Abstract group 1889568.ij: $C_3^9.C_2^4.C_6$

• Label: 1889568.ij
• Order: \( 2^{5} \cdot 3^{10} \)
• Exponent: \( 2^{2} \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2 \cdot 3 \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{8} \cdot 3^{12} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \cdot 3^{2} \)
• Perm deg.: $27$
• Trans deg.: $36$
• Rank: $2$

Group information

Description:	$C_3^9.C_2^4.C_6$
Order:	\(1889568\)\(\medspace = 2^{5} \cdot 3^{10} \)
Exponent:	\(36\)\(\medspace = 2^{2} \cdot 3^{2} \)
Automorphism group:	Group of order \(136048896\)\(\medspace = 2^{8} \cdot 3^{12} \)
Composition factors:	$C_2$ x 5, $C_3$ x 10
Derived length:	$3$

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
Elements	1	3699	43010	78732	580230	606528	157464	419904	1889568
Conjugacy classes	1	4	377	1	652	28	1	4	1068
Divisions	1	4	214	1	349	14	1	2	586
Autjugacy classes	1	4	56	1	73	7	1	1	144

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:	${\langle a, b, c, d, e, f, g, h, i, j \mid a^{6}=b^{6}=c^{6}=d^{6}=e^{6}= \!\cdots\! \rangle}$
Permutation group:	Degree $27$ $\langle(1,3,2)(4,6,5,8,13,14,17,7,11)(9,15,18,10,16,12)(19,21,23)(20,22,25,24,27,26) \!\cdots\! \rangle$
Transitive group:	36T43050	36T43058			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:	$(C_3^9.C_2^3)$ . $A_4$	$C_3^9$ . $(C_2^3:A_4)$	$(C_3^9.C_2^4)$ . $C_6$	$(C_3^8.C_2^4:S_3)$ . $C_3$	all 19

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

Homology

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

Subgroups

There are 21 normal subgroups, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $C_3^9.C_2^4.C_6$
Commutator:	$G' \simeq$ $C_3^9.C_2^4$	$G/G' \simeq$ $C_6$
Frattini:	$\Phi \simeq$ $C_3^2$	$G/\Phi \simeq$ $C_3^7.C_2^3:A_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_3^9$	$G/\operatorname{Fit} \simeq$ $C_2^3:A_4$
Radical:	$R \simeq$ $C_3^9.C_2^4.C_6$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3^7$	$G/\operatorname{soc} \simeq$ $(C_2^2\times C_6^2):C_6$
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 8 (order at least 236196) are shown, as well as all normal subgroups of any index.

Order 1889568: $C_3^9.C_2^4.C_6$

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

Order 629856: $C_3^8.C_2^4:S_3$, $C_3^8.C_2^3:A_4$

Order 472392: $C_3^8.(S_3\times A_4)$

Order 314928: $C_3^9.C_2^4$, $C_3^9.C_2^3.C_2$, $C_3^9.C_2^3.C_2$, $C_3^2\wr C_2^2.A_4$, $C_3^2\wr C_2^2.A_4$

Order 236196: $C_3^3.C_3\wr A_4$ x 2, $C_3^7.C_6^2.C_3$

Order 157464: $C_3^9.C_2^3$

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

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

Order 59049: $C_3^9.C_3$

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

Order 26244: $C_3^2\wr C_2^2$

Order 19683: $C_3^9$

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

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

Order 6561: $C_3^8$

Order 2916: $C_3^5:D_6$

Order 2592: $(C_2\times C_6^3):C_6$

Order 2187: $C_3^7$

Order 729: $C_3^6$

Order 32: $C_2^2\wr C_2$

Order 27: $C_3^3$

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 8 (order at least 236196) are shown, as well as all normal subgroups of any index.

Order 1889568: $C_3^9.C_2^4.C_6$

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

Order 629856: $C_3^8.C_2^4:S_3$, $C_3^8.C_2^3:A_4$

Order 472392: $C_3^8.(S_3\times A_4)$

Order 314928: $C_3^9.C_2^4$, $C_3^9.C_2^3.C_2$, $C_3^9.C_2^3.C_2$, $C_3^2\wr C_2^2.A_4$, $C_3^2\wr C_2^2.A_4$

Order 236196: $C_3^3.C_3\wr A_4$ x 2, $C_3^7.C_6^2.C_3$

Order 157464: $C_3^9.C_2^3$

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

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

Order 59049: $C_3^9.C_3$

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

Order 26244: $C_3^2\wr C_2^2$

Order 19683: $C_3^9$

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

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

Order 6561: $C_3^8$

Order 2916: $C_3^5:D_6$

Order 2592: $(C_2\times C_6^3):C_6$

Order 2187: $C_3^7$

Order 729: $C_3^6$

Order 32: $C_2^2\wr C_2$

Order 27: $C_3^3$

Order 9: $C_3^2$

Order 3: $C_3$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1889568: $C_3^9.C_2^4.C_6$ ($C_1$)

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

Order 629856: $C_3^8.C_2^4:S_3$ ($C_3$)

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

Order 157464: $C_3^9.C_2^3$ ($A_4$)

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

Order 78732: $C_3^9.C_2^2$ ($C_2\times A_4$)

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

Order 26244: $C_3^2\wr C_2^2$ ($S_3\times A_4$)

Order 19683: $C_3^9$ ($C_2^3:A_4$)

Order 11664: $C_3^6.C_2^4$ ($C_3^3:C_6$)

Order 8748: $C_3^7:C_2^2$ ($C_6^2:C_6$)

Order 6561: $C_3^8$ ($(C_2\times D_6):A_4$)

Order 2916: $C_3^5:D_6$ ($(C_3\times C_6^2):C_6$)

Order 2187: $C_3^7$ ($(C_2^2\times C_6^2):C_6$)

Order 729: $C_3^6$ ($(C_2\times C_6^3):C_6$)

Order 27: $C_3^3$ ($C_3^6:C_2^3:A_4$)

Order 9: $C_3^2$ ($C_3^7.C_2^3:A_4$)

Order 3: $C_3$ ($C_3^8.C_2^3:A_4$)

Order 1: $C_1$ ($C_3^9.C_2^4.C_6$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1889568: $C_3^9.C_2^4.C_6$ ($C_1$)

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

Order 629856: $C_3^8.C_2^4:S_3$ ($C_3$)

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

Order 157464: $C_3^9.C_2^3$ ($A_4$)

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

Order 78732: $C_3^9.C_2^2$ ($C_2\times A_4$)

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

Order 26244: $C_3^2\wr C_2^2$ ($S_3\times A_4$)

Order 19683: $C_3^9$ ($C_2^3:A_4$)

Order 11664: $C_3^6.C_2^4$ ($C_3^3:C_6$)

Order 8748: $C_3^7:C_2^2$ ($C_6^2:C_6$)

Order 6561: $C_3^8$ ($(C_2\times D_6):A_4$)

Order 2916: $C_3^5:D_6$ ($(C_3\times C_6^2):C_6$)

Order 2187: $C_3^7$ ($(C_2^2\times C_6^2):C_6$)

Order 729: $C_3^6$ ($(C_2\times C_6^3):C_6$)

Order 27: $C_3^3$ ($C_3^6:C_2^3:A_4$)

Order 9: $C_3^2$ ($C_3^7.C_2^3:A_4$)

Order 3: $C_3$ ($C_3^8.C_2^3:A_4$)

Order 1: $C_1$ ($C_3^9.C_2^4.C_6$)

Series

Derived series

$C_3^9.C_2^4.C_6$

$\rhd$

$C_3^9.C_2^4$

$\rhd$

$C_3^6$

$\rhd$

$C_1$

Chief series

$C_3^9.C_2^4.C_6$

$\rhd$

$C_3^7.C_6^2:A_4$

$\rhd$

$C_3^9.C_2^4$

$\rhd$

$C_3^9.C_2^2$

$\rhd$

$C_3^9$

$\rhd$

$C_3^8$

$\rhd$

$C_3^7$

$\rhd$

$C_3^6$

$\rhd$

$C_1$

Lower central series

$C_3^9.C_2^4.C_6$

$\rhd$

$C_3^9.C_2^4$

Upper central series

$C_1$

Supergroups

This group is a maximal subgroup of 10 larger groups in the database.

This group is a maximal quotient of 3 larger groups in the database.

Character theory

Complex character table

The $1068 \times 1068$ character table is not available for this group.

Rational character table

The $586 \times 586$ rational character table is not available for this group.