Abstract group 78732.fx: $C_3^8.D_6$

• Label: 78732.fx
• 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.: $27$
• Trans deg.: $36$
• Rank: $3$

Group information

Description:	$C_3^8.D_6$
Order:	\(78732\)\(\medspace = 2^{2} \cdot 3^{9} \)
Exponent:	\(18\)\(\medspace = 2 \cdot 3^{2} \)
Automorphism group:	$C_5\times C_{11}^2:C_{40}$, 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	495	8018	46890	11664	11664	78732
Conjugacy classes	1	3	655	244	16	8	927
Divisions	1	3	349	130	10	5	498
Autjugacy classes	1	3	43	26	6	3	82

Minimal presentations

Permutation degree:	$27$
Transitive degree:	$36$
Rank:	$3$
Inequivalent generating triples:	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:	${\langle a, b, c, d, e, f, g, h \mid a^{6}=b^{6}=c^{3}=d^{3}=e^{3}=f^{3}= \!\cdots\! \rangle}$
Permutation group:	Degree $27$ $\langle(1,2,5)(4,9,7,13,11,17)(8,14,16,18,12,15)(20,22)(21,24,23,26,25,27), (1,4) \!\cdots\! \rangle$
Transitive group:	36T18810				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:	$C_3^8$ . $D_6$	$C_3^7$ . $S_3^2$	$(C_3^8.C_2)$ . $S_3$ (4)	$C_3^7$ . $(C_6:S_3)$	all 70

Elements of the group are displayed as permutations of degree 27.

Homology

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

Subgroups

There are 144 normal subgroups (60 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_3$	$G/Z \simeq$ $C_3^6.(C_6\times S_3)$
Commutator:	$G' \simeq$ $C_3^7.C_3$	$G/G' \simeq$ $C_2\times C_6$
Frattini:	$\Phi \simeq$ $C_3^5$	$G/\Phi \simeq$ $C_3^2:S_3^2$
Fitting:	$\operatorname{Fit} \simeq$ $C_3^8.C_3$	$G/\operatorname{Fit} \simeq$ $C_2^2$
Radical:	$R \simeq$ $C_3^8.D_6$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3^3$	$G/\operatorname{soc} \simeq$ $C_3^5:D_6$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^8.C_3$

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 6 (order at least 13122) are shown, as well as all normal subgroups of any index.

Order 78732: $C_3^8.D_6$

Order 39366: $C_3^5.\He_3.C_6$, $C_3^7.C_3.C_6$, $C_3^8.C_6$

Order 26244: $C_3^7.D_6$ x 2, $C_3^8.C_2^2$, $C_3^7.D_6$, $C_3^7.D_6$, $C_3^7.D_6$

Order 19683: $C_3^8.C_3$

Order 13122: $C_3^7.C_6$ x 5, $C_3^5.C_3^3.C_2$ x 4, $C_3^7.C_6$ x 3, $C_3^8.C_2$ x 2, $C_3^6.C_3.C_6$ x 2, $C_3^4.C_3^4.C_2$ x 2, $C_3^4.\He_3.C_6$ x 2, $C_3^5.C_3^3.C_2$, $C_3^7.C_6$, $C_3^8.C_2$, $C_3^4.\He_3.C_6$, $C_3^4.C_3^4.C_2$, $C_3^6.C_3.C_6$, $C_3^7.S_3$

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

Order 4374: $C_3^6.C_6$ x 2, $C_3^7.C_2$ x 2, $C_3^6.S_3$, $C_3^6.C_6$

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

Order 1458: $C_3^5:C_6$ x 7

Order 729: $C_3^6$ x 10, $C_3^5.C_3$ x 2, $C_3^5:C_3$

Order 486: $C_3^4:C_6$ x 8

Order 243: $C_3^5$ x 17

Order 162: $C_3^2\wr C_2$ x 7

Order 81: $C_3^4$ x 22

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

Order 27: $C_3^3$ x 17

Order 18: $C_3:S_3$

Order 9: $C_3^2$ x 10

Order 4: $C_2^2$

Order 3: $C_3$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

All subgroups of index up to 6 (order at least 13122) are shown, as well as all normal subgroups of any index.

Order 78732: $C_3^8.D_6$

Order 39366: $C_3^5.\He_3.C_6$, $C_3^7.C_3.C_6$, $C_3^8.C_6$

Order 26244: $C_3^8.C_2^2$, $C_3^7.D_6$, $C_3^7.D_6$, $C_3^7.D_6$, $C_3^7.D_6$

Order 19683: $C_3^8.C_3$

Order 13122: $C_3^7.C_6$ x 3, $C_3^7.C_6$ x 3, $C_3^5.C_3^3.C_2$ x 3, $C_3^8.C_2$ x 2, $C_3^6.C_3.C_6$ x 2, $C_3^4.C_3^4.C_2$, $C_3^5.C_3^3.C_2$, $C_3^4.\He_3.C_6$, $C_3^7.C_6$, $C_3^8.C_2$, $C_3^4.\He_3.C_6$, $C_3^4.C_3^4.C_2$, $C_3^6.C_3.C_6$, $C_3^7.S_3$

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

Order 4374: $C_3^7.C_2$ x 2, $C_3^6.C_6$, $C_3^6.S_3$, $C_3^6.C_6$

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

Order 1458: $C_3^5:C_6$ x 3

Order 729: $C_3^6$ x 6, $C_3^5.C_3$, $C_3^5:C_3$

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

Order 243: $C_3^5$ x 9

Order 162: $C_3^2\wr C_2$ x 3

Order 81: $C_3^4$ x 10

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

Order 27: $C_3^3$ x 9

Order 18: $C_3:S_3$

Order 9: $C_3^2$ x 6

Order 4: $C_2^2$

Order 3: $C_3$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 78732: $C_3^8.D_6$ ($C_1$)

Order 39366: $C_3^5.\He_3.C_6$ ($C_2$), $C_3^7.C_3.C_6$ ($C_2$), $C_3^8.C_6$ ($C_2$)

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

Order 19683: $C_3^8.C_3$ ($C_2^2$)

Order 13122: $C_3^7.C_6$ ($S_3$) x 2, $C_3^7.C_6$ ($C_6$), $C_3^7.C_6$ ($S_3$), $C_3^6.C_3.C_6$ ($S_3$), $C_3^8.C_2$ ($S_3$), $C_3^6.C_3.C_6$ ($C_6$), $C_3^7.S_3$ ($C_6$)

Order 6561: $C_3^7.C_3$ ($D_6$) x 2, $C_3^7.C_3$ ($C_2\times C_6$), $C_3^7.C_3$ ($D_6$), $C_3\times C_3^6.C_3$ ($D_6$), $C_3^8$ ($D_6$)

Order 4374: $C_3^6.C_6$ ($C_3\times S_3$) x 2, $C_3^6.S_3$ ($C_3\times S_3$), $C_3^7.C_2$ ($C_3:S_3$), $C_3^7.C_2$ ($C_3\times S_3$), $C_3^6.C_6$ ($C_3\times S_3$)

Order 2187: $C_3\times C_3^5.C_3$ ($S_3^2$) x 2, $C_3^6.C_3$ ($C_6\times S_3$) x 2, $C_3^7$ ($C_6:S_3$), $C_3^7$ ($C_6\times S_3$), $C_3^7$ ($S_3^2$), $C_3^6.C_3$ ($C_6\times S_3$), $C_3^6.C_3$ ($S_3^2$), $C_3^6.C_3$ ($C_6\times S_3$)

Order 1458: $C_3^5:C_6$ ($C_3^2:S_3$) x 3, $C_3^5:C_6$ ($C_3^2:C_6$) x 3, $C_3^5:C_6$ ($C_3^2:C_6$)

Order 729: $C_3^6$ ($C_3^2:D_6$) x 3, $C_3^6$ ($C_3^2:D_6$) x 3, $C_3^5.C_3$ ($C_3\times S_3^2$) x 2, $C_3^6$ ($C_3^2:D_6$), $C_3^6$ ($C_3:S_3^2$), $C_3^6$ ($C_3\times S_3^2$), $C_3^6$ ($C_3^2:D_6$), $C_3^5:C_3$ ($C_3\times S_3^2$)

Order 486: $C_3^4:C_6$ ($C_3^3:S_3$) x 3, $C_3^4:C_6$ ($C_3\wr S_3$) x 3, $C_3^4:C_6$ ($C_3^3:S_3$), $C_3^4:C_6$ ($C_3^3:C_6$)

Order 243: $C_3^5$ ($C_3^3:D_6$) x 3, $C_3^5$ ($C_3^3:D_6$) x 3, $C_3^5$ ($C_3^2:S_3^2$) x 3, $C_3^5$ ($C_3^2:S_3^2$) x 3, $C_3^5$ ($C_3^2:S_3^2$), $C_3^5$ ($C_3^3:D_6$), $C_3^5$ ($C_3^3:D_6$), $C_3^5$ ($C_3^2:S_3^2$), $C_3^5$ ($C_3^3:D_6$)

Order 162: $C_3^2\wr C_2$ ($C_3^4:S_3$) x 3, $C_3^2\wr C_2$ ($C_3^4:S_3$) x 3, $C_3^2\wr C_2$ ($C_3^4:S_3$)

Order 81: $C_3^4$ ($C_3^4:D_6$) x 3, $C_3^4$ ($C_3^4:D_6$) x 3, $C_3^4$ ($C_3^3:S_3^2$) x 3, $C_3^4$ ($C_3^3:S_3^2$) x 3, $C_3^4$ ($C_3^3:S_3^2$) x 3, $C_3^4$ ($C_3^4:D_6$) x 3, $C_3^4$ ($C_3^3:S_3^2$), $C_3^4$ ($C_3^3:S_3^2$), $C_3^4$ ($C_3^3:S_3^2$), $C_3^4$ ($C_3^4:D_6$)

Order 54: $C_3^2:C_6$ ($C_3^5:S_3$), $C_3^2:C_6$ ($C_3^5:S_3$)

Order 27: $C_3^3$ ($C_3^5.D_6$) x 3, $C_3^3$ ($C_3^5:D_6$) x 3, $C_3^3$ ($C_3^5:D_6$) x 3, $C_3^3$ ($C_3^5:D_6$) x 3, $C_3^3$ ($C_3^4.(C_6\times S_3)$), $C_3^3$ ($C_3^4:C_3.C_6.C_2$), $C_3^3$ ($C_3^4.(C_6\times S_3)$), $C_3^3$ ($C_3^4.(C_6\times S_3)$), $C_3^3$ ($C_3^5:D_6$)

Order 18: $C_3:S_3$ ($C_3^5.C_3.C_6$)

Order 9: $C_3^2$ ($C_3^5.(C_6\times S_3)$) x 3, $C_3^2$ ($C_3^6:D_6$) x 3, $C_3^2$ ($C_3^5.(C_6\times S_3)$), $C_3^2$ ($C_3^5.(C_6\times S_3)$), $C_3^2$ ($C_3^6:D_6$), $C_3^2$ ($C_3^6.D_6$)

Order 3: $C_3$ ($C_3^6.(C_6\times S_3)$), $C_3$ ($C_3^6.(C_6\times S_3)$), $C_3$ ($C_3^7.D_6$)

Order 1: $C_1$ ($C_3^8.D_6$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 78732: $C_3^8.D_6$ ($C_1$)

Order 39366: $C_3^5.\He_3.C_6$ ($C_2$), $C_3^7.C_3.C_6$ ($C_2$), $C_3^8.C_6$ ($C_2$)

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

Order 19683: $C_3^8.C_3$ ($C_2^2$)

Order 13122: $C_3^7.C_6$ ($C_6$), $C_3^7.C_6$ ($S_3$), $C_3^6.C_3.C_6$ ($S_3$), $C_3^7.C_6$ ($S_3$), $C_3^8.C_2$ ($S_3$), $C_3^6.C_3.C_6$ ($C_6$), $C_3^7.S_3$ ($C_6$)

Order 6561: $C_3^7.C_3$ ($D_6$), $C_3^7.C_3$ ($C_2\times C_6$), $C_3^7.C_3$ ($D_6$), $C_3\times C_3^6.C_3$ ($D_6$), $C_3^8$ ($D_6$)

Order 4374: $C_3^6.C_6$ ($C_3\times S_3$), $C_3^6.S_3$ ($C_3\times S_3$), $C_3^7.C_2$ ($C_3:S_3$), $C_3^7.C_2$ ($C_3\times S_3$), $C_3^6.C_6$ ($C_3\times S_3$)

Order 2187: $C_3^7$ ($C_6:S_3$), $C_3^7$ ($C_6\times S_3$), $C_3^7$ ($S_3^2$), $C_3\times C_3^5.C_3$ ($S_3^2$), $C_3^6.C_3$ ($C_6\times S_3$), $C_3^6.C_3$ ($S_3^2$), $C_3^6.C_3$ ($C_6\times S_3$), $C_3^6.C_3$ ($C_6\times S_3$)

Order 1458: $C_3^5:C_6$ ($C_3^2:S_3$), $C_3^5:C_6$ ($C_3^2:C_6$), $C_3^5:C_6$ ($C_3^2:C_6$)

Order 729: $C_3^6$ ($C_3^2:D_6$), $C_3^6$ ($C_3:S_3^2$), $C_3^6$ ($C_3\times S_3^2$), $C_3^6$ ($C_3^2:D_6$), $C_3^6$ ($C_3^2:D_6$), $C_3^6$ ($C_3^2:D_6$), $C_3^5.C_3$ ($C_3\times S_3^2$), $C_3^5:C_3$ ($C_3\times S_3^2$)

Order 486: $C_3^4:C_6$ ($C_3^3:S_3$), $C_3^4:C_6$ ($C_3^3:C_6$), $C_3^4:C_6$ ($C_3^3:S_3$), $C_3^4:C_6$ ($C_3\wr S_3$)

Order 243: $C_3^5$ ($C_3^3:D_6$), $C_3^5$ ($C_3^3:D_6$), $C_3^5$ ($C_3^2:S_3^2$), $C_3^5$ ($C_3^3:D_6$), $C_3^5$ ($C_3^3:D_6$), $C_3^5$ ($C_3^2:S_3^2$), $C_3^5$ ($C_3^2:S_3^2$), $C_3^5$ ($C_3^3:D_6$), $C_3^5$ ($C_3^2:S_3^2$)

Order 162: $C_3^2\wr C_2$ ($C_3^4:S_3$), $C_3^2\wr C_2$ ($C_3^4:S_3$), $C_3^2\wr C_2$ ($C_3^4:S_3$)

Order 81: $C_3^4$ ($C_3^3:S_3^2$), $C_3^4$ ($C_3^3:S_3^2$), $C_3^4$ ($C_3^3:S_3^2$), $C_3^4$ ($C_3^4:D_6$), $C_3^4$ ($C_3^4:D_6$), $C_3^4$ ($C_3^4:D_6$), $C_3^4$ ($C_3^3:S_3^2$), $C_3^4$ ($C_3^3:S_3^2$), $C_3^4$ ($C_3^3:S_3^2$), $C_3^4$ ($C_3^4:D_6$)

Order 54: $C_3^2:C_6$ ($C_3^5:S_3$), $C_3^2:C_6$ ($C_3^5:S_3$)

Order 27: $C_3^3$ ($C_3^4.(C_6\times S_3)$), $C_3^3$ ($C_3^5.D_6$), $C_3^3$ ($C_3^4:C_3.C_6.C_2$), $C_3^3$ ($C_3^4.(C_6\times S_3)$), $C_3^3$ ($C_3^4.(C_6\times S_3)$), $C_3^3$ ($C_3^5:D_6$), $C_3^3$ ($C_3^5:D_6$), $C_3^3$ ($C_3^5:D_6$), $C_3^3$ ($C_3^5:D_6$)

Order 18: $C_3:S_3$ ($C_3^5.C_3.C_6$)

Order 9: $C_3^2$ ($C_3^5.(C_6\times S_3)$), $C_3^2$ ($C_3^5.(C_6\times S_3)$), $C_3^2$ ($C_3^5.(C_6\times S_3)$), $C_3^2$ ($C_3^6:D_6$), $C_3^2$ ($C_3^6:D_6$), $C_3^2$ ($C_3^6.D_6$)

Order 3: $C_3$ ($C_3^6.(C_6\times S_3)$), $C_3$ ($C_3^6.(C_6\times S_3)$), $C_3$ ($C_3^7.D_6$)

Order 1: $C_1$ ($C_3^8.D_6$)

Series

Derived series

$C_3^8.D_6$

$\rhd$

$C_3^7.C_3$

$\rhd$

$C_3^4$

$\rhd$

$C_1$

Chief series

$C_3^8.D_6$

$\rhd$

$C_3^7.C_3.C_6$

$\rhd$

$C_3^8.C_3$

$\rhd$

$C_3^7.C_3$

$\rhd$

$C_3^7$

$\rhd$

$C_3^6$

$\rhd$

$C_3^5$

$\rhd$

$C_3^4$

$\rhd$

$C_3^3$

$\rhd$

$C_3^2$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_3^8.D_6$

$\rhd$

$C_3^7.C_3$

Upper central series

$C_1$

$\lhd$

$C_3$

Supergroups

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

This group is a maximal quotient of 1 larger groups in the database.

Character theory

Complex character table

The $927 \times 927$ character table is not available for this group.

Rational character table

The $498 \times 498$ rational character table is not available for this group.