Abstract group 17006112.cy: $C_3^7.C_6^4:C_6$

• Label: 17006112.cy
• Order: \( 2^{5} \cdot 3^{12} \)
• Exponent: \( 2^{2} \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2 \cdot 3^{2} \)
• $\card{Z(G)}$: 3
• $\card{\Aut(G)}$: \( 2^{9} \cdot 3^{15} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \cdot 3^{4} \)
• Perm deg.: not computed
• Trans deg.: $36$
• Rank: $2$

Group information

Description:	$C_3^7.C_6^4:C_6$
Order:	\(17006112\)\(\medspace = 2^{5} \cdot 3^{12} \)
Exponent:	\(36\)\(\medspace = 2^{2} \cdot 3^{2} \)
Automorphism group:	Group of order \(7346640384\)\(\medspace = 2^{9} \cdot 3^{15} \)
Composition factors:	$C_2$ x 5, $C_3$ x 12
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	387098	78732	5251662	5458752	2047032	3779136	17006112
Conjugacy classes	1	4	3329	1	5876	84	17	12	9324
Divisions	1	4	1690	1	2961	42	9	6	4714

Minimal presentations

Permutation degree:	not computed
Transitive degree:	$36$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of linear representations for this group have not been computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g, h, i, j, k, l \mid b^{6}=c^{6}=d^{6}=e^{6}= \!\cdots\! \rangle}$
Permutation group:	Degree $36$ $\langle(1,13,25,2,15,27,3,14,26)(4,17,32,5,16,33,6,18,31)(7,20,29,8,21,30,9,19,28) \!\cdots\! \rangle$
Transitive group:	36T64614				more information
Direct product:	not computed
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$C_3^7$ . $(C_6^4:C_6)$	$C_3^7$ . $(C_6^4:C_6)$	$(C_3^{11}.C_2^3)$ . $A_4$	$C_3^{11}$ . $(C_2^3:A_4)$	all 56

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

Homology

Abelianization:	$C_{3} \times C_{6} \simeq C_{2} \times C_{3}^{2}$
Schur multiplier:	$C_{3} \times C_{6}^{2}$
Commutator length:	not computed

Subgroups

There are 66 normal subgroups (57 characteristic).

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

Special subgroups

Center:	a subgroup isomorphic to $C_3$
Commutator:	not computed
Frattini:	a subgroup isomorphic to $C_3^3$
Fitting:	not computed
Radical:	not computed
Socle:	not computed
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^{11}.C_3$

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 48 (order at least 354294) are shown, as well as all normal subgroups of any index.

Order 17006112: $C_3^7.C_6^4:C_6$

Order 8503056: $C_3^9.C_2^4.C_3^3$

Order 5668704: $C_3^6.C_6^4:C_6$ x 3, $C_3^8.C_6^3.C_2^2$, $C_3^6.C_6^4:C_6$

Order 4251528: $C_3^8.S_3\wr C_3$

Order 2834352: $C_3^8.C_2^4.C_3^3$ x 8, $C_3^{11}.C_2^4$, $C_3^9.C_6^2.C_2^2$, $C_3^8.C_6^3.C_2$

Order 2125764: $C_3^9.C_6^2.C_3$ x 2, $C_3^8.C_6^2.C_3^2$

Order 1889568: $C_3^7.C_6^3.C_2^2$ x 7, $C_3^2\wr C_2^2.(C_6\times A_4)$ x 3, $C_3^9.C_2^4.C_6$ x 3, $C_3^2\wr C_2^2.(C_6\times A_4)$

Order 1417176: $C_3^8.C_6^3$ x 3, $C_3^7.S_3\wr C_3$ x 3, $C_3^{11}.C_4.C_2$, $C_3^{11}.C_2^3$, $C_3^9.C_6^2.C_2$, $C_3^8.C_6.C_6^2$, $C_3^7.S_3\wr C_3$

Order 1062882: $C_3^{11}.C_6$

Order 944784: $C_3^{10}.C_2^4$ x 26, $C_3^7.C_6^3.C_2$ x 20, $C_3^7.C_6^2:A_4$ x 15, $C_3^8.C_6^2.C_2^2$ x 13, $C_3^7.C_2^4.C_3^3$ x 8, $C_3^7.C_2^4.C_3^2.C_3$ x 3, $C_3^9.C_6.C_2^3$ x 2

Order 708588: $C_3^7.C_6^2.C_3^2$ x 39, $C_3^8.C_6^2.C_3$ x 20, $C_3^9.C_6^2$ x 10, $C_3^{11}.C_4$

Order 629856: $C_3^6.C_6^3.C_2^2$ x 24, $C_3^8.C_2^3:A_4$ x 3, $C_3^8.C_2^3:A_4$ x 3, $C_3^9.C_2^4.C_2$, $C_3^8.C_2^3:A_4$

Order 531441: $C_3^{11}.C_3$

Order 472392: $C_3^7.C_6^3$ x 449, $C_3^8.C_6^2.C_2$ x 30, $C_3^7.C_6.C_6^2$ x 27, $C_3^{10}.C_4.C_2$ x 15, $C_3^{10}.C_2^3$ x 8, $C_3^8.(C_6\times A_4)$ x 3, $C_3^8.(S_3\times A_4)$ x 3, $C_3^9.C_6.C_2^2$, $C_3\times C_3^9.C_2^3$, $C_3^8.(C_6\times A_4)$

Order 354294: $C_3^{10}.C_6$ x 16, $C_3^{11}.C_2$ x 4

Order 314928: $C_3^9.C_2^4$ x 3

Order 236196: $C_3^8.C_6^2$ x 2

Order 177147: $C_3^{11}$

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

Order 104976: $C_3^8.C_2^4$ x 3

Order 78732: $C_3^9.C_2^2$ x 3

Order 59049: $C_3^{10}$ x 2

Order 34992: $C_3^7.C_2^4$ x 2

Order 26244: $C_3^2\wr C_2^2$ x 3

Order 19683: $C_3^9$ x 3

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

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

Order 6561: $C_3^8$ x 3

Order 2916: $C_3^5:D_6$

Order 2187: $C_3^7$ x 2

Order 729: $C_3^6$

Order 243: $C_3^5$

Order 81: $C_3^4$ x 2

Order 27: $C_3^3$ x 3

Order 9: $C_3^2$ x 3

Order 3: $C_3$ x 2

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 17006112: $C_3^7.C_6^4:C_6$ ($C_1$)

Order 8503056: $C_3^9.C_2^4.C_3^3$ ($C_2$)

Order 5668704: $C_3^6.C_6^4:C_6$ ($C_3$) x 3, $C_3^8.C_6^3.C_2^2$ ($C_3$)

Order 2834352: $C_3^8.C_2^4.C_3^3$ ($C_6$) x 3, $C_3^{11}.C_2^4$ ($C_6$), $C_3^8.C_2^4.C_3^3$ ($S_3$)

Order 1889568: $C_3^7.C_6^3.C_2^2$ ($C_3^2$)

Order 1417176: $C_3^{11}.C_2^3$ ($A_4$)

Order 944784: $C_3^7.C_2^4.C_3^3$ ($C_3\times S_3$) x 3, $C_3^{10}.C_2^4$ ($C_3\times C_6$), $C_3^{10}.C_2^4$ ($C_3\times S_3$)

Order 708588: $C_3^9.C_6^2$ ($C_2\times A_4$)

Order 629856: $C_3^9.C_2^4.C_2$ ($\He_3$)

Order 472392: $C_3^{10}.C_2^3$ ($C_3\times A_4$)

Order 314928: $C_3^9.C_2^4$ ($C_3^2:C_6$), $C_3^9.C_2^4$ ($S_3\times C_3^2$), $C_3^9.C_2^4$ ($C_2\times \He_3$)

Order 236196: $C_3^8.C_6^2$ ($C_6\times A_4$), $C_3^8.C_6^2$ ($S_3\times A_4$)

Order 177147: $C_3^{11}$ ($C_2^3:A_4$)

Order 157464: $C_3^9.C_2^3$ ($C_3^2:A_4$)

Order 104976: $C_3^8.C_2^4$ ($S_3\times \He_3$), $C_3^8.C_2^4$ ($C_3^3:C_6$), $C_3^8.C_2^4$ ($C_3^3:C_6$)

Order 78732: $C_3^9.C_2^2$ ($C_6^2:C_6$), $C_3^9.C_2^2$ ($C_6^2:C_6$), $C_3^9.C_2^2$ ($C_6^2:C_6$)

Order 59049: $C_3^{10}$ ($C_3\times C_2^3:A_4$), $C_3^{10}$ ($(C_2\times D_6):A_4$)

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

Order 26244: $C_3^2\wr C_2^2$ ($S_3\times C_3^2:A_4$), $C_3^2\wr C_2^2$ ($C_3\times C_6^2:C_6$), $C_3^2\wr C_2^2$ ($(C_3\times C_6^2):C_6$)

Order 19683: $C_3^9$ ($(C_2^2\times C_6^2):C_6$), $C_3^9$ ($(C_6\times D_6):A_4$), $C_3^9$ ($(C_2\times C_6^2):A_4$)

Order 11664: $C_3^6.C_2^4$ ($C_3^5:C_6$)

Order 8748: $C_3^7:C_2^2$ ($(C_3^2\times C_6^2):C_6$), $C_3^7:C_2^2$ ($C_3^3:(C_6\times A_4)$)

Order 6561: $C_3^8$ ($(C_2\times C_6^3):C_6$), $C_3^8$ ($(C_2\times C_6^3):C_6$), $C_3^8$ ($(C_2\times C_6^3):C_6$)

Order 2916: $C_3^5:D_6$ ($(C_3^3\times C_6^2):C_6$)

Order 2187: $C_3^7$ ($C_6^4:C_6$), $C_3^7$ ($C_6^4:C_6$)

Order 729: $C_3^6$ ($C_6^4.(C_3\times C_6)$)

Order 243: $C_3^5$ ($C_3^6:C_2^3:A_4$)

Order 81: $C_3^4$ ($C_3^7.C_2^3:A_4$), $C_3^4$ ($C_3^7.C_2^3:A_4$)

Order 27: $C_3^3$ ($C_3^8.C_2^3:A_4$), $C_3^3$ ($C_3^8.C_2^3:A_4$), $C_3^3$ ($C_3^8.C_2^3:A_4$)

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

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

Order 1: $C_1$ ($C_3^7.C_6^4:C_6$)

Normal subgroups up to automorphism (quotient in parentheses)

not computed

Series

Derived series	not computed
Chief series	not computed
Lower central series	not computed
Upper central series	not computed

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 $9324 \times 9324$ character table is not available for this group.

Rational character table

The $4714 \times 4714$ rational character table is not available for this group.