Abstract group 864.2515: $D_4\times C_9\times A_4$

• Label: 864.2515
• Order: \( 2^{5} \cdot 3^{3} \)
• Exponent: \( 2^{2} \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 3^{3} \)
• $\card{Z(G)}$: \( 2 \cdot 3^{2} \)
• $\card{\Aut(G)}$: \( 2^{7} \cdot 3^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \cdot 3^{2} \)
• Perm deg.: $17$
• Trans deg.: $108$
• Rank: $2$

Group information

Description:	$D_4\times C_9\times A_4$
Order:	\(864\)\(\medspace = 2^{5} \cdot 3^{3} \)
Exponent:	\(36\)\(\medspace = 2^{2} \cdot 3^{2} \)
Automorphism group:	$C_{12}:D_6\times S_4$, of order \(3456\)\(\medspace = 2^{7} \cdot 3^{3} \)
Composition factors:	$C_2$ x 5, $C_3$ x 3
Derived length:	$2$

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

Group statistics

Order	1	2	3	4	6	9	12	18	36
Elements	1	23	26	8	166	54	64	378	144	864
Conjugacy classes	1	7	8	2	32	18	10	78	24	180
Divisions	1	7	4	2	16	3	5	13	4	55
Autjugacy classes	1	5	2	2	7	2	3	7	3	32

Dimension	1	2	3	4	6	12	18	36
Irr. complex chars.	108	27	36	0	9	0	0	0	180
Irr. rational chars.	4	17	4	4	17	4	4	1	55

Minimal presentations

Permutation degree:	$17$
Transitive degree:	$108$
Rank:	$2$
Inequivalent generating pairs:	$36$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	6	12	36
Arbitrary	5	7	11

Constructions

Presentation:	$\langle a, b, c, d \mid a^{2}=b^{12}=c^{2}=d^{18}=[a,c]=[a,d]=[c,d]=1, b^{a}=b^{7}, c^{b}=d^{9}, d^{b}=cd \rangle$
Permutation group:	Degree $17$ $\langle(10,11)(12,13), (11,13), (1,2,4)(3,5,7)(6,8,9)(15,17,16), (1,3,6,4,7,9,2,5,8), (10,12)(11,13), (1,4,2)(3,7,5)(6,9,8), (14,15)(16,17), (14,16)(15,17)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 35 & 16 \\ 28 & 3 \end{array}\right), \left(\begin{array}{rr} 11 & 57 \\ 19 & 68 \end{array}\right), \left(\begin{array}{rr} 75 & 0 \\ 38 & 75 \end{array}\right), \left(\begin{array}{rr} 37 & 0 \\ 0 & 37 \end{array}\right), \left(\begin{array}{rr} 49 & 0 \\ 0 & 49 \end{array}\right), \left(\begin{array}{rr} 25 & 8 \\ 12 & 13 \end{array}\right), \left(\begin{array}{rr} 61 & 0 \\ 0 & 61 \end{array}\right), \left(\begin{array}{rr} 37 & 38 \\ 38 & 37 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/76\Z)$
Direct product:	$C_9$ $\, \times\, $ $D_4$ $\, \times\, $ $A_4$
Semidirect product:	$(A_4\times C_{36})$ $\,\rtimes\,$ $C_2$	$(C_4\times A_4)$ $\,\rtimes\,$ $C_{18}$ (3)	$C_{36}$ $\,\rtimes\,$ $(C_2\times A_4)$	$C_4$ $\,\rtimes\,$ $(A_4\times C_{18})$	all 20
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_{12}\times A_4)$ . $C_6$	$C_{12}$ . $(C_6\times A_4)$	$(A_4\times C_{18})$ . $C_2^2$	$C_{18}$ . $(C_2^2\times A_4)$	all 22

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

Homology

Abelianization:	$C_{6} \times C_{18} \simeq C_{2}^{2} \times C_{3} \times C_{9}$
Schur multiplier:	$C_{2} \times C_{6}$
Commutator length:	$1$

Subgroups

There are 824 subgroups in 262 conjugacy classes, 78 normal (32 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_{18}$	$G/Z \simeq$ $C_2^2\times A_4$
Commutator:	$G' \simeq$ $C_2^3$	$G/G' \simeq$ $C_6\times C_{18}$
Frattini:	$\Phi \simeq$ $C_6$	$G/\Phi \simeq$ $C_2^2:C_6^2$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^4:C_{18}$	$G/\operatorname{Fit} \simeq$ $C_3$
Radical:	$R \simeq$ $D_4\times C_9\times A_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^2\times C_6$	$G/\operatorname{soc} \simeq$ $C_6^2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2\times D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3\times C_9$

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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Subgroup information

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Classes of subgroups up to conjugation

Order 864: $D_4\times C_9\times A_4$

Order 432: $C_2\times A_4\times C_{18}$ x 2, $A_4\times C_{36}$

Order 288: $C_2^4:C_{18}$ x 2, $C_2^3.C_6^2$, $C_2^4:C_{18}$

Order 216: $A_4\times C_{18}$ x 3, $C_{12}:C_{18}$

Order 144: $D_4\times C_{18}$ x 4, $C_2^3:C_{18}$ x 4, $C_2^2:C_{36}$ x 2, $C_2^2:C_6^2$ x 2, $C_2^3\times C_{18}$ x 2, $C_2^2\times C_{36}$, $C_{12}\times A_4$

Order 108: $C_6\times C_{18}$ x 2, $C_9\times A_4$, $C_3\times C_{36}$

Order 96: $D_4\times A_4$ x 3, $C_{12}:C_2^3$

Order 72: $C_2^2\times C_{18}$ x 9, $D_4\times C_9$ x 8, $C_2^2:C_{18}$ x 6, $C_6\times A_4$ x 3, $C_2\times C_{36}$ x 2, $D_4\times C_3^2$

Order 54: $C_3\times C_{18}$ x 3

Order 48: $C_2^2\times A_4$ x 6, $C_6\times D_4$ x 4, $C_4\times A_4$ x 3, $C_2^3\times C_6$ x 2, $C_2^2\times C_{12}$

Order 36: $C_2\times C_{18}$ x 19, $C_{36}$ x 4, $C_2^2:C_9$ x 2, $C_6^2$ x 2, $C_3\times C_{12}$, $C_3\times A_4$

Order 32: $C_2^2\times D_4$

Order 27: $C_3\times C_9$

Order 24: $C_2^2\times C_6$ x 9, $C_2\times A_4$ x 9, $C_3\times D_4$ x 9, $C_2\times C_{12}$ x 2

Order 18: $C_{18}$ x 13, $C_3\times C_6$ x 3

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

Order 12: $C_2\times C_6$ x 21, $C_{12}$ x 5, $A_4$ x 3

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

Order 8: $C_2^3$ x 9, $D_4$ x 6, $C_2\times C_4$ x 2

Order 6: $C_6$ x 16

Order 4: $C_2^2$ x 15, $C_4$ x 2

Order 3: $C_3$ x 4

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 864: $D_4\times C_9\times A_4$

Order 432: $C_2\times A_4\times C_{18}$, $A_4\times C_{36}$

Order 288: $C_2^3.C_6^2$, $C_2^4:C_{18}$, $C_2^4:C_{18}$

Order 216: $A_4\times C_{18}$ x 2, $C_{12}:C_{18}$

Order 144: $D_4\times C_{18}$ x 3, $C_2^2\times C_{36}$, $C_2^2:C_{36}$, $C_2^2:C_6^2$, $C_{12}\times A_4$, $C_2^3\times C_{18}$, $C_2^3:C_{18}$

Order 108: $C_6\times C_{18}$, $C_9\times A_4$, $C_3\times C_{36}$

Order 96: $C_{12}:C_2^3$, $D_4\times A_4$

Order 72: $C_2^2\times C_{18}$ x 5, $D_4\times C_9$ x 5, $C_2\times C_{36}$ x 2, $C_6\times A_4$ x 2, $C_2^2:C_{18}$ x 2, $D_4\times C_3^2$

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

Order 48: $C_6\times D_4$ x 3, $C_2^3\times C_6$, $C_2^2\times A_4$, $C_2^2\times C_{12}$, $C_4\times A_4$

Order 36: $C_2\times C_{18}$ x 10, $C_{36}$ x 3, $C_3\times C_{12}$, $C_2^2:C_9$, $C_6^2$, $C_3\times A_4$

Order 32: $C_2^2\times D_4$

Order 27: $C_3\times C_9$

Order 24: $C_2^2\times C_6$ x 5, $C_3\times D_4$ x 5, $C_2\times C_{12}$ x 2, $C_2\times A_4$ x 2

Order 18: $C_{18}$ x 7, $C_3\times C_6$ x 2

Order 16: $C_2\times D_4$ x 3, $C_2^4$, $C_2^2\times C_4$

Order 12: $C_2\times C_6$ x 10, $C_{12}$ x 3, $A_4$

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

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

Order 6: $C_6$ x 7

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

Order 3: $C_3$ x 2

Order 2: $C_2$ x 5

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 864: $D_4\times C_9\times A_4$ ($C_1$)

Order 432: $C_2\times A_4\times C_{18}$ ($C_2$) x 2, $A_4\times C_{36}$ ($C_2$)

Order 288: $C_2^4:C_{18}$ ($C_3$) x 2, $C_2^3.C_6^2$ ($C_3$), $C_2^4:C_{18}$ ($C_3$)

Order 216: $A_4\times C_{18}$ ($C_2^2$)

Order 144: $C_2^3:C_{18}$ ($C_6$) x 4, $C_2^2:C_{36}$ ($C_6$) x 2, $C_2^2:C_6^2$ ($C_6$) x 2, $C_2^3\times C_{18}$ ($C_6$) x 2, $C_2^2\times C_{36}$ ($C_6$), $C_{12}\times A_4$ ($C_6$)

Order 108: $C_9\times A_4$ ($D_4$)

Order 96: $D_4\times A_4$ ($C_9$) x 3, $C_{12}:C_2^3$ ($C_3^2$)

Order 72: $C_2^2:C_{18}$ ($C_2\times C_6$) x 2, $C_6\times A_4$ ($C_2\times C_6$), $C_2^2\times C_{18}$ ($C_2\times C_6$), $D_4\times C_9$ ($A_4$)

Order 48: $C_2^2\times A_4$ ($C_{18}$) x 6, $C_4\times A_4$ ($C_{18}$) x 3, $C_2^3\times C_6$ ($C_3\times C_6$) x 2, $C_2^2\times C_{12}$ ($C_3\times C_6$)

Order 36: $C_2\times C_{18}$ ($C_2\times A_4$) x 2, $C_2^2:C_9$ ($C_3\times D_4$) x 2, $C_2\times C_{18}$ ($C_3\times D_4$), $C_{36}$ ($C_2\times A_4$), $C_3\times A_4$ ($C_3\times D_4$)

Order 32: $C_2^2\times D_4$ ($C_3\times C_9$)

Order 24: $C_2\times A_4$ ($C_2\times C_{18}$) x 3, $C_2^2\times C_6$ ($C_6^2$), $C_3\times D_4$ ($C_3\times A_4$)

Order 18: $C_{18}$ ($C_2^2\times A_4$)

Order 16: $C_2^4$ ($C_3\times C_{18}$) x 2, $C_2^2\times C_4$ ($C_3\times C_{18}$)

Order 12: $A_4$ ($D_4\times C_9$) x 3, $C_2\times C_6$ ($C_6\times A_4$) x 2, $C_2\times C_6$ ($D_4\times C_3^2$), $C_{12}$ ($C_6\times A_4$)

Order 9: $C_9$ ($D_4\times A_4$)

Order 8: $C_2^3$ ($C_6\times C_{18}$), $D_4$ ($C_9\times A_4$)

Order 6: $C_6$ ($C_2^2:C_6^2$)

Order 4: $C_2^2$ ($A_4\times C_{18}$) x 2, $C_2^2$ ($C_{12}:C_{18}$), $C_4$ ($A_4\times C_{18}$)

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

Order 2: $C_2$ ($C_2\times A_4\times C_{18}$)

Order 1: $C_1$ ($D_4\times C_9\times A_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 864: $D_4\times C_9\times A_4$ ($C_1$)

Order 432: $C_2\times A_4\times C_{18}$ ($C_2$), $A_4\times C_{36}$ ($C_2$)

Order 288: $C_2^3.C_6^2$ ($C_3$), $C_2^4:C_{18}$ ($C_3$), $C_2^4:C_{18}$ ($C_3$)

Order 216: $A_4\times C_{18}$ ($C_2^2$)

Order 144: $C_2^2\times C_{36}$ ($C_6$), $C_2^2:C_{36}$ ($C_6$), $C_2^2:C_6^2$ ($C_6$), $C_{12}\times A_4$ ($C_6$), $C_2^3\times C_{18}$ ($C_6$), $C_2^3:C_{18}$ ($C_6$)

Order 108: $C_9\times A_4$ ($D_4$)

Order 96: $C_{12}:C_2^3$ ($C_3^2$), $D_4\times A_4$ ($C_9$)

Order 72: $C_6\times A_4$ ($C_2\times C_6$), $C_2^2\times C_{18}$ ($C_2\times C_6$), $C_2^2:C_{18}$ ($C_2\times C_6$), $D_4\times C_9$ ($A_4$)

Order 48: $C_2^3\times C_6$ ($C_3\times C_6$), $C_2^2\times A_4$ ($C_{18}$), $C_2^2\times C_{12}$ ($C_3\times C_6$), $C_4\times A_4$ ($C_{18}$)

Order 36: $C_2\times C_{18}$ ($C_2\times A_4$), $C_2\times C_{18}$ ($C_3\times D_4$), $C_2^2:C_9$ ($C_3\times D_4$), $C_{36}$ ($C_2\times A_4$), $C_3\times A_4$ ($C_3\times D_4$)

Order 32: $C_2^2\times D_4$ ($C_3\times C_9$)

Order 24: $C_2^2\times C_6$ ($C_6^2$), $C_2\times A_4$ ($C_2\times C_{18}$), $C_3\times D_4$ ($C_3\times A_4$)

Order 18: $C_{18}$ ($C_2^2\times A_4$)

Order 16: $C_2^4$ ($C_3\times C_{18}$), $C_2^2\times C_4$ ($C_3\times C_{18}$)

Order 12: $C_2\times C_6$ ($C_6\times A_4$), $C_2\times C_6$ ($D_4\times C_3^2$), $A_4$ ($D_4\times C_9$), $C_{12}$ ($C_6\times A_4$)

Order 9: $C_9$ ($D_4\times A_4$)

Order 8: $C_2^3$ ($C_6\times C_{18}$), $D_4$ ($C_9\times A_4$)

Order 6: $C_6$ ($C_2^2:C_6^2$)

Order 4: $C_2^2$ ($C_{12}:C_{18}$), $C_2^2$ ($A_4\times C_{18}$), $C_4$ ($A_4\times C_{18}$)

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

Order 2: $C_2$ ($C_2\times A_4\times C_{18}$)

Order 1: $C_1$ ($D_4\times C_9\times A_4$)

Series

Derived series

$D_4\times C_9\times A_4$

$\rhd$

$C_2^3$

$\rhd$

$C_1$

Chief series

$D_4\times C_9\times A_4$

$\rhd$

$C_2\times A_4\times C_{18}$

$\rhd$

$A_4\times C_{18}$

$\rhd$

$C_9\times A_4$

$\rhd$

$C_2\times C_{18}$

$\rhd$

$C_9$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$D_4\times C_9\times A_4$

$\rhd$

$C_2^3$

$\rhd$

$C_2^2$

Upper central series

$C_1$

$\lhd$

$C_{18}$

$\lhd$

$D_4\times C_9$

Supergroups

This group is a maximal subgroup of 16 larger groups in the database.

This group is a maximal quotient of 12 larger groups in the database.

Character theory

Complex character table

See the $180 \times 180$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

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