Abstract group 5668704.fa: $C_3^9.C_2^4.C_{18}$

• Label: 5668704.fa
• Order: \( 2^{5} \cdot 3^{11} \)
• Exponent: \( 2^{2} \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2 \cdot 3^{2} \)
• $\card{Z(G)}$: \( 3 \)
• $\card{\Aut(G)}$: \( 2^{8} \cdot 3^{13} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \cdot 3^{3} \)
• Perm deg.: $36$
• Trans deg.: $36$
• Rank: $2$

Group information

Description:	$C_3^9.C_2^4.C_{18}$
Order:	\(5668704\)\(\medspace = 2^{5} \cdot 3^{11} \)
Exponent:	\(36\)\(\medspace = 2^{2} \cdot 3^{2} \)
Automorphism group:	Group of order \(408146688\)\(\medspace = 2^{8} \cdot 3^{13} \)
Composition factors:	$C_2$ x 5, $C_3$ x 11
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	59048	78732	1118232	1889568	629856	1889568	5668704
Conjugacy classes	1	4	1127	1	1958	90	5	18	3204
Divisions	1	4	589	1	1002	15	3	3	1618
Autjugacy classes	1	4	115	1	168	8	3	2	302

Minimal presentations

Permutation degree:	$36$
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:	${\langle a, b, c, d, e, f, g, h, i, j \mid a^{18}=b^{6}=c^{6}=d^{6}=e^{6}= \!\cdots\! \rangle}$
Permutation group:	Degree $36$ $\langle(1,10,23)(2,11,24)(3,12,22)(4,17,32,5,16,33,6,18,31)(7,21,28,9,20,30,8,19,29) \!\cdots\! \rangle$
Transitive group:	36T54812				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:	$C_3^6$ . $(C_6^4.C_6)$	$(C_3^{10}.C_2^3)$ . $A_4$	$C_3^{10}$ . $(C_2^3:A_4)$	$(C_3^{10}.C_2^4)$ . $C_6$	all 37

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

Homology

Abelianization:	$C_{18} \simeq C_{2} \times C_{9}$
Schur multiplier:	$C_{2} \times C_{6}$
Commutator length:	$1$

Subgroups

There are 39 normal subgroups, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_3$	$G/Z \simeq$ $C_3^9.C_2^4.C_6$
Commutator:	$G' \simeq$ $C_3^9.C_2^4$	$G/G' \simeq$ $C_{18}$
Frattini:	$\Phi \simeq$ $C_3^3$	$G/\Phi \simeq$ $C_3^7.C_2^3:A_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_3^{10}$	$G/\operatorname{Fit} \simeq$ $C_2^3:A_4$
Radical:	$R \simeq$ $C_3^9.C_2^4.C_{18}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3^8$	$G/\operatorname{soc} \simeq$ $(C_2^2\times C_6^2):C_6$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2\wr C_2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^{10}.C_3$

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 1417176) are shown, as well as all normal subgroups of any index.

Order 5668704: $C_3^9.C_2^4.C_{18}$

Order 2834352: $C_3^8.C_2^4.C_3^2.C_3$

Order 1889568: $C_3^7.C_6^3.C_2^2$, $C_3^8.C_2^4:C_{18}$

Order 1417176: $C_3^7.S_3\wr C_3$

Order 944784: $C_3^{10}.C_2^4$, $C_3^7.C_2^4.C_3^2.C_3$

Order 629856: $C_3^9.C_2^4.C_2$

Order 472392: $C_3^{10}.C_2^3$

Order 314928: $C_3^9.C_2^4$ x 2

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

Order 177147: $C_3^{10}.C_3$

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

Order 104976: $C_3^8.C_2^4$ x 2

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

Order 59049: $C_3^{10}$

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

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

Order 19683: $C_3^9$ x 2

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

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

Order 7776: $C_6^4.C_6$

Order 6561: $C_3^8$ x 2

Order 2916: $C_3^5:D_6$

Order 2187: $C_3^7$ x 2

Order 729: $C_3^6$

Order 81: $C_3^4$

Order 32: $C_2^2\wr C_2$

Order 27: $C_3^3$ x 2

Order 9: $C_3^2$ x 2

Order 3: $C_3$ x 2

Order 1: $C_1$

Classes of subgroups up to automorphism

All subgroups of index up to 4 (order at least 1417176) are shown, as well as all normal subgroups of any index.

Order 5668704: $C_3^9.C_2^4.C_{18}$

Order 2834352: $C_3^8.C_2^4.C_3^2.C_3$

Order 1889568: $C_3^7.C_6^3.C_2^2$, $C_3^8.C_2^4:C_{18}$

Order 1417176: $C_3^7.S_3\wr C_3$

Order 944784: $C_3^{10}.C_2^4$, $C_3^7.C_2^4.C_3^2.C_3$

Order 629856: $C_3^9.C_2^4.C_2$

Order 472392: $C_3^{10}.C_2^3$

Order 314928: $C_3^9.C_2^4$ x 2

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

Order 177147: $C_3^{10}.C_3$

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

Order 104976: $C_3^8.C_2^4$ x 2

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

Order 59049: $C_3^{10}$

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

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

Order 19683: $C_3^9$ x 2

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

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

Order 7776: $C_6^4.C_6$

Order 6561: $C_3^8$ x 2

Order 2916: $C_3^5:D_6$

Order 2187: $C_3^7$ x 2

Order 729: $C_3^6$

Order 81: $C_3^4$

Order 32: $C_2^2\wr C_2$

Order 27: $C_3^3$ x 2

Order 9: $C_3^2$ x 2

Order 3: $C_3$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 5668704: $C_3^9.C_2^4.C_{18}$ ($C_1$)

Order 2834352: $C_3^8.C_2^4.C_3^2.C_3$ ($C_2$)

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

Order 944784: $C_3^{10}.C_2^4$ ($C_6$), $C_3^7.C_2^4.C_3^2.C_3$ ($S_3$)

Order 629856: $C_3^9.C_2^4.C_2$ ($C_9$)

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

Order 314928: $C_3^9.C_2^4$ ($C_3\times S_3$), $C_3^9.C_2^4$ ($C_{18}$)

Order 236196: $C_3^8.C_6^2$ ($C_2\times A_4$)

Order 157464: $C_3^9.C_2^3$ ($C_2^2:C_9$)

Order 104976: $C_3^8.C_2^4$ ($C_3^2:C_6$), $C_3^8.C_2^4$ ($S_3\times C_9$)

Order 78732: $C_3^9.C_2^2$ ($S_3\times A_4$), $C_3^9.C_2^2$ ($C_2^2:C_{18}$)

Order 59049: $C_3^{10}$ ($C_2^3:A_4$)

Order 34992: $C_3^7.C_2^4$ ($C_3^2:C_{18}$), $C_3^7.C_2^4$ ($C_3^3:C_6$)

Order 26244: $C_3^2\wr C_2^2$ ($C_6^2:C_6$), $C_3^2\wr C_2^2$ ($C_6^2.C_6$)

Order 19683: $C_3^9$ ($C_2^4:C_{18}$), $C_3^9$ ($(C_2\times D_6):A_4$)

Order 11664: $C_3^6.C_2^4$ ($C_3^3:C_{18}$)

Order 8748: $C_3^7:C_2^2$ ($(C_3\times C_6^2):C_6$), $C_3^7:C_2^2$ ($C_6^2:C_{18}$)

Order 6561: $C_3^8$ ($(C_2^2\times C_6^2):C_6$), $C_3^8$ ($C_2^4:(S_3\times C_9)$)

Order 2916: $C_3^5:D_6$ ($(C_3\times C_6^2):C_{18}$)

Order 2187: $C_3^7$ ($(C_2\times C_6^3):C_6$), $C_3^7$ ($(C_2^2\times C_6^2):C_{18}$)

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

Order 81: $C_3^4$ ($C_3^6:C_2^3:A_4$)

Order 27: $C_3^3$ ($C_3^7.C_2^3:A_4$), $C_3^3$ ($C_3^6.C_2^4:C_{18}$)

Order 9: $C_3^2$ ($C_3^7.C_2^4:C_{18}$), $C_3^2$ ($C_3^8.C_2^3:A_4$)

Order 3: $C_3$ ($C_3^8.C_2^4:C_{18}$), $C_3$ ($C_3^9.C_2^4.C_6$)

Order 1: $C_1$ ($C_3^9.C_2^4.C_{18}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 5668704: $C_3^9.C_2^4.C_{18}$ ($C_1$)

Order 2834352: $C_3^8.C_2^4.C_3^2.C_3$ ($C_2$)

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

Order 944784: $C_3^{10}.C_2^4$ ($C_6$), $C_3^7.C_2^4.C_3^2.C_3$ ($S_3$)

Order 629856: $C_3^9.C_2^4.C_2$ ($C_9$)

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

Order 314928: $C_3^9.C_2^4$ ($C_3\times S_3$), $C_3^9.C_2^4$ ($C_{18}$)

Order 236196: $C_3^8.C_6^2$ ($C_2\times A_4$)

Order 157464: $C_3^9.C_2^3$ ($C_2^2:C_9$)

Order 104976: $C_3^8.C_2^4$ ($C_3^2:C_6$), $C_3^8.C_2^4$ ($S_3\times C_9$)

Order 78732: $C_3^9.C_2^2$ ($S_3\times A_4$), $C_3^9.C_2^2$ ($C_2^2:C_{18}$)

Order 59049: $C_3^{10}$ ($C_2^3:A_4$)

Order 34992: $C_3^7.C_2^4$ ($C_3^2:C_{18}$), $C_3^7.C_2^4$ ($C_3^3:C_6$)

Order 26244: $C_3^2\wr C_2^2$ ($C_6^2:C_6$), $C_3^2\wr C_2^2$ ($C_6^2.C_6$)

Order 19683: $C_3^9$ ($C_2^4:C_{18}$), $C_3^9$ ($(C_2\times D_6):A_4$)

Order 11664: $C_3^6.C_2^4$ ($C_3^3:C_{18}$)

Order 8748: $C_3^7:C_2^2$ ($(C_3\times C_6^2):C_6$), $C_3^7:C_2^2$ ($C_6^2:C_{18}$)

Order 6561: $C_3^8$ ($(C_2^2\times C_6^2):C_6$), $C_3^8$ ($C_2^4:(S_3\times C_9)$)

Order 2916: $C_3^5:D_6$ ($(C_3\times C_6^2):C_{18}$)

Order 2187: $C_3^7$ ($(C_2\times C_6^3):C_6$), $C_3^7$ ($(C_2^2\times C_6^2):C_{18}$)

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

Order 81: $C_3^4$ ($C_3^6:C_2^3:A_4$)

Order 27: $C_3^3$ ($C_3^7.C_2^3:A_4$), $C_3^3$ ($C_3^6.C_2^4:C_{18}$)

Order 9: $C_3^2$ ($C_3^7.C_2^4:C_{18}$), $C_3^2$ ($C_3^8.C_2^3:A_4$)

Order 3: $C_3$ ($C_3^8.C_2^4:C_{18}$), $C_3$ ($C_3^9.C_2^4.C_6$)

Order 1: $C_1$ ($C_3^9.C_2^4.C_{18}$)

Series

Derived series

$C_3^9.C_2^4.C_{18}$

$\rhd$

$C_3^9.C_2^4$

$\rhd$

$C_3^6$

$\rhd$

$C_1$

Chief series

$C_3^9.C_2^4.C_{18}$

$\rhd$

$C_3^8.C_2^4.C_3^2.C_3$

$\rhd$

$C_3^{10}.C_2^4$

$\rhd$

$C_3^9.C_2^4$

$\rhd$

$C_3^9.C_2^2$

$\rhd$

$C_3^9$

$\rhd$

$C_3^8$

$\rhd$

$C_3^7$

$\rhd$

$C_3^6$

$\rhd$

$C_1$

Lower central series

$C_3^9.C_2^4.C_{18}$

$\rhd$

$C_3^9.C_2^4$

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 $3204 \times 3204$ character table is not available for this group.

Rational character table

The $1618 \times 1618$ rational character table is not available for this group.