.C_6$ x 36, $C_3^4:C_{18}$ x 29, $C_3^3\wr C_2$ x 28, $C_3^5:S_3$ x 22, $C_3^4:C_{18}$ x 19, $C_3^5:C_6$ x 14, $C_3^5:S_3$ x 14, $C_3^4.(C_3\times S_3)$ x 10, $C_3^4.(C_3\times S_3)$ x 10, $C_3^5:C_6$ x 9, $C_3^5:S_3$ x 8, $C_3^5.C_6$ x 7, $C_3^5:C_6$ x 7, $C_3^4:C_{18}$ x 5, $C_3^5.C_6$ x 5, $C_3^5:C_6$ x 5, $C_3^5:C_6$ x 5, $S_3\times C_3^3.C_3^2$ x 3, $C_3^5.C_6$ x 3, $C_3^4:C_{18}$ x 3, $C_3^4:C_{18}$ x 3, $C_3^5:C_6$, $C_3^4:C_{18}$

Order 972: $C_3^3:S_3^2$ x 19, $C_3^3:S_3^2$ x 18, $C_3^3:S_3^2$ x 12, $C_3^3:S_3^2$ x 6, $S_3\times C_3^2:C_{18}$ x 6, $C_3^3:S_3^2$ x 3, $C_3^3:S_3^2$ x 3, $C_3^3:S_3^2$ x 3, $C_3^3:S_3^2$ x 2, $C_3^3:S_3^2$ x 2, $C_9\times C_3:S_3^2$, $C_3^3.(C_6\times S_3)$

Order 729: $C_3^6$ x 3, $C_3^3\times \He_3$, $C_3^4:C_9$

Order 486: $C_3^4:C_6$ x 2

Order 243: $C_3^5$ x 5

Order 162: $C_3^3:S_3$

Order 81: $C_3^4$ x 8

Order 27: $C_3^3$ x 7

Order 9: $C_3^2$ x 6

Order 4: $C_2^2$

Order 3: $C_3$ x 4

Order 1: $C_1$

Classes of subgroups up to automorphism

All subgroups of index up to 81 (order at least 972) are shown, as well as all normal subgroups of any index.

Order 78732: $C_3^7.(C_6\times S_3)$

Order 39366: $C_3^6.C_3^3.C_2$, $C_3^7.C_3.C_6$

Order 26244: $C_3^6.(C_6\times S_3)$, $C_3^7.D_6$, $C_3^5.(S_3\times C_{18})$

Order 19683: $C_3^7.C_3^2$

Order 13122: $C_3^7.C_6$ x 2, $C_3^6.C_3.C_6$ x 2, $C_3^5.C_3^3.C_2$ x 2, $C_3^6.C_3.C_6$, $C_3^6.C_3.C_6$, $C_3^6.C_3.C_6$, $C_3^7.C_6$, $C_3^7.C_6$

Order 8748: $C_3^6.C_6.C_2$ x 2, $C_3^6.D_6$, $C_3^7:C_2^2$, $C_3^5.(C_6\times S_3)$, $C_3^6:D_6$, $C_3^6:D_6$

Order 6561: $C_3\times C_3^5.C_3^2$ x 5, $C_3^6.C_3^2$ x 4, $C_3\times C_3^6.C_3$, $C_3^7.C_3$, $C_3^7.C_3$

Order 4374: $C_3.C_3^5.C_6$ x 5, $C_3^6.C_6$ x 5, $C_3^6.C_6$ x 3, $C_3^6.C_6$ x 3, $C_3^7.C_2$ x 2, $C_3^4.C_3^3.C_2$ x 2, $C_3^5.C_3.C_6$ x 2, $C_3^6:C_6$ x 2, $C_3^6.C_6$, $C_3^6.S_3$, $C_3^6.C_6$, $C_3^6.C_6$, $C_3^6.C_6$, $C_3^3.\He_3.C_6$, $C_3.C_3^5.C_6$, $C_3^5.C_3.C_6$, $C_3^5.C_3.C_6$, $C_3^6.C_6$

Order 2916: $C_3^5:D_6$ x 4, $C_3^5.C_6.C_2$ x 2, $C_3^5.C_6.C_2$ x 2, $C_3^5.D_6$, $(C_3^2\times \He_3).D_6$, $C_3^5:D_6$, $C_3^4.(C_6\times S_3)$, $C_3^5:D_6$, $C_3^5:D_6$, $C_3^5:D_6$, $C_3^5:D_6$

Order 2187: $C_3^3.C_3^4$ x 15, $C_3^2\times C_3^4.C_3$ x 14, $C_3^2\times C_3^4:C_3$ x 7, $C_3^5.C_3^2$ x 7, $C_3\times C_3^5.C_3$ x 7, $C_3^6.C_3$ x 3, $C_3^4\times C_9:C_3$ x 2, $C_3^3\times C_3^2:C_9$ x 2, $C_3^7$

Order 1458: $C_3^3\wr C_2$ x 12, $C_3^4:C_{18}$ x 11, $C_3^5:C_6$ x 9, $C_3^4:C_{18}$ x 7, $C_3^5:S_3$ x 6, $C_3^4.(C_3\times S_3)$ x 5, $C_3^4.(C_3\times S_3)$ x 5, $C_3^5:S_3$ x 5, $C_3^5:S_3$ x 5, $C_3^5.C_6$ x 4, $C_3^5:S_3$ x 4, $C_3^5:C_6$ x 4, $C_3^4:C_{18}$ x 3, $C_3^5.C_6$ x 3, $C_3^5:C_6$ x 3, $S_3\times C_3^3.C_3^2$ x 2, $C_3^5.C_6$ x 2, $C_3^5:C_6$ x 2, $C_3^5:C_6$ x 2, $C_3^4:C_{18}$ x 2, $C_3^5.C_6$, $C_3^5:C_6$, $C_3^4:C_{18}$, $C_3^4:C_{18}$

Order 972: $C_3^3:S_3^2$ x 8, $C_3^3:S_3^2$ x 4, $S_3\times C_3^2:C_{18}$ x 3, $C_3^3:S_3^2$ x 2, $C_3^3:S_3^2$ x 2, $C_3^3:S_3^2$, $C_3^3:S_3^2$, $C_3^3:S_3^2$, $C_9\times C_3:S_3^2$, $C_3^3:S_3^2$, $C_3^3:S_3^2$, $C_3^3.(C_6\times S_3)$

Order 729: $C_3^6$ x 2, $C_3^3\times \He_3$, $C_3^4:C_9$

Order 486: $C_3^4:C_6$ x 2

Order 243: $C_3^5$ x 4

Order 162: $C_3^3:S_3$

Order 81: $C_3^4$ x 5

Order 27: $C_3^3$ x 5

Order 9: $C_3^2$ x 4

Order 4: $C_2^2$

Order 3: $C_3$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 78732: $C_3^7.(C_6\times S_3)$ ($C_1$)

Order 39366: $C_3^6.C_3^3.C_2$ ($C_2$) x 2, $C_3^7.C_3.C_6$ ($C_2$)

Order 26244: $C_3^7.D_6$ ($C_3$)

Order 19683: $C_3^7.C_3^2$ ($C_2^2$)

Order 13122: $C_3^6.C_3.C_6$ ($S_3$) x 2, $C_3^6.C_3.C_6$ ($C_6$) x 2, $C_3^7.C_6$ ($S_3$), $C_3^7.C_6$ ($C_6$)

Order 6561: $C_3^6.C_3^2$ ($D_6$) x 2, $C_3\times C_3^6.C_3$ ($C_2\times C_6$), $C_3^7.C_3$ ($D_6$)

Order 4374: $C_3^6.S_3$ ($C_3\times S_3$) x 2, $C_3^7.C_2$ ($C_3\times S_3$)

Order 2187: $C_3^2\times C_3^4:C_3$ ($C_6\times S_3$) x 2, $C_3\times C_3^5.C_3$ ($S_3^2$) x 2, $C_3^7$ ($C_6\times S_3$), $C_3^3.C_3^4$ ($S_3^2$)

Order 1458: $C_3^5:C_6$ ($C_3^2:C_6$)

Order 729: $C_3^6$ ($C_3\times S_3^2$) x 2, $C_3^6$ ($C_3^2:D_6$), $C_3^3\times \He_3$ ($C_3\times S_3^2$), $C_3^4:C_9$ ($C_3:S_3^2$)

Order 486: $C_3^4:C_6$ ($\He_3.C_6$), $C_3^4:C_6$ ($C_3^3:C_6$)

Order 243: $C_3^5$ ($C_3^2:S_3^2$) x 2, $C_3^5$ ($(C_3\times C_{18}):S_3$), $C_3^5$ ($\He_3:D_6$), $C_3^5$ ($C_3\wr C_2^2$)

Order 162: $C_3^3:S_3$ ($C_3^3.(C_3\times S_3)$)

Order 81: $C_3^4$ ($C_3^3:S_3^2$) x 2, $C_3^4$ ($(C_3\times C_9):S_3^2$) x 2, $C_3^4$ ($\He_3:S_3^2$) x 2, $C_3^4$ ($C_3^3:S_3^2$), $C_3^4$ ($C_3^3.(C_6\times S_3)$)

Order 27: $C_3^3$ ($C_3^4.(C_6\times S_3)$) x 2, $C_3^3$ ($C_3^4.(C_6\times S_3)$) x 2, $C_3^3$ ($\He_3.C_3^3.C_2^2$), $C_3^3$ ($C_3^4:(C_6\times S_3)$), $C_3^3$ ($C_3^4:S_3^2$)

Order 9: $C_3^2$ ($C_3^5.(C_6\times S_3)$) x 2, $C_3^2$ ($C_3^5.(C_6\times S_3)$) x 2, $C_3^2$ ($C_3^5.(C_6\times S_3)$), $C_3^2$ ($(C_3^2\times \He_3):S_3^2$)

Order 3: $C_3$ ($C_3^5.C_3^3.C_2^2$) x 2, $C_3$ ($C_3^5.C_3^3.C_2^2$), $C_3$ ($C_3^6.(C_6\times S_3)$)

Order 1: $C_1$ ($C_3^7.(C_6\times S_3)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 78732: $C_3^7.(C_6\times S_3)$ ($C_1$)

Order 39366: $C_3^6.C_3^3.C_2$ ($C_2$), $C_3^7.C_3.C_6$ ($C_2$)

Order 26244: $C_3^7.D_6$ ($C_3$)

Order 19683: $C_3^7.C_3^2$ ($C_2^2$)

Order 13122: $C_3^7.C_6$ ($S_3$), $C_3^6.C_3.C_6$ ($S_3$), $C_3^6.C_3.C_6$ ($C_6$), $C_3^7.C_6$ ($C_6$)

Order 6561: $C_3^6.C_3^2$ ($D_6$), $C_3\times C_3^6.C_3$ ($C_2\times C_6$), $C_3^7.C_3$ ($D_6$)

Order 4374: $C_3^6.S_3$ ($C_3\times S_3$), $C_3^7.C_2$ ($C_3\times S_3$)

Order 2187: $C_3^7$ ($C_6\times S_3$), $C_3^2\times C_3^4:C_3$ ($C_6\times S_3$), $C_3^3.C_3^4$ ($S_3^2$), $C_3\times C_3^5.C_3$ ($S_3^2$)

Order 1458: $C_3^5:C_6$ ($C_3^2:C_6$)

Order 729: $C_3^6$ ($C_3\times S_3^2$), $C_3^6$ ($C_3^2:D_6$), $C_3^3\times \He_3$ ($C_3\times S_3^2$), $C_3^4:C_9$ ($C_3:S_3^2$)

Order 486: $C_3^4:C_6$ ($\He_3.C_6$), $C_3^4:C_6$ ($C_3^3:C_6$)

Order 243: $C_3^5$ ($(C_3\times C_{18}):S_3$), $C_3^5$ ($\He_3:D_6$), $C_3^5$ ($C_3\wr C_2^2$), $C_3^5$ ($C_3^2:S_3^2$)

Order 162: $C_3^3:S_3$ ($C_3^3.(C_3\times S_3)$)

Order 81: $C_3^4$ ($C_3^3:S_3^2$), $C_3^4$ ($C_3^3:S_3^2$), $C_3^4$ ($(C_3\times C_9):S_3^2$), $C_3^4$ ($\He_3:S_3^2$), $C_3^4$ ($C_3^3.(C_6\times S_3)$)

Order 27: $C_3^3$ ($C_3^4.(C_6\times S_3)$), $C_3^3$ ($\He_3.C_3^3.C_2^2$), $C_3^3$ ($C_3^4.(C_6\times S_3)$), $C_3^3$ ($C_3^4:(C_6\times S_3)$), $C_3^3$ ($C_3^4:S_3^2$)

Order 9: $C_3^2$ ($C_3^5.(C_6\times S_3)$), $C_3^2$ ($C_3^5.(C_6\times S_3)$), $C_3^2$ ($C_3^5.(C_6\times S_3)$), $C_3^2$ ($(C_3^2\times \He_3):S_3^2$)

Order 3: $C_3$ ($C_3^5.C_3^3.C_2^2$), $C_3$ ($C_3^6.(C_6\times S_3)$), $C_3$ ($C_3^5.C_3^3.C_2^2$)

Order 1: $C_1$ ($C_3^7.(C_6\times S_3)$)

Series

Derived series

$C_3^7.(C_6\times S_3)$

$\rhd$

$C_3\times C_3^6.C_3$

$\rhd$

$C_3^3$

$\rhd$

$C_1$

Chief series

$C_3^7.(C_6\times S_3)$

$\rhd$

$C_3^6.C_3^3.C_2$

$\rhd$

$C_3^7.C_3^2$

$\rhd$

$C_3\times C_3^6.C_3$

$\rhd$

$C_3^7$

$\rhd$

$C_3^6$

$\rhd$

$C_3^5$

$\rhd$

$C_3^4$

$\rhd$

$C_3^3$

$\rhd$

$C_3^2$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_3^7.(C_6\times S_3)$

$\rhd$

$C_3\times C_3^6.C_3$

Upper central series

$C_1$

$\lhd$

$C_3$

Supergroups

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

This group is a maximal quotient of 0 larger groups in the database.

Character theory

Complex character table

The $294 \times 294$ character table is not available for this group.

Rational character table

The $151 \times 151$ rational character table is not available for this group.