Abstract group 648.23: $\He_3.C_{24}$

• Label: 648.23
• Order: \( 2^{3} \cdot 3^{4} \)
• Exponent: \( 2^{3} \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 3 \)
• $\card{Z(G)}$: \( 2^{2} \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{4} \cdot 3^{4} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \cdot 3 \)
• Perm deg.: $35$
• Trans deg.: $216$
• Rank: $2$

Group information

Description:	$\He_3.C_{24}$
Order:	\(648\)\(\medspace = 2^{3} \cdot 3^{4} \)
Exponent:	\(72\)\(\medspace = 2^{3} \cdot 3^{2} \)
Automorphism group:	$C_6^2.S_3^2$, of order \(1296\)\(\medspace = 2^{4} \cdot 3^{4} \)
Composition factors:	$C_2$ x 3, $C_3$ x 4
Derived length:	$3$

This group is nonabelian and supersolvable (hence solvable and monomial).

Group statistics

Order	1	2	3	4	6	8	9	12	18	24	36	72
Elements	1	1	62	2	62	36	18	124	18	72	36	216	648
Conjugacy classes	1	1	6	2	6	4	6	12	6	8	12	24	88
Divisions	1	1	4	1	4	1	1	4	1	1	1	1	21
Autjugacy classes	1	1	4	1	4	1	1	4	1	1	1	1	21

Dimension	1	2	3	4	6	8	12	18	36	72
Irr. complex chars.	24	12	48	0	4	0	0	0	0	0	88
Irr. rational chars.	2	5	0	5	2	2	1	2	1	1	21

Minimal presentations

Permutation degree:	$35$
Transitive degree:	$216$
Rank:	$2$
Inequivalent generating pairs:	$144$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	3	6	72
Arbitrary	3	6	22

Constructions

Presentation:	$\langle a, b, c, d \mid a^{8}=b^{3}=c^{3}=d^{9}=[a,d]=[c,d]=1, b^{a}=b^{2}, c^{a}=c^{2}d^{6}, c^{b}=cd^{3}, d^{b}=c^{2}d^{7} \rangle$
Permutation group:	Degree $35$ $\langle(3,10)(7,17)(11,20)(12,21)(13,23)(14,24)(18,25)(19,26)(22,27)(28,29,30,32,31,33,34,35) \!\cdots\! \rangle$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(\He_3:C_8)$ $\,\rtimes\,$ $C_3$	$(\He_3:C_3)$ $\,\rtimes\,$ $C_8$	$(C_3\times C_9)$ $\,\rtimes\,$ $(C_3:C_8)$		more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$\He_3$ . $C_{24}$	$(C_3\times C_{36})$ . $S_3$	$(\He_3:C_6)$ . $C_4$	$(C_4\times \He_3)$ . $C_6$	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:	$C_1$
Commutator length:	$1$

Subgroups

There are 206 subgroups in 49 conjugacy classes, 20 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_{12}$	$G/Z \simeq$ $C_3^2:C_6$
Commutator:	$G' \simeq$ $\He_3$	$G/G' \simeq$ $C_{24}$
Frattini:	$\Phi \simeq$ $C_3\times C_{12}$	$G/\Phi \simeq$ $C_3\times S_3$
Fitting:	$\operatorname{Fit} \simeq$ $\He_3:C_{12}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $\He_3.C_{24}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_6$	$G/\operatorname{soc} \simeq$ $C_3^2:C_{12}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_8$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $\He_3:C_3$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

Classes of subgroups up to automorphism

Normal subgroups

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 648: $\He_3.C_{24}$

Order 324: $\He_3:C_{12}$

Order 216: $\He_3:C_8$, $C_3:C_{72}$

Order 162: $\He_3:C_6$

Order 108: $C_4\times \He_3$ x 2, $C_3\times C_{36}$

Order 81: $\He_3:C_3$

Order 72: $C_3:C_{24}$ x 2, $C_{72}$

Order 54: $C_2\times \He_3$ x 2, $C_3\times C_{18}$

Order 36: $C_3\times C_{12}$ x 3, $C_{36}$

Order 27: $\He_3$ x 2, $C_3\times C_9$

Order 24: $C_3:C_8$ x 2, $C_{24}$

Order 18: $C_3\times C_6$ x 3, $C_{18}$

Order 12: $C_{12}$ x 4

Order 9: $C_3^2$ x 3, $C_9$

Order 8: $C_8$

Order 6: $C_6$ x 4

Order 4: $C_4$

Order 3: $C_3$ x 4

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 648: $\He_3.C_{24}$

Order 324: $\He_3:C_{12}$

Order 216: $\He_3:C_8$, $C_3:C_{72}$

Order 162: $\He_3:C_6$

Order 108: $C_4\times \He_3$ x 2, $C_3\times C_{36}$

Order 81: $\He_3:C_3$

Order 72: $C_3:C_{24}$ x 2, $C_{72}$

Order 54: $C_2\times \He_3$ x 2, $C_3\times C_{18}$

Order 36: $C_3\times C_{12}$ x 3, $C_{36}$

Order 27: $\He_3$ x 2, $C_3\times C_9$

Order 24: $C_3:C_8$ x 2, $C_{24}$

Order 18: $C_3\times C_6$ x 3, $C_{18}$

Order 12: $C_{12}$ x 4

Order 9: $C_3^2$ x 3, $C_9$

Order 8: $C_8$

Order 6: $C_6$ x 4

Order 4: $C_4$

Order 3: $C_3$ x 4

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 648: $\He_3.C_{24}$ ($C_1$)

Order 324: $\He_3:C_{12}$ ($C_2$)

Order 216: $\He_3:C_8$ ($C_3$)

Order 162: $\He_3:C_6$ ($C_4$)

Order 108: $C_4\times \He_3$ ($C_6$), $C_3\times C_{36}$ ($S_3$)

Order 81: $\He_3:C_3$ ($C_8$)

Order 54: $C_3\times C_{18}$ ($C_3:C_4$), $C_2\times \He_3$ ($C_{12}$)

Order 36: $C_3\times C_{12}$ ($C_3\times S_3$)

Order 27: $\He_3$ ($C_{24}$), $C_3\times C_9$ ($C_3:C_8$)

Order 18: $C_3\times C_6$ ($C_3:C_{12}$)

Order 12: $C_{12}$ ($C_3^2:C_6$)

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

Order 6: $C_6$ ($C_3^2:C_{12}$)

Order 4: $C_4$ ($\He_3.C_6$)

Order 3: $C_3$ ($C_3^2:C_{24}$)

Order 2: $C_2$ ($\He_3.C_{12}$)

Order 1: $C_1$ ($\He_3.C_{24}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 648: $\He_3.C_{24}$ ($C_1$)

Order 324: $\He_3:C_{12}$ ($C_2$)

Order 216: $\He_3:C_8$ ($C_3$)

Order 162: $\He_3:C_6$ ($C_4$)

Order 108: $C_4\times \He_3$ ($C_6$), $C_3\times C_{36}$ ($S_3$)

Order 81: $\He_3:C_3$ ($C_8$)

Order 54: $C_3\times C_{18}$ ($C_3:C_4$), $C_2\times \He_3$ ($C_{12}$)

Order 36: $C_3\times C_{12}$ ($C_3\times S_3$)

Order 27: $\He_3$ ($C_{24}$), $C_3\times C_9$ ($C_3:C_8$)

Order 18: $C_3\times C_6$ ($C_3:C_{12}$)

Order 12: $C_{12}$ ($C_3^2:C_6$)

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

Order 6: $C_6$ ($C_3^2:C_{12}$)

Order 4: $C_4$ ($\He_3.C_6$)

Order 3: $C_3$ ($C_3^2:C_{24}$)

Order 2: $C_2$ ($\He_3.C_{12}$)

Order 1: $C_1$ ($\He_3.C_{24}$)

Series

Derived series

$\He_3.C_{24}$

$\rhd$

$\He_3$

$\rhd$

$C_3$

$\rhd$

$C_1$

Chief series

$\He_3.C_{24}$

$\rhd$

$\He_3:C_{12}$

$\rhd$

$\He_3:C_6$

$\rhd$

$\He_3:C_3$

$\rhd$

$C_3\times C_9$

$\rhd$

$C_3^2$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$\He_3.C_{24}$

$\rhd$

$\He_3$

Upper central series

$C_1$

$\lhd$

$C_{12}$

Supergroups

This group is a maximal subgroup of 22 larger groups in the database.

This group is a maximal quotient of 14 larger groups in the database.

Character theory

Complex character table

See the $88 \times 88$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $21 \times 21$ rational character table.