Abstract group 708588.je: $C_3^7.C_3\wr C_2^2$

• Label: 708588.je
• Order: \( 2^{2} \cdot 3^{11} \)
• Exponent: \( 2 \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 3 \)
• $\card{Z(G)}$: \( 3 \)
• $\card{\Aut(G)}$: \( 2^{4} \cdot 3^{15} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \cdot 3^{5} \)
• Perm deg.: $36$
• Trans deg.: $36$
• Rank: $3$

Group information

Description:	$C_3^7.C_3\wr C_2^2$
Order:	\(708588\)\(\medspace = 2^{2} \cdot 3^{11} \)
Exponent:	\(18\)\(\medspace = 2 \cdot 3^{2} \)
Automorphism group:	$C_3^5.C_3^5.C_3.C_6.C_6^3$, of order \(229582512\)\(\medspace = 2^{4} \cdot 3^{15} \)
Composition factors:	$C_2$ x 2, $C_3$ x 11
Derived length:	$3$

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

Group statistics

Order	1	2	3	6	9	18
Elements	1	2187	54674	214326	122472	314928	708588
Conjugacy classes	1	3	326	30	158	41	559
Divisions	1	3	181	18	80	14	297
Autjugacy classes	1	2	36	9	32	9	89

Minimal presentations

Permutation degree:	$36$
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:	${\langle a, b, c, d, e, f, g, h, i, j \mid c^{3}=d^{9}=e^{3}=f^{3}=g^{3}= \!\cdots\! \rangle}$
Permutation group:	Degree $36$ $\langle(1,26,14,3,25,13,2,27,15)(4,29,17,6,28,16,5,30,18)(7,8,9)(10,24,34)(11,22,35) \!\cdots\! \rangle$
Transitive group:	36T34087				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:	$(C_3^6.C_3^4)$ . $D_6$ (4)	$(C_3^7.C_3^2)$ . $S_3^2$ (4)	$C_3^6$ . $(C_3^3:S_3^2)$	$C_3^6$ . $(C_3^3:S_3^2)$	all 31

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 91 normal subgroups (35 characteristic).

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

Special subgroups

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

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 12 (order at least 59049) are shown, as well as all normal subgroups of any index.

Order 708588: $C_3^7.C_3\wr C_2^2$

Order 354294: $C_3^7.C_3^4.C_2$ x 2, $C_3^6.C_3^5.C_2$

Order 236196: $C_3^6.C_3\wr C_2^2$ x 2, $C_3^7.(C_3\times S_3^2)$ x 2, $C_3^6.C_3\wr C_2^2$, $C_3^7.C_3:S_3^2$, $C_3^7.(C_3\times S_3^2)$

Order 177147: $C_3^7.C_3^4$

Order 118098: $C_3^6.C_3^4.C_2$ x 6, $C_3^7.C_3^3.C_2$ x 4, $C_3^6.C_3^4.C_2$ x 4, $C_3^7.C_3^3.C_2$ x 4, $C_3^7.C_3^3.C_2$ x 2, $C_3^6.C_3^4.C_2$ x 2, $C_3^5.C_3^4.C_6$ x 2, $C_3^6.C_3^4.C_2$ x 2, $C_3^6.C_3^4.C_2$ x 2, $C_3^6.C_3^4.C_2$ x 2, $C_3^7.C_3^3.C_2$ x 2, $C_3^6.C_3^4.C_2$, $C_3^5.C_3^4.C_6$, $C_3^6.C_3^4.C_2$, $C_3^6.C_3^4.C_2$

Order 78732: $C_3^6.(C_3\times S_3^2)$ x 4, $C_3^6.C_3^3.C_2^2$ x 2, $C_3^5.\He_3.D_6$ x 2, $C_3^7.(C_6\times S_3)$ x 2, $C_3^7.S_3^2$ x 2, $C_3^6.C_3^3.C_2^2$, $C_3^6.C_3^2.D_6$, $C_3^7.(C_6\times S_3)$, $C_3^7.S_3^2$

Order 59049: $C_3^6.C_3^4$ x 8, $C_3^6.C_3^4$ x 7, $C_3^6.C_3^4$ x 6, $C_3^7.C_3^3$ x 5, $C_3^7.C_3^3$ x 4, $C_3^7.C_3^3$ x 4, $C_3^6.C_3^4$ x 3, $C_3^6.C_3^4$ x 2, $C_3^5.C_3^5$, $C_3^7.C_3^3$, $C_3^6.C_3^4$, $C_3^6.C_3^4$

Order 39366: $C_3^6.C_3^3.C_2$ x 3, $C_3^7.C_3.S_3$ x 2, $C_3^6.C_3^3.C_2$, $C_3^5.C_3^3.C_6$

Order 19683: $C_3^6.C_3^3$ x 6, $C_3^6.C_3^3$ x 4, $C_3^7.C_3^2$ x 4, $C_3^7.C_3^2$ x 2

Order 13122: $C_3^6.C_3.S_3$

Order 6561: $C_3^6.C_3^2$ x 8, $C_3^7.C_3$ x 4, $C_3^5.C_3^3$ x 2, $C_3^6.C_3^2$ x 2

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

Order 729: $C_3^6$ x 3

Order 243: $C_3^5$ x 3

Order 81: $C_3^4$

Order 27: $C_3^3$ x 4

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 12 (order at least 59049) are shown, as well as all normal subgroups of any index.

Order 708588: $C_3^7.C_3\wr C_2^2$

Order 354294: $C_3^6.C_3^5.C_2$, $C_3^7.C_3^4.C_2$

Order 236196: $C_3^6.C_3\wr C_2^2$, $C_3^6.C_3\wr C_2^2$, $C_3^7.C_3:S_3^2$, $C_3^7.(C_3\times S_3^2)$, $C_3^7.(C_3\times S_3^2)$

Order 177147: $C_3^7.C_3^4$

Order 118098: $C_3^7.C_3^3.C_2$ x 2, $C_3^5.C_3^4.C_6$ x 2, $C_3^7.C_3^3.C_2$ x 2, $C_3^6.C_3^4.C_2$ x 2, $C_3^7.C_3^3.C_2$, $C_3^6.C_3^4.C_2$, $C_3^6.C_3^4.C_2$, $C_3^6.C_3^4.C_2$, $C_3^6.C_3^4.C_2$, $C_3^6.C_3^4.C_2$, $C_3^5.C_3^4.C_6$, $C_3^6.C_3^4.C_2$, $C_3^6.C_3^4.C_2$, $C_3^7.C_3^3.C_2$, $C_3^6.C_3^4.C_2$

Order 78732: $C_3^6.C_3^3.C_2^2$, $C_3^6.C_3^3.C_2^2$, $C_3^6.C_3^2.D_6$, $C_3^5.\He_3.D_6$, $C_3^6.(C_3\times S_3^2)$, $C_3^7.(C_6\times S_3)$, $C_3^7.(C_6\times S_3)$, $C_3^7.S_3^2$, $C_3^7.S_3^2$

Order 59049: $C_3^7.C_3^3$ x 4, $C_3^6.C_3^4$ x 4, $C_3^6.C_3^4$ x 3, $C_3^7.C_3^3$ x 3, $C_3^6.C_3^4$ x 2, $C_3^6.C_3^4$ x 2, $C_3^7.C_3^3$ x 2, $C_3^5.C_3^5$, $C_3^7.C_3^3$, $C_3^6.C_3^4$, $C_3^6.C_3^4$, $C_3^6.C_3^4$

Order 39366: $C_3^6.C_3^3.C_2$ x 2, $C_3^6.C_3^3.C_2$, $C_3^5.C_3^3.C_6$, $C_3^7.C_3.S_3$

Order 19683: $C_3^6.C_3^3$ x 4, $C_3^7.C_3^2$ x 2, $C_3^7.C_3^2$ x 2, $C_3^6.C_3^3$

Order 13122: $C_3^6.C_3.S_3$

Order 6561: $C_3^6.C_3^2$ x 3, $C_3^7.C_3$ x 3, $C_3^6.C_3^2$ x 2, $C_3^5.C_3^3$

Order 2187: $C_3\times C_3^5.C_3$ x 4, $C_3^7$

Order 729: $C_3^6$ x 2

Order 243: $C_3^5$ x 2

Order 81: $C_3^4$

Order 27: $C_3^3$ x 3

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 708588: $C_3^7.C_3\wr C_2^2$ ($C_1$)

Order 354294: $C_3^7.C_3^4.C_2$ ($C_2$) x 2, $C_3^6.C_3^5.C_2$ ($C_2$)

Order 236196: $C_3^7.C_3:S_3^2$ ($C_3$)

Order 177147: $C_3^7.C_3^4$ ($C_2^2$)

Order 118098: $C_3^7.C_3^3.C_2$ ($C_6$) x 2, $C_3^7.C_3^3.C_2$ ($S_3$) x 2, $C_3^6.C_3^4.C_2$ ($S_3$) x 2, $C_3^5.C_3^4.C_6$ ($C_6$), $C_3^5.C_3^4.C_6$ ($S_3$), $C_3^6.C_3^4.C_2$ ($S_3$)

Order 59049: $C_3^7.C_3^3$ ($D_6$) x 3, $C_3^6.C_3^4$ ($D_6$) x 2, $C_3^7.C_3^3$ ($C_2\times C_6$), $C_3^6.C_3^4$ ($D_6$)

Order 39366: $C_3^6.C_3^3.C_2$ ($C_3\times S_3$) x 3, $C_3^7.C_3.S_3$ ($C_3\times S_3$) x 2, $C_3^6.C_3^3.C_2$ ($C_3\times S_3$), $C_3^5.C_3^3.C_6$ ($C_3:S_3$)

Order 19683: $C_3^6.C_3^3$ ($S_3^2$) x 4, $C_3^6.C_3^3$ ($C_6\times S_3$) x 3, $C_3^6.C_3^3$ ($S_3^2$) x 2, $C_3^7.C_3^2$ ($C_6\times S_3$) x 2, $C_3^7.C_3^2$ ($S_3^2$) x 2, $C_3^7.C_3^2$ ($C_6\times S_3$), $C_3^7.C_3^2$ ($S_3^2$), $C_3^6.C_3^3$ ($C_6:S_3$)

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

Order 6561: $C_3^6.C_3^2$ ($C_3\times S_3^2$) x 6, $C_3^7.C_3$ ($C_3\times S_3^2$) x 3, $C_3^6.C_3^2$ ($C_3:S_3^2$) x 2, $C_3^5.C_3^3$ ($C_3:S_3^2$) x 2, $C_3^7.C_3$ ($C_3:S_3^2$), $C_3^6.C_3^2$ ($C_3^2:D_6$), $C_3^6.C_3^2$ ($C_3:S_3^2$)

Order 2187: $C_3\times C_3^5.C_3$ ($C_3\wr C_2^2$) x 3, $C_3\times C_3^5.C_3$ ($C_3^2:S_3^2$) x 2, $C_3^7$ ($C_3\wr C_2^2$), $C_3\times C_3^5.C_3$ ($C_3^2:S_3^2$)

Order 729: $C_3^6$ ($C_3^3:S_3^2$) x 2, $C_3^6$ ($C_3^3:S_3^2$)

Order 243: $C_3^5$ ($C_3.C_3^5.C_2^2$) x 2, $C_3^5$ ($C_3^4:S_3^2$)

Order 81: $C_3^4$ ($C_3^5:S_3^2$)

Order 27: $C_3^3$ ($C_3^4.C_3^4.C_2^2$) x 2, $C_3^3$ ($C_3^3.C_3^5.C_2^2$), $C_3^3$ ($C_3^3.C_3^5.C_2^2$)

Order 9: $C_3^2$ ($C_3^5.C_3^4.C_2^2$) x 2, $C_3^2$ ($C_3^4.C_3^5.C_2^2$) x 2, $C_3^2$ ($C_3^4.C_3^5.C_2^2$), $C_3^2$ ($C_3^5.C_3^4.C_2^2$)

Order 3: $C_3$ ($C_3^5.C_3^5.C_2^2$) x 2, $C_3$ ($C_3^5.C_3^5.C_2^2$), $C_3$ ($C_3^6.C_3^4.C_2^2$)

Order 1: $C_1$ ($C_3^7.C_3\wr C_2^2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 708588: $C_3^7.C_3\wr C_2^2$ ($C_1$)

Order 354294: $C_3^6.C_3^5.C_2$ ($C_2$), $C_3^7.C_3^4.C_2$ ($C_2$)

Order 236196: $C_3^7.C_3:S_3^2$ ($C_3$)

Order 177147: $C_3^7.C_3^4$ ($C_2^2$)

Order 118098: $C_3^7.C_3^3.C_2$ ($C_6$), $C_3^7.C_3^3.C_2$ ($S_3$), $C_3^5.C_3^4.C_6$ ($C_6$), $C_3^5.C_3^4.C_6$ ($S_3$), $C_3^6.C_3^4.C_2$ ($S_3$), $C_3^6.C_3^4.C_2$ ($S_3$)

Order 59049: $C_3^7.C_3^3$ ($D_6$) x 2, $C_3^7.C_3^3$ ($C_2\times C_6$), $C_3^6.C_3^4$ ($D_6$), $C_3^6.C_3^4$ ($D_6$)

Order 39366: $C_3^6.C_3^3.C_2$ ($C_3\times S_3$) x 2, $C_3^6.C_3^3.C_2$ ($C_3\times S_3$), $C_3^5.C_3^3.C_6$ ($C_3:S_3$), $C_3^7.C_3.S_3$ ($C_3\times S_3$)

Order 19683: $C_3^6.C_3^3$ ($C_6\times S_3$) x 2, $C_3^7.C_3^2$ ($C_6\times S_3$), $C_3^7.C_3^2$ ($S_3^2$), $C_3^6.C_3^3$ ($C_6:S_3$), $C_3^6.C_3^3$ ($S_3^2$), $C_3^6.C_3^3$ ($S_3^2$), $C_3^7.C_3^2$ ($C_6\times S_3$), $C_3^7.C_3^2$ ($S_3^2$)

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

Order 6561: $C_3^6.C_3^2$ ($C_3\times S_3^2$) x 2, $C_3^7.C_3$ ($C_3\times S_3^2$) x 2, $C_3^6.C_3^2$ ($C_3:S_3^2$), $C_3^7.C_3$ ($C_3:S_3^2$), $C_3^5.C_3^3$ ($C_3:S_3^2$), $C_3^6.C_3^2$ ($C_3^2:D_6$), $C_3^6.C_3^2$ ($C_3:S_3^2$)

Order 2187: $C_3\times C_3^5.C_3$ ($C_3\wr C_2^2$) x 2, $C_3^7$ ($C_3\wr C_2^2$), $C_3\times C_3^5.C_3$ ($C_3^2:S_3^2$), $C_3\times C_3^5.C_3$ ($C_3^2:S_3^2$)

Order 729: $C_3^6$ ($C_3^3:S_3^2$), $C_3^6$ ($C_3^3:S_3^2$)

Order 243: $C_3^5$ ($C_3.C_3^5.C_2^2$), $C_3^5$ ($C_3^4:S_3^2$)

Order 81: $C_3^4$ ($C_3^5:S_3^2$)

Order 27: $C_3^3$ ($C_3^3.C_3^5.C_2^2$), $C_3^3$ ($C_3^3.C_3^5.C_2^2$), $C_3^3$ ($C_3^4.C_3^4.C_2^2$)

Order 9: $C_3^2$ ($C_3^4.C_3^5.C_2^2$), $C_3^2$ ($C_3^5.C_3^4.C_2^2$), $C_3^2$ ($C_3^5.C_3^4.C_2^2$), $C_3^2$ ($C_3^4.C_3^5.C_2^2$)

Order 3: $C_3$ ($C_3^5.C_3^5.C_2^2$), $C_3$ ($C_3^5.C_3^5.C_2^2$), $C_3$ ($C_3^6.C_3^4.C_2^2$)

Order 1: $C_1$ ($C_3^7.C_3\wr C_2^2$)

Series

Derived series

$C_3^7.C_3\wr C_2^2$

$\rhd$

$C_3^7.C_3^3$

$\rhd$

$C_3^6$

$\rhd$

$C_1$

Chief series

$C_3^7.C_3\wr C_2^2$

$\rhd$

$C_3^6.C_3^5.C_2$

$\rhd$

$C_3^7.C_3^4$

$\rhd$

$C_3^7.C_3^3$

$\rhd$

$C_3^6.C_3^3$

$\rhd$

$C_3^6.C_3^2$

$\rhd$

$C_3\times C_3^5.C_3$

$\rhd$

$C_3^6$

$\rhd$

$C_3^5$

$\rhd$

$C_3^4$

$\rhd$

$C_3^3$

$\rhd$

$C_3^2$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_3^7.C_3\wr C_2^2$

$\rhd$

$C_3^7.C_3^3$

Upper central series

$C_1$

$\lhd$

$C_3$

Supergroups

This group is a maximal subgroup of 4 larger groups in the database.

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

Character theory

Complex character table

The $559 \times 559$ character table is not available for this group.

Rational character table

The $297 \times 297$ rational character table is not available for this group.