Abstract group 1728.46424: $D_{12}\times F_9$

• Label: 1728.46424
• Order: \( 2^{6} \cdot 3^{3} \)
• Exponent: \( 2^{3} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{5} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{9} \cdot 3^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \)
• Perm deg.: $16$
• Trans deg.: $96$
• Rank: $3$

Group information

Description:	$D_{12}\times F_9$
Order:	\(1728\)\(\medspace = 2^{6} \cdot 3^{3} \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Automorphism group:	$(C_6\times C_3:S_3.C_2).C_2^6$, of order \(13824\)\(\medspace = 2^{9} \cdot 3^{3} \)
Composition factors:	$C_2$ x 6, $C_3$ x 3
Derived length:	$2$

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

Group statistics

Order	1	2	3	4	6	8	12	24
Elements	1	139	26	308	158	576	232	288	1728
Conjugacy classes	1	7	3	12	7	20	15	16	81
Divisions	1	7	3	7	7	5	7	3	40
Autjugacy classes	1	5	3	6	6	6	7	4	38

Dimension	1	2	4	8	16	32
Irr. complex chars.	32	40	0	4	5	0	81
Irr. rational chars.	8	10	9	8	4	1	40

Minimal presentations

Permutation degree:	$16$
Transitive degree:	$96$
Rank:	$3$
Inequivalent generating triples:	$107520$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	16	16	32
Arbitrary	10	10	12

Constructions

Presentation:	$\langle a, b, c, d \mid a^{8}=b^{3}=c^{6}=d^{12}=[a,d]=[b,c]=[b,d]=1, b^{a}=bc^{2}, c^{a}=bc, d^{c}=d^{11} \rangle$
Permutation group:	Degree $16$ $\langle(2,4,5,3,6,7,9,8), (11,13)(15,16), (10,11,12,13), (2,5,6,9)(3,7,8,4), (10,12) \!\cdots\! \rangle$
Direct product:	$D_{12}$ $\, \times\, $ $F_9$
Semidirect product:	$(C_4\times F_9)$ $\,\rtimes\,$ $S_3$	$C_4$ $\,\rtimes\,$ $(S_3\times F_9)$	$(D_6\times F_9)$ $\,\rtimes\,$ $C_2$ (2)	$(D_6:F_9)$ $\,\rtimes\,$ $C_2$ (2)	all 21
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_2\times F_9)$ . $D_6$ (2)	$C_2$ . $(D_6\times F_9)$	$(C_6\times F_9)$ . $C_2^2$ (2)	$(C_6:F_9)$ . $C_2^2$ (2)	all 32

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

Homology

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

Subgroups

There are 4784 subgroups in 338 conjugacy classes, 72 normal (46 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $D_6\times F_9$
Commutator:	$G' \simeq$ $C_3^2\times C_6$	$G/G' \simeq$ $C_2^2\times C_8$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $D_6\times F_9$
Fitting:	$\operatorname{Fit} \simeq$ $C_3^2\times C_{12}$	$G/\operatorname{Fit} \simeq$ $C_2\times C_8$
Radical:	$R \simeq$ $D_{12}\times F_9$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3^2\times C_6$	$G/\operatorname{soc} \simeq$ $C_2^2\times C_8$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_8\times D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^3$

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 1728: $D_{12}\times F_9$

Order 864: $D_6\times F_9$ x 2, $D_6:F_9$ x 2, $C_3^2:C_4\times D_{12}$, $C_{12}:F_9$, $C_{12}\times F_9$

Order 576: $D_4\times F_9$

Order 432: $S_3\times F_9$ x 4, $C_3^2:C_4\times D_6$ x 2, $C_6:F_9$ x 2, $C_6\times F_9$ x 2, $(C_3^2\times D_6):C_4$ x 2, $C_{12}:S_3^2$, $(C_3^2\times C_{12}):C_4$, $C_3^3:C_4^2$

Order 288: $C_6^2:C_8$ x 2, $C_2^2\times F_9$ x 2, $C_3^2:C_4\times D_4$, $C_4:F_9$, $C_4\times F_9$

Order 216: $C_3^2:C_4\times S_3$ x 4, $C_3\times F_9$ x 3, $C_6:S_3^2$ x 2, $C_2\times C_3^3:C_4$ x 2, $C_2\times C_3^2:C_{12}$ x 2, $C_3:F_9$ x 2, $C_3^2:D_{12}$ x 2, $C_3^2:D_{12}$, $C_3:S_3\times C_{12}$, $C_3^2\times D_{12}$

Order 192: $C_8\times D_{12}$

Order 144: $C_2\times F_9$ x 8, $C_6^2:C_4$ x 2, $C_6^2:C_4$ x 2, $C_{12}:D_6$, $S_3\times D_{12}$, $(C_3\times C_{12}):C_4$, $C_3^2:C_4^2$

Order 108: $C_3:S_3^2$ x 4, $C_3^2:C_{12}$ x 3, $C_3^2:D_6$ x 2, $C_3^2\times D_6$ x 2, $C_3^3:C_4$ x 2, $C_3^2:D_6$, $C_3^2\times C_{12}$, $C_3^2:C_{12}$

Order 96: $D_6:C_8$ x 2, $C_8\times D_6$ x 2, $C_4\times D_{12}$, $C_4\times C_{24}$, $C_{12}:C_8$

Order 72: $C_2\times C_3^2:C_4$ x 8, $F_9$ x 5, $C_3:D_{12}$ x 3, $C_6:D_6$ x 2, $S_3\times D_6$ x 2, $C_6^2:C_2$ x 2, $C_3:D_{12}$ x 2, $D_4\times C_3^2$, $C_{12}:S_3$, $C_3\times D_{12}$, $S_3\times C_{12}$

Order 64: $C_8\times D_4$

Order 54: $C_3^2:S_3$ x 2, $C_3^2:C_6$ x 2, $S_3\times C_3^2$ x 2, $C_3^2\times C_6$

Order 48: $S_3\times C_8$ x 4, $C_6:C_8$ x 2, $C_4\times D_6$ x 2, $C_2\times C_{24}$ x 2, $D_6:C_4$ x 2, $S_3\times D_4$, $C_2\times D_{12}$, $C_4\times C_{12}$, $C_{12}:C_4$

Order 36: $C_6:S_3$ x 11, $C_3^2:C_4$ x 5, $S_3^2$ x 4, $C_3\times C_{12}$ x 3, $C_6\times S_3$ x 3, $C_6^2$ x 2, $C_3^2:C_4$, $C_3:C_{12}$

Order 32: $C_2^2:C_8$ x 2, $C_2^2\times C_8$ x 2, $C_4\times C_8$, $C_4\times D_4$, $C_4:C_8$

Order 27: $C_3^3$

Order 24: $D_{12}$ x 6, $C_4\times S_3$ x 5, $C_2\times D_6$ x 4, $C_2\times C_{12}$ x 3, $C_{24}$ x 3, $C_3:D_4$ x 2, $C_6:C_4$ x 2, $C_3:C_8$ x 2, $C_3\times D_4$

Order 18: $C_3:S_3$ x 8, $C_3\times C_6$ x 5, $C_3\times S_3$ x 4

Order 16: $C_2\times C_8$ x 8, $C_2^2:C_4$ x 2, $C_2^2\times C_4$ x 2, $C_4:C_4$, $C_4^2$, $C_2\times D_4$

Order 12: $D_6$ x 17, $C_{12}$ x 7, $C_2\times C_6$ x 3, $C_3:C_4$ x 3

Order 9: $C_3^2$ x 3

Order 8: $C_2\times C_4$ x 9, $C_8$ x 5, $D_4$ x 4, $C_2^3$ x 2

Order 6: $S_3$ x 10, $C_6$ x 7

Order 4: $C_2^2$ x 9, $C_4$ x 7

Order 3: $C_3$ x 3

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1728: $D_{12}\times F_9$

Order 864: $D_6\times F_9$, $C_3^2:C_4\times D_{12}$, $C_{12}:F_9$, $C_{12}\times F_9$, $D_6:F_9$

Order 576: $D_4\times F_9$

Order 432: $C_6\times F_9$ x 2, $C_3^2:C_4\times D_6$, $C_6:F_9$, $S_3\times F_9$, $C_{12}:S_3^2$, $(C_3^2\times C_{12}):C_4$, $C_3^3:C_4^2$, $(C_3^2\times D_6):C_4$

Order 288: $C_3^2:C_4\times D_4$, $C_6^2:C_8$, $C_4:F_9$, $C_4\times F_9$, $C_2^2\times F_9$

Order 216: $C_2\times C_3^2:C_{12}$ x 2, $C_3^2:C_4\times S_3$ x 2, $C_3\times F_9$ x 2, $C_6:S_3^2$, $C_2\times C_3^3:C_4$, $C_3:F_9$, $C_3^2:D_{12}$, $C_3:S_3\times C_{12}$, $C_3^2\times D_{12}$, $C_3^2:D_{12}$

Order 192: $C_8\times D_{12}$

Order 144: $C_2\times F_9$ x 4, $C_6^2:C_4$, $C_{12}:D_6$, $S_3\times D_{12}$, $C_6^2:C_4$, $(C_3\times C_{12}):C_4$, $C_3^2:C_4^2$

Order 108: $C_3^2:C_{12}$ x 3, $C_3:S_3^2$ x 2, $C_3^2:D_6$, $C_3^2:D_6$, $C_3^2\times D_6$, $C_3^3:C_4$, $C_3^2\times C_{12}$, $C_3^2:C_{12}$

Order 96: $C_4\times D_{12}$, $C_4\times C_{24}$, $D_6:C_8$, $C_{12}:C_8$, $C_8\times D_6$

Order 72: $C_2\times C_3^2:C_4$ x 5, $F_9$ x 3, $C_3:D_{12}$ x 3, $C_6:D_6$, $S_3\times D_6$, $D_4\times C_3^2$, $C_6^2:C_2$, $C_{12}:S_3$, $C_3\times D_{12}$, $S_3\times C_{12}$, $C_3:D_{12}$

Order 64: $C_8\times D_4$

Order 54: $C_3^2:C_6$ x 2, $C_3^2\times C_6$, $C_3^2:S_3$, $S_3\times C_3^2$

Order 48: $C_2\times C_{24}$ x 2, $C_6:C_8$, $S_3\times C_8$, $S_3\times D_4$, $C_2\times D_{12}$, $C_4\times D_6$, $C_4\times C_{12}$, $D_6:C_4$, $C_{12}:C_4$

Order 36: $C_6:S_3$ x 6, $C_3^2:C_4$ x 4, $C_3\times C_{12}$ x 3, $C_6\times S_3$ x 2, $S_3^2$ x 2, $C_3^2:C_4$, $C_3:C_{12}$, $C_6^2$

Order 32: $C_2^2:C_8$, $C_2^2\times C_8$, $C_4\times C_8$, $C_4\times D_4$, $C_4:C_8$

Order 27: $C_3^3$

Order 24: $D_{12}$ x 5, $C_2\times C_{12}$ x 3, $C_4\times S_3$ x 3, $C_{24}$ x 2, $C_2\times D_6$ x 2, $C_3:D_4$, $C_6:C_4$, $C_3\times D_4$, $C_3:C_8$

Order 18: $C_3:S_3$ x 5, $C_3\times C_6$ x 4, $C_3\times S_3$ x 3

Order 16: $C_2\times C_8$ x 4, $C_4:C_4$, $C_2^2:C_4$, $C_4^2$, $C_2\times D_4$, $C_2^2\times C_4$

Order 12: $D_6$ x 9, $C_{12}$ x 7, $C_2\times C_6$ x 2, $C_3:C_4$ x 2

Order 9: $C_3^2$ x 3

Order 8: $C_2\times C_4$ x 6, $D_4$ x 3, $C_8$ x 3, $C_2^3$

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

Order 4: $C_4$ x 6, $C_2^2$ x 5

Order 3: $C_3$ x 3

Order 2: $C_2$ x 5

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1728: $D_{12}\times F_9$ ($C_1$)

Order 864: $D_6\times F_9$ ($C_2$) x 2, $D_6:F_9$ ($C_2$) x 2, $C_3^2:C_4\times D_{12}$ ($C_2$), $C_{12}:F_9$ ($C_2$), $C_{12}\times F_9$ ($C_2$)

Order 432: $C_3^2:C_4\times D_6$ ($C_2^2$) x 2, $C_6:F_9$ ($C_2^2$) x 2, $C_6\times F_9$ ($C_2^2$) x 2, $(C_3^2\times D_6):C_4$ ($C_4$) x 2, $C_{12}:S_3^2$ ($C_4$), $(C_3^2\times C_{12}):C_4$ ($C_4$), $C_3^3:C_4^2$ ($C_2^2$)

Order 288: $C_4\times F_9$ ($S_3$)

Order 216: $C_6:S_3^2$ ($C_2\times C_4$) x 2, $C_2\times C_3^3:C_4$ ($C_2\times C_4$) x 2, $C_3\times F_9$ ($D_4$) x 2, $C_3^2:D_{12}$ ($C_8$) x 2, $C_2\times C_3^2:C_{12}$ ($C_2^3$), $C_2\times C_3^2:C_{12}$ ($C_2\times C_4$), $C_3^2:D_{12}$ ($C_8$), $C_3:S_3\times C_{12}$ ($C_2\times C_4$), $C_3^2\times D_{12}$ ($C_8$)

Order 144: $C_2\times F_9$ ($D_6$) x 2, $C_3^2:C_4^2$ ($D_6$)

Order 108: $C_3^2:D_6$ ($C_2\times C_8$) x 2, $C_3^2\times D_6$ ($C_2\times C_8$) x 2, $C_3^2:D_6$ ($C_2^2\times C_4$), $C_3^2:C_{12}$ ($D_4:C_2$), $C_3^2:C_{12}$ ($C_2\times D_4$), $C_3^2\times C_{12}$ ($C_2\times C_8$), $C_3^2:C_{12}$ ($C_2\times C_8$)

Order 72: $F_9$ ($D_{12}$) x 2, $C_2\times C_3^2:C_4$ ($C_4\times S_3$), $C_2\times C_3^2:C_4$ ($C_2\times D_6$), $C_{12}:S_3$ ($C_4\times S_3$)

Order 54: $C_3^2\times C_6$ ($C_2^2\times C_8$), $C_3^2:C_6$ ($\OD_{16}:C_2$), $C_3^2:C_6$ ($C_4\times D_4$)

Order 36: $C_3^2:C_4$ ($D_{12}:C_2$), $C_3^2:C_4$ ($C_2\times D_{12}$), $C_3\times C_{12}$ ($S_3\times C_8$), $C_3^2:C_4$ ($S_3\times C_8$), $C_6:S_3$ ($C_4\times D_6$)

Order 27: $C_3^3$ ($C_8\times D_4$)

Order 24: $D_{12}$ ($F_9$)

Order 18: $C_3\times C_6$ ($C_8\times D_6$), $C_3:S_3$ ($C_4\times D_{12}$), $C_3:S_3$ ($C_8.D_6$)

Order 12: $D_6$ ($C_2\times F_9$) x 2, $C_{12}$ ($C_2\times F_9$)

Order 9: $C_3^2$ ($C_8\times D_{12}$)

Order 6: $C_6$ ($C_2^2\times F_9$)

Order 4: $C_4$ ($S_3\times F_9$)

Order 3: $C_3$ ($D_4\times F_9$)

Order 2: $C_2$ ($D_6\times F_9$)

Order 1: $C_1$ ($D_{12}\times F_9$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1728: $D_{12}\times F_9$ ($C_1$)

Order 864: $D_6\times F_9$ ($C_2$), $C_3^2:C_4\times D_{12}$ ($C_2$), $C_{12}:F_9$ ($C_2$), $C_{12}\times F_9$ ($C_2$), $D_6:F_9$ ($C_2$)

Order 432: $C_6\times F_9$ ($C_2^2$) x 2, $C_3^2:C_4\times D_6$ ($C_2^2$), $C_6:F_9$ ($C_2^2$), $C_{12}:S_3^2$ ($C_4$), $(C_3^2\times C_{12}):C_4$ ($C_4$), $C_3^3:C_4^2$ ($C_2^2$), $(C_3^2\times D_6):C_4$ ($C_4$)

Order 288: $C_4\times F_9$ ($S_3$)

Order 216: $C_6:S_3^2$ ($C_2\times C_4$), $C_2\times C_3^3:C_4$ ($C_2\times C_4$), $C_2\times C_3^2:C_{12}$ ($C_2^3$), $C_2\times C_3^2:C_{12}$ ($C_2\times C_4$), $C_3\times F_9$ ($D_4$), $C_3^2:D_{12}$ ($C_8$), $C_3:S_3\times C_{12}$ ($C_2\times C_4$), $C_3^2\times D_{12}$ ($C_8$), $C_3^2:D_{12}$ ($C_8$)

Order 144: $C_2\times F_9$ ($D_6$) x 2, $C_3^2:C_4^2$ ($D_6$)

Order 108: $C_3^2:D_6$ ($C_2\times C_8$), $C_3^2:D_6$ ($C_2^2\times C_4$), $C_3^2\times D_6$ ($C_2\times C_8$), $C_3^2:C_{12}$ ($D_4:C_2$), $C_3^2:C_{12}$ ($C_2\times D_4$), $C_3^2\times C_{12}$ ($C_2\times C_8$), $C_3^2:C_{12}$ ($C_2\times C_8$)

Order 72: $C_2\times C_3^2:C_4$ ($C_4\times S_3$), $C_2\times C_3^2:C_4$ ($C_2\times D_6$), $F_9$ ($D_{12}$), $C_{12}:S_3$ ($C_4\times S_3$)

Order 54: $C_3^2\times C_6$ ($C_2^2\times C_8$), $C_3^2:C_6$ ($\OD_{16}:C_2$), $C_3^2:C_6$ ($C_4\times D_4$)

Order 36: $C_3^2:C_4$ ($D_{12}:C_2$), $C_3^2:C_4$ ($C_2\times D_{12}$), $C_3\times C_{12}$ ($S_3\times C_8$), $C_3^2:C_4$ ($S_3\times C_8$), $C_6:S_3$ ($C_4\times D_6$)

Order 27: $C_3^3$ ($C_8\times D_4$)

Order 24: $D_{12}$ ($F_9$)

Order 18: $C_3\times C_6$ ($C_8\times D_6$), $C_3:S_3$ ($C_4\times D_{12}$), $C_3:S_3$ ($C_8.D_6$)

Order 12: $D_6$ ($C_2\times F_9$), $C_{12}$ ($C_2\times F_9$)

Order 9: $C_3^2$ ($C_8\times D_{12}$)

Order 6: $C_6$ ($C_2^2\times F_9$)

Order 4: $C_4$ ($S_3\times F_9$)

Order 3: $C_3$ ($D_4\times F_9$)

Order 2: $C_2$ ($D_6\times F_9$)

Order 1: $C_1$ ($D_{12}\times F_9$)

Series

Derived series

$D_{12}\times F_9$

$\rhd$

$C_3^2\times C_6$

$\rhd$

$C_1$

Chief series

$D_{12}\times F_9$

$\rhd$

$D_6\times F_9$

$\rhd$

$C_6\times F_9$

$\rhd$

$C_2\times C_3^2:C_{12}$

$\rhd$

$C_3^2:C_{12}$

$\rhd$

$C_3^2:C_6$

$\rhd$

$C_3^3$

$\rhd$

$C_3^2$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$D_{12}\times F_9$

$\rhd$

$C_3^2\times C_6$

$\rhd$

$C_3^3$

Upper central series

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_4$

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 $40 \times 40$ rational character table.