Abstract group 1889568.jn: $C_3^7.C_6^2:S_4$

• Label: 1889568.jn
• Order: \( 2^{5} \cdot 3^{10} \)
• Exponent: \( 2^{2} \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2 \cdot 3 \)
• $\card{Z(G)}$: \( 3 \)
• $\card{\Aut(G)}$: \( 2^{9} \cdot 3^{10} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \cdot 3 \)
• Perm deg.: $27$
• Trans deg.: $36$
• Rank: $2$

Group information

Description:	$C_3^7.C_6^2:S_4$
Order:	\(1889568\)\(\medspace = 2^{5} \cdot 3^{10} \)
Exponent:	\(36\)\(\medspace = 2^{2} \cdot 3^{2} \)
Automorphism group:	$C_3\times C_3^7.C_2^4.D_6^2.C_2$, of order \(30233088\)\(\medspace = 2^{9} \cdot 3^{10} \)
Composition factors:	$C_2$ x 5, $C_3$ x 10
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
Elements	1	3699	43010	8748	527742	606528	699840	1889568
Conjugacy classes	1	5	414	3	751	26	186	1386
Divisions	1	5	230	3	403	17	62	721
Autjugacy classes	1	4	74	2	123	5	19	228

Minimal presentations

Permutation degree:	$27$
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:	${\langle a, b, c, d, e, f, g, h, i, j \mid c^{6}=d^{6}=e^{6}=f^{6}=g^{3}= \!\cdots\! \rangle}$
Permutation group:	Degree $27$ $\langle(1,3,7,13)(2,5,10,15)(4,8,14,11)(6,12,9)(19,20,22,26)(21,24,27)(23,25), (1,2,4)(3,6,11,16)(5,9,13,17)(7,10,14)(8,12,15,18)(20,21,23,27,26,24)(22,25)\rangle$
Transitive group:	36T43124				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:	$(C_3^9.C_2^4)$ . $S_3$	$(C_3^9.C_2^2)$ . $S_4$ (2)	$C_3^9$ . $(C_2^2:S_4)$	$C_3^7$ . $(C_6^2:S_4)$	all 29

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

Homology

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

Subgroups

There are 44 normal subgroups (20 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_3$	$G/Z \simeq$ $C_3^2\wr C_2^2.S_4$
Commutator:	$G' \simeq$ $C_3^2\wr C_2^2.A_4$	$G/G' \simeq$ $C_6$
Frattini:	$\Phi \simeq$ $C_3^2$	$G/\Phi \simeq$ $C_3^7.C_2^2:S_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_3^9$	$G/\operatorname{Fit} \simeq$ $C_2^2:S_4$
Radical:	$R \simeq$ $C_3^7.C_6^2:S_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3^7$	$G/\operatorname{soc} \simeq$ $C_6^2:S_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2\wr C_2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^9.C_3$

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 4 (order at least 472392) are shown, as well as all normal subgroups of any index.

Order 1889568: $C_3^7.C_6^2:S_4$

Order 944784: $C_3^7.C_6^2:A_4$

Order 629856: $C_3^6.C_6^3.C_2^2$, $C_3^2\wr C_2^2.S_4$

Order 472392: $C_3^8.(C_3\times S_4)$ x 2, $C_3^7.C_6^2:C_3.C_2$

Order 314928: $C_3^9.C_2^4$, $C_3^2\wr C_2^2.A_4$

Order 104976: $C_3^8.C_2^4$

Order 78732: $C_3^9.C_2^2$ x 2, $C_3^9.C_2^2$

Order 69984: $C_6^2.C_3^4:S_4$ x 2

Order 59049: $C_3^9.C_3$

Order 34992: $C_3^7.C_2^4$

Order 26244: $C_3^8.C_2^2$ x 2, $C_3^2\wr C_2^2$

Order 19683: $C_3^9$

Order 11664: $C_3^6.C_2^4$

Order 8748: $C_3^7.C_2^2$ x 2, $C_3^7:C_2^2$

Order 6561: $C_3^8$

Order 2916: $C_3^6.C_2^2$ x 2, $C_3^3\times C_3^3:C_2^2$ x 2, $C_3^5:D_6$

Order 2187: $C_3^7$

Order 972: $C_3^3:C_6^2$ x 2

Order 729: $C_3^6$ x 3

Order 324: $C_3\wr C_2^2$ x 2

Order 243: $C_3^5$ x 2

Order 108: $C_3:S_3^2$ x 2

Order 81: $C_3^4$ x 2

Order 32: $C_2^2\wr C_2$

Order 27: $C_3^3$ x 3

Order 9: $C_3^2$

Order 3: $C_3$

Order 1: $C_1$

Classes of subgroups up to automorphism

All subgroups of index up to 4 (order at least 472392) are shown, as well as all normal subgroups of any index.

Order 1889568: $C_3^7.C_6^2:S_4$

Order 944784: $C_3^7.C_6^2:A_4$

Order 629856: $C_3^6.C_6^3.C_2^2$, $C_3^2\wr C_2^2.S_4$

Order 472392: $C_3^7.C_6^2:C_3.C_2$, $C_3^8.(C_3\times S_4)$

Order 314928: $C_3^9.C_2^4$, $C_3^2\wr C_2^2.A_4$

Order 104976: $C_3^8.C_2^4$

Order 78732: $C_3^9.C_2^2$, $C_3^9.C_2^2$

Order 69984: $C_6^2.C_3^4:S_4$

Order 59049: $C_3^9.C_3$

Order 34992: $C_3^7.C_2^4$

Order 26244: $C_3^8.C_2^2$, $C_3^2\wr C_2^2$

Order 19683: $C_3^9$

Order 11664: $C_3^6.C_2^4$

Order 8748: $C_3^7.C_2^2$, $C_3^7:C_2^2$

Order 6561: $C_3^8$

Order 2916: $C_3^6.C_2^2$, $C_3^3\times C_3^3:C_2^2$, $C_3^5:D_6$

Order 2187: $C_3^7$

Order 972: $C_3^3:C_6^2$

Order 729: $C_3^6$ x 2

Order 324: $C_3\wr C_2^2$

Order 243: $C_3^5$

Order 108: $C_3:S_3^2$

Order 81: $C_3^4$

Order 32: $C_2^2\wr C_2$

Order 27: $C_3^3$ x 2

Order 9: $C_3^2$

Order 3: $C_3$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1889568: $C_3^7.C_6^2:S_4$ ($C_1$)

Order 944784: $C_3^7.C_6^2:A_4$ ($C_2$)

Order 629856: $C_3^2\wr C_2^2.S_4$ ($C_3$)

Order 314928: $C_3^9.C_2^4$ ($S_3$), $C_3^2\wr C_2^2.A_4$ ($C_6$)

Order 104976: $C_3^8.C_2^4$ ($C_3\times S_3$)

Order 78732: $C_3^9.C_2^2$ ($S_4$) x 2, $C_3^9.C_2^2$ ($S_4$)

Order 34992: $C_3^7.C_2^4$ ($C_3^2:C_6$)

Order 26244: $C_3^8.C_2^2$ ($C_3\times S_4$) x 2, $C_3^2\wr C_2^2$ ($C_3\times S_4$)

Order 19683: $C_3^9$ ($C_2^2:S_4$)

Order 11664: $C_3^6.C_2^4$ ($C_3\wr S_3$)

Order 8748: $C_3^7.C_2^2$ ($C_3^2:S_4$) x 2, $C_3^7:C_2^2$ ($C_3^2:S_4$)

Order 6561: $C_3^8$ ($C_3\times C_2^2:S_4$)

Order 2916: $C_3^6.C_2^2$ ($C_3^3:S_4$) x 2, $C_3^3\times C_3^3:C_2^2$ ($C_3^3:S_4$) x 2, $C_3^5:D_6$ ($C_3^3:S_4$)

Order 2187: $C_3^7$ ($C_6^2:S_4$)

Order 972: $C_3^3:C_6^2$ ($C_3^4:S_4$) x 2

Order 729: $C_3^6$ ($C_3^3:C_2^2.S_4$) x 2, $C_3^6$ ($(C_3\times C_6^2):S_4$)

Order 324: $C_3\wr C_2^2$ ($C_3^3.C_6^2:C_3.C_2$) x 2

Order 243: $C_3^5$ ($(C_3^2\times C_6^2):S_4$) x 2

Order 108: $C_3:S_3^2$ ($C_3^4.C_6^2:C_3.C_2$) x 2

Order 81: $C_3^4$ ($(C_3^3\times C_6^2):S_4$) x 2

Order 27: $C_3^3$ ($C_6^2.C_3^4:S_4$) x 2, $C_3^3$ ($C_3^6.C_2^4.S_3$)

Order 9: $C_3^2$ ($C_3^7.C_2^2:S_4$)

Order 3: $C_3$ ($C_3^2\wr C_2^2.S_4$)

Order 1: $C_1$ ($C_3^7.C_6^2:S_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1889568: $C_3^7.C_6^2:S_4$ ($C_1$)

Order 944784: $C_3^7.C_6^2:A_4$ ($C_2$)

Order 629856: $C_3^2\wr C_2^2.S_4$ ($C_3$)

Order 314928: $C_3^9.C_2^4$ ($S_3$), $C_3^2\wr C_2^2.A_4$ ($C_6$)

Order 104976: $C_3^8.C_2^4$ ($C_3\times S_3$)

Order 78732: $C_3^9.C_2^2$ ($S_4$), $C_3^9.C_2^2$ ($S_4$)

Order 34992: $C_3^7.C_2^4$ ($C_3^2:C_6$)

Order 26244: $C_3^8.C_2^2$ ($C_3\times S_4$), $C_3^2\wr C_2^2$ ($C_3\times S_4$)

Order 19683: $C_3^9$ ($C_2^2:S_4$)

Order 11664: $C_3^6.C_2^4$ ($C_3\wr S_3$)

Order 8748: $C_3^7.C_2^2$ ($C_3^2:S_4$), $C_3^7:C_2^2$ ($C_3^2:S_4$)

Order 6561: $C_3^8$ ($C_3\times C_2^2:S_4$)

Order 2916: $C_3^6.C_2^2$ ($C_3^3:S_4$), $C_3^3\times C_3^3:C_2^2$ ($C_3^3:S_4$), $C_3^5:D_6$ ($C_3^3:S_4$)

Order 2187: $C_3^7$ ($C_6^2:S_4$)

Order 972: $C_3^3:C_6^2$ ($C_3^4:S_4$)

Order 729: $C_3^6$ ($C_3^3:C_2^2.S_4$), $C_3^6$ ($(C_3\times C_6^2):S_4$)

Order 324: $C_3\wr C_2^2$ ($C_3^3.C_6^2:C_3.C_2$)

Order 243: $C_3^5$ ($(C_3^2\times C_6^2):S_4$)

Order 108: $C_3:S_3^2$ ($C_3^4.C_6^2:C_3.C_2$)

Order 81: $C_3^4$ ($(C_3^3\times C_6^2):S_4$)

Order 27: $C_3^3$ ($C_3^6.C_2^4.S_3$), $C_3^3$ ($C_6^2.C_3^4:S_4$)

Order 9: $C_3^2$ ($C_3^7.C_2^2:S_4$)

Order 3: $C_3$ ($C_3^2\wr C_2^2.S_4$)

Order 1: $C_1$ ($C_3^7.C_6^2:S_4$)

Series

Derived series

$C_3^7.C_6^2:S_4$

$\rhd$

$C_3^2\wr C_2^2.A_4$

$\rhd$

$C_3^7.C_2^4$

$\rhd$

$C_3^6$

$\rhd$

$C_1$

Chief series

$C_3^7.C_6^2:S_4$

$\rhd$

$C_3^7.C_6^2:A_4$

$\rhd$

$C_3^2\wr C_2^2.A_4$

$\rhd$

$C_3^8.C_2^4$

$\rhd$

$C_3^7.C_2^4$

$\rhd$

$C_3^7.C_2^2$

$\rhd$

$C_3^7$

$\rhd$

$C_3^6$

$\rhd$

$C_3^3$

$\rhd$

$C_1$

Lower central series

$C_3^7.C_6^2:S_4$

$\rhd$

$C_3^2\wr C_2^2.A_4$

Upper central series

$C_1$

$\lhd$

$C_3$

Supergroups

This group is a maximal subgroup of 9 larger groups in the database.

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

Character theory

Complex character table

The $1386 \times 1386$ character table is not available for this group.

Rational character table

The $721 \times 721$ rational character table is not available for this group.