Abstract group 1296.292: $\He_3.D_{24}$

• Label: 1296.292
• Order: \( 2^{4} \cdot 3^{4} \)
• Exponent: \( 2^{3} \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 3 \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{7} \cdot 3^{5} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \cdot 3 \)
• Perm deg.: $35$
• Trans deg.: $216$
• Rank: $2$

Group information

Description:	$\He_3.D_{24}$
Order:	\(1296\)\(\medspace = 2^{4} \cdot 3^{4} \)
Exponent:	\(72\)\(\medspace = 2^{3} \cdot 3^{2} \)
Automorphism group:	$\He_3.C_3.C_{12}.C_2^5$, of order \(31104\)\(\medspace = 2^{7} \cdot 3^{5} \)
Composition factors:	$C_2$ x 4, $C_3$ x 4
Derived length:	$2$

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

Group statistics

Order	1	2	3	4	6	8	9	12	18	24	36	72
Elements	1	217	26	2	458	4	54	52	54	104	108	216	1296
Conjugacy classes	1	3	4	1	8	2	5	6	5	12	10	20	77
Divisions	1	3	3	1	5	1	2	3	2	3	2	2	28
Autjugacy classes	1	2	3	1	4	1	2	3	2	3	2	2	26

Dimension	1	2	4	6	8	12	16	18	24	36	72
Irr. complex chars.	12	33	0	32	0	0	0	0	0	0	0	77
Irr. rational chars.	4	7	5	2	3	1	1	2	1	1	1	28

Minimal presentations

Permutation degree:	$35$
Transitive degree:	$216$
Rank:	$2$
Inequivalent generating pairs:	$12$

Minimal degrees of faithful linear representations

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

Constructions

Presentation:	$\langle a, b, c \mid a^{6}=b^{9}=c^{24}=[b,c]=1, b^{a}=b^{8}c^{16}, c^{a}=b^{3}c^{23} \rangle$
Permutation group:	Degree $35$ $\langle(2,5)(3,9)(4,11)(6,12)(7,18)(8,20)(10,21)(13,16)(14,24)(15,26)(17,25)(19,27) \!\cdots\! \rangle$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_3:D_{72})$ $\,\rtimes\,$ $C_3$	$(C_3:D_{36})$ $\,\rtimes\,$ $C_6$ (2)	$(C_3\times C_{72})$ $\,\rtimes\,$ $C_6$	$C_8$ $\,\rtimes\,$ $(\He_3.S_3)$	all 8
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$\He_3$ . $D_{24}$	$(C_8\times \He_3)$ . $S_3$	$(C_6.\He_3)$ . $D_4$	$(C_4\times \He_3)$ . $D_6$	all 18

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

Homology

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

Subgroups

There are 1868 subgroups in 118 conjugacy classes, 30 normal (26 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $\He_3.D_{12}$
Commutator:	$G' \simeq$ $C_3\times C_{36}$	$G/G' \simeq$ $C_2\times C_6$
Frattini:	$\Phi \simeq$ $C_3\times C_{12}$	$G/\Phi \simeq$ $C_6\times S_3$
Fitting:	$\operatorname{Fit} \simeq$ $C_{24}.\He_3$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $\He_3.D_{24}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_6$	$G/\operatorname{soc} \simeq$ $C_3^2:D_{12}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_8$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3.\He_3$

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

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 1296: $\He_3.D_{24}$

Order 648: $\He_3.D_{12}$ x 2, $C_{24}.\He_3$

Order 432: $C_3:D_{72}$, $C_3^2:D_{24}$

Order 324: $\He_3.D_6$ x 2, $C_{12}.\He_3$

Order 216: $C_3:D_{36}$ x 2, $C_3^2:D_{12}$ x 2, $C_9:C_{24}$, $C_8\times \He_3$, $C_3\times C_{72}$

Order 162: $\He_3.S_3$ x 2, $C_6.\He_3$

Order 144: $C_3:D_{24}$, $D_{72}$, $C_3\times D_{24}$

Order 108: $C_3:D_{18}$ x 2, $C_3^2:D_6$ x 2, $C_9:C_{12}$, $C_4\times \He_3$, $C_3\times C_{36}$

Order 81: $C_3.\He_3$

Order 72: $D_{36}$ x 2, $C_3:D_{12}$ x 2, $C_3\times D_{12}$ x 2, $C_{72}$ x 2, $C_3\times C_{24}$ x 2

Order 54: $C_3:D_9$ x 2, $C_3^2:C_6$ x 2, $C_3\times C_{18}$, $C_9:C_6$, $C_2\times \He_3$

Order 48: $D_{24}$ x 2, $C_3\times D_8$

Order 36: $C_3\times C_{12}$ x 2, $D_{18}$ x 2, $C_{36}$ x 2, $C_6:S_3$ x 2, $C_6\times S_3$ x 2

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

Order 24: $D_{12}$ x 4, $C_{24}$ x 3, $C_3\times D_4$ x 2

Order 18: $C_3\times C_6$ x 2, $C_3:S_3$ x 2, $C_3\times S_3$ x 2, $C_{18}$ x 2, $D_9$ x 2

Order 16: $D_8$

Order 12: $D_6$ x 4, $C_{12}$ x 3, $C_2\times C_6$ x 2

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

Order 8: $D_4$ x 2, $C_8$

Order 6: $C_6$ x 5, $S_3$ x 4

Order 4: $C_2^2$ x 2, $C_4$

Order 3: $C_3$ x 3

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1296: $\He_3.D_{24}$

Order 648: $C_{24}.\He_3$, $\He_3.D_{12}$

Order 432: $C_3:D_{72}$, $C_3^2:D_{24}$

Order 324: $\He_3.D_6$, $C_{12}.\He_3$

Order 216: $C_3:D_{36}$, $C_3^2:D_{12}$, $C_9:C_{24}$, $C_8\times \He_3$, $C_3\times C_{72}$

Order 162: $C_6.\He_3$, $\He_3.S_3$

Order 144: $C_3:D_{24}$, $D_{72}$, $C_3\times D_{24}$

Order 108: $C_3:D_{18}$, $C_3^2:D_6$, $C_9:C_{12}$, $C_4\times \He_3$, $C_3\times C_{36}$

Order 81: $C_3.\He_3$

Order 72: $C_{72}$ x 2, $C_3\times C_{24}$ x 2, $D_{36}$, $C_3:D_{12}$, $C_3\times D_{12}$

Order 54: $C_3\times C_{18}$, $C_3:D_9$, $C_3^2:C_6$, $C_9:C_6$, $C_2\times \He_3$

Order 48: $D_{24}$ x 2, $C_3\times D_8$

Order 36: $C_3\times C_{12}$ x 2, $C_{36}$ x 2, $D_{18}$, $C_6:S_3$, $C_6\times S_3$

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

Order 24: $C_{24}$ x 3, $D_{12}$ x 2, $C_3\times D_4$

Order 18: $C_3\times C_6$ x 2, $C_{18}$ x 2, $C_3:S_3$, $C_3\times S_3$, $D_9$

Order 16: $D_8$

Order 12: $C_{12}$ x 3, $D_6$ x 2, $C_2\times C_6$

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

Order 8: $D_4$, $C_8$

Order 6: $C_6$ x 4, $S_3$ x 2

Order 4: $C_2^2$, $C_4$

Order 3: $C_3$ x 3

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1296: $\He_3.D_{24}$ ($C_1$)

Order 648: $\He_3.D_{12}$ ($C_2$) x 2, $C_{24}.\He_3$ ($C_2$)

Order 432: $C_3:D_{72}$ ($C_3$)

Order 324: $C_{12}.\He_3$ ($C_2^2$)

Order 216: $C_3:D_{36}$ ($C_6$) x 2, $C_8\times \He_3$ ($S_3$), $C_3\times C_{72}$ ($C_6$)

Order 162: $C_6.\He_3$ ($D_4$)

Order 108: $C_4\times \He_3$ ($D_6$), $C_3\times C_{36}$ ($C_2\times C_6$)

Order 81: $C_3.\He_3$ ($D_8$)

Order 72: $C_3\times C_{24}$ ($C_3\times S_3$)

Order 54: $C_3\times C_{18}$ ($C_3\times D_4$), $C_2\times \He_3$ ($D_{12}$)

Order 36: $C_3\times C_{12}$ ($C_6\times S_3$)

Order 27: $\He_3$ ($D_{24}$), $C_3\times C_9$ ($C_3\times D_8$)

Order 24: $C_{24}$ ($C_3^2:C_6$)

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

Order 12: $C_{12}$ ($C_3^2:D_6$)

Order 9: $C_3^2$ ($C_3\times D_{24}$)

Order 8: $C_8$ ($\He_3.S_3$)

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

Order 4: $C_4$ ($\He_3.D_6$)

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

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

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

Normal subgroups up to automorphism (quotient in parentheses)

Order 1296: $\He_3.D_{24}$ ($C_1$)

Order 648: $C_{24}.\He_3$ ($C_2$), $\He_3.D_{12}$ ($C_2$)

Order 432: $C_3:D_{72}$ ($C_3$)

Order 324: $C_{12}.\He_3$ ($C_2^2$)

Order 216: $C_3:D_{36}$ ($C_6$), $C_8\times \He_3$ ($S_3$), $C_3\times C_{72}$ ($C_6$)

Order 162: $C_6.\He_3$ ($D_4$)

Order 108: $C_4\times \He_3$ ($D_6$), $C_3\times C_{36}$ ($C_2\times C_6$)

Order 81: $C_3.\He_3$ ($D_8$)

Order 72: $C_3\times C_{24}$ ($C_3\times S_3$)

Order 54: $C_3\times C_{18}$ ($C_3\times D_4$), $C_2\times \He_3$ ($D_{12}$)

Order 36: $C_3\times C_{12}$ ($C_6\times S_3$)

Order 27: $\He_3$ ($D_{24}$), $C_3\times C_9$ ($C_3\times D_8$)

Order 24: $C_{24}$ ($C_3^2:C_6$)

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

Order 12: $C_{12}$ ($C_3^2:D_6$)

Order 9: $C_3^2$ ($C_3\times D_{24}$)

Order 8: $C_8$ ($\He_3.S_3$)

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

Order 4: $C_4$ ($\He_3.D_6$)

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

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

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

Series

Derived series

$\He_3.D_{24}$

$\rhd$

$C_3\times C_{36}$

$\rhd$

$C_1$

Chief series

$\He_3.D_{24}$

$\rhd$

$\He_3.D_{12}$

$\rhd$

$C_{12}.\He_3$

$\rhd$

$C_3\times C_{36}$

$\rhd$

$C_3\times C_{18}$

$\rhd$

$C_3\times C_9$

$\rhd$

$C_3^2$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$\He_3.D_{24}$

$\rhd$

$C_3\times C_{36}$

$\rhd$

$C_3\times C_{18}$

$\rhd$

$C_3\times C_9$

Upper central series

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_4$

$\lhd$

$C_8$

Character theory

Complex character table

See the $77 \times 77$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $28 \times 28$ rational character table.