Abstract group 17496.gg: $C_3^6.C_{24}$

• Label: 17496.gg
• Order: \( 2^{3} \cdot 3^{7} \)
• Exponent: \( 2^{3} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 3 \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{8} \cdot 3^{8} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \cdot 3 \)
• Perm deg.: $36$
• Trans deg.: $36$
• Rank: $2$

Group information

Description:	$C_3^6.C_{24}$
Order:	\(17496\)\(\medspace = 2^{3} \cdot 3^{7} \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Automorphism group:	$C_3^6.C_{12}^2.C_2^4$, of order \(1679616\)\(\medspace = 2^{8} \cdot 3^{8} \)
Composition factors:	$C_2$ x 3, $C_3$ x 7
Derived length:	$2$

This group is nonabelian and metabelian (hence solvable). Whether it is monomial has not been computed.

Group statistics

Order	1	2	3	4	6	8	12	24
Elements	1	729	2186	1458	1458	2916	2916	5832	17496
Conjugacy classes	1	1	59	2	2	4	4	8	81
Divisions	1	1	48	1	1	1	1	1	55
Autjugacy classes	1	1	9	1	1	2	1	2	18

Dimension	1	2	4	8	16	24
Irr. complex chars.	24	0	0	30	0	27	81
Irr. rational chars.	2	3	2	11	10	27	55

Minimal presentations

Permutation degree:	$36$
Transitive degree:	$36$
Rank:	$2$
Inequivalent generating pairs:	$96$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	24	24	24
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g \mid a^{24}=b^{3}=c^{3}=d^{3}=e^{3}=f^{3}=g^{3}= \!\cdots\! \rangle}$
Permutation group:	Degree $36$ $\langle(1,8,15,21,26,33,2,9,14,20,25,31)(3,7,13,19,27,32)(4,11,18,24,28,36,6,10,17,23,29,34) \!\cdots\! \rangle$
Transitive group:	36T10379	36T10754			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:	$C_3^5$ . $F_9$	$C_3^6$ . $C_{24}$	$(C_3^6.C_8)$ . $C_3$	$(C_3^6.C_6)$ . $C_4$	all 15

Elements of the group are displayed as words in the presentation generators from the presentation above.

Homology

Abelianization:	$C_{24} \simeq C_{8} \times C_{3}$
Schur multiplier:	$C_{3}^{2}$
Commutator length:	$1$

Subgroups

There are 2157248 subgroups in 12000 conjugacy classes, 16 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $C_3^6.C_{24}$
Commutator:	$G' \simeq$ $C_3^6$	$G/G' \simeq$ $C_{24}$
Frattini:	$\Phi \simeq$ $C_3^2$	$G/\Phi \simeq$ $C_3^4:C_{24}$
Fitting:	$\operatorname{Fit} \simeq$ $C_3^2\times C_3^4:C_3$	$G/\operatorname{Fit} \simeq$ $C_8$
Radical:	$R \simeq$ $C_3^6.C_{24}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3^4$	$G/\operatorname{soc} \simeq$ $C_3\times F_9$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_8$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2\times C_3^4:C_3$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

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Classes of subgroups up to automorphism

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Normal subgroups

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Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 17496: $C_3^6.C_{24}$

Order 8748: $C_3^6.C_{12}$

Order 5832: $C_3^6.C_8$

Order 4374: $C_3^6.C_6$

Order 2916: $C_3^6.C_4$

Order 2187: $C_3^2\times C_3^4:C_3$

Order 1944: $C_3^4:C_{24}$, $C_3^4:C_{24}$

Order 1458: $C_3^5:C_6$ x 9, $C_3^5:S_3$, $C_3^5:C_6$

Order 972: $C_3^4:C_{12}$ x 5, $C_3^4:C_{12}$

Order 729: $C_3^5:C_3$ x 18, $C_3^3\times \He_3$ x 2, $C_3^6$

Order 648: $C_3^2:F_9$ x 4, $C_3^2:F_9$

Order 486: $C_3^4:C_6$ x 39, $C_3^4:S_3$ x 37, $C_3^4:C_6$ x 23, $C_3^4:C_6$

Order 324: $C_3^4:C_4$ x 21

Order 243: $C_3^2\times \He_3$ x 114, $C_3^4:C_3$ x 104, $C_3^5$ x 39

Order 216: $C_3\times F_9$ x 2

Order 162: $C_3^3:S_3$ x 969, $C_3^3:C_6$ x 117, $C_3^3:C_6$ x 10

Order 108: $C_3^2:C_{12}$ x 6

Order 81: $C_3^4$ x 1028, $C_3\times \He_3$ x 666

Order 72: $F_9$ x 5

Order 54: $C_3^2:S_3$ x 2908, $C_3^2:C_6$ x 36, $C_3^2:C_6$ x 27

Order 36: $C_3^2:C_4$ x 21

Order 27: $C_3^3$ x 3191, $\He_3$ x 378

Order 24: $C_{24}$

Order 18: $C_3:S_3$ x 969, $C_3\times S_3$ x 10

Order 12: $C_{12}$

Order 9: $C_3^2$ x 1136

Order 8: $C_8$

Order 6: $S_3$ x 37, $C_6$

Order 4: $C_4$

Order 3: $C_3$ x 48

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 17496: $C_3^6.C_{24}$

Order 8748: $C_3^6.C_{12}$

Order 5832: $C_3^6.C_8$

Order 4374: $C_3^6.C_6$

Order 2916: $C_3^6.C_4$

Order 2187: $C_3^2\times C_3^4:C_3$

Order 1944: $C_3^4:C_{24}$, $C_3^4:C_{24}$

Order 1458: $C_3^5:C_6$ x 2, $C_3^5:S_3$, $C_3^5:C_6$

Order 972: $C_3^4:C_{12}$ x 2, $C_3^4:C_{12}$

Order 729: $C_3^5:C_3$ x 4, $C_3^3\times \He_3$ x 2, $C_3^6$

Order 648: $C_3^2:F_9$ x 2, $C_3^2:F_9$

Order 486: $C_3^4:S_3$ x 5, $C_3^4:C_6$ x 5, $C_3^4:C_6$ x 4, $C_3^4:C_6$

Order 324: $C_3^4:C_4$ x 5

Order 243: $C_3^4:C_3$ x 13, $C_3^2\times \He_3$ x 12, $C_3^5$ x 7

Order 216: $C_3\times F_9$ x 2

Order 162: $C_3^3:S_3$ x 39, $C_3^3:C_6$ x 6, $C_3^3:C_6$ x 3

Order 108: $C_3^2:C_{12}$ x 3

Order 81: $C_3^4$ x 51, $C_3\times \He_3$ x 30

Order 72: $F_9$ x 3

Order 54: $C_3^2:S_3$ x 65, $C_3^2:C_6$ x 8, $C_3^2:C_6$ x 2

Order 36: $C_3^2:C_4$ x 5

Order 27: $C_3^3$ x 98, $\He_3$ x 14

Order 24: $C_{24}$

Order 18: $C_3:S_3$ x 39, $C_3\times S_3$ x 3

Order 12: $C_{12}$

Order 9: $C_3^2$ x 59

Order 8: $C_8$

Order 6: $S_3$ x 5, $C_6$

Order 4: $C_4$

Order 3: $C_3$ x 9

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 17496: $C_3^6.C_{24}$ ($C_1$)

Order 8748: $C_3^6.C_{12}$ ($C_2$)

Order 5832: $C_3^6.C_8$ ($C_3$)

Order 4374: $C_3^6.C_6$ ($C_4$)

Order 2916: $C_3^6.C_4$ ($C_6$)

Order 2187: $C_3^2\times C_3^4:C_3$ ($C_8$)

Order 1458: $C_3^5:S_3$ ($C_{12}$)

Order 729: $C_3^6$ ($C_{24}$)

Order 243: $C_3^5$ ($F_9$), $C_3^4:C_3$ ($F_9$)

Order 81: $C_3^4$ ($C_3\times F_9$) x 2

Order 27: $C_3^3$ ($C_3^2:F_9$)

Order 9: $C_3^2$ ($C_3^4:C_{24}$), $C_3^2$ ($C_3^4:C_{24}$)

Order 1: $C_1$ ($C_3^6.C_{24}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 17496: $C_3^6.C_{24}$ ($C_1$)

Order 8748: $C_3^6.C_{12}$ ($C_2$)

Order 5832: $C_3^6.C_8$ ($C_3$)

Order 4374: $C_3^6.C_6$ ($C_4$)

Order 2916: $C_3^6.C_4$ ($C_6$)

Order 2187: $C_3^2\times C_3^4:C_3$ ($C_8$)

Order 1458: $C_3^5:S_3$ ($C_{12}$)

Order 729: $C_3^6$ ($C_{24}$)

Order 243: $C_3^5$ ($F_9$), $C_3^4:C_3$ ($F_9$)

Order 81: $C_3^4$ ($C_3\times F_9$) x 2

Order 27: $C_3^3$ ($C_3^2:F_9$)

Order 9: $C_3^2$ ($C_3^4:C_{24}$), $C_3^2$ ($C_3^4:C_{24}$)

Order 1: $C_1$ ($C_3^6.C_{24}$)

Series

Derived series

$C_3^6.C_{24}$

$\rhd$

$C_3^6$

$\rhd$

$C_1$

Chief series

$C_3^6.C_{24}$

$\rhd$

$C_3^6.C_{12}$

$\rhd$

$C_3^6.C_6$

$\rhd$

$C_3^2\times C_3^4:C_3$

$\rhd$

$C_3^6$

$\rhd$

$C_3^4$

$\rhd$

$C_3^2$

$\rhd$

$C_3^2$

$\rhd$

$C_1$

Lower central series

$C_3^6.C_{24}$

$\rhd$

$C_3^6$

Upper central series

$C_1$

Supergroups

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

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

Character theory

Complex character table

See the $81 \times 81$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $55 \times 55$ rational character table.