Abstract group 576.4970: $C_2^4:D_{18}$

• Label: 576.4970
• Order: \( 2^{6} \cdot 3^{2} \)
• Exponent: \( 2^{2} \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{7} \cdot 3^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \cdot 3 \)
• Perm deg.: $17$
• Trans deg.: $36$
• Rank: $3$

Group information

Description:	$C_2^4:D_{18}$
Order:	\(576\)\(\medspace = 2^{6} \cdot 3^{2} \)
Exponent:	\(36\)\(\medspace = 2^{2} \cdot 3^{2} \)
Automorphism group:	$(C_2\times C_3^2.S_4).C_2^3$, of order \(3456\)\(\medspace = 2^{7} \cdot 3^{3} \)
Composition factors:	$C_2$ x 6, $C_3$ x 2
Derived length:	$3$

This group is nonabelian and monomial (hence solvable).

Group statistics

Order	1	2	3	4	6	9	12	18	36
Elements	1	131	2	188	46	24	16	120	48	576
Conjugacy classes	1	11	1	8	7	3	2	9	3	45
Divisions	1	11	1	8	7	1	2	3	1	35
Autjugacy classes	1	7	1	6	5	1	2	2	1	26

Dimension	1	2	3	4	6	12
Irr. complex chars.	8	18	8	4	6	1	45
Irr. rational chars.	8	6	8	1	10	2	35

Minimal presentations

Permutation degree:	$17$
Transitive degree:	$36$
Rank:	$3$
Inequivalent generating triples:	$15120$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	12	12	12
Arbitrary	7	7	8

Constructions

Presentation:	${\langle a, b, c, d, e \mid a^{2}=b^{2}=c^{36}=d^{2}=e^{2}=[a,b]=[a,e]=[b,d]= \!\cdots\! \rangle}$
Permutation group:	Degree $17$ $\langle(2,3)(6,8)(7,11)(9,12)(10,13), (14,15)(16,17), (15,17), (14,16)(15,17), (2,3,4) \!\cdots\! \rangle$
Transitive group:	36T783				more information
Direct product:	$D_4$ $\, \times\, $ $(C_2^2:D_9)$
Semidirect product:	$C_2^4$ $\,\rtimes\,$ $D_{18}$ (2)	$(C_{12}.S_4)$ $\,\rtimes\,$ $C_2$	$(C_6.S_4)$ $\,\rtimes\,$ $C_2^2$	$(C_2^2\times D_4)$ $\,\rtimes\,$ $D_9$	all 16
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_3\times D_4)$ . $S_4$	$C_3$ . $(D_4\times S_4)$	$C_{12}$ . $(C_2\times S_4)$	$C_6$ . $(C_2^2\times S_4)$	all 14

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

Homology

Abelianization:	$C_{2}^{3} $
Schur multiplier:	$C_{2}^{4}$
Commutator length:	$1$

Subgroups

There are 3263 subgroups in 414 conjugacy classes, 43 normal (25 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_2^3:D_{18}$
Commutator:	$G' \simeq$ $C_2^2:C_{18}$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $C_6$	$G/\Phi \simeq$ $C_2^2\times S_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_{12}:C_2^3$	$G/\operatorname{Fit} \simeq$ $S_3$
Radical:	$R \simeq$ $C_2^4:D_{18}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^2\times C_6$	$G/\operatorname{soc} \simeq$ $C_2\times D_6$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_4^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_9$

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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Classes of subgroups up to conjugation

Order 576: $C_2^4:D_{18}$

Order 288: $C_2^3:D_{18}$ x 2, $C_2^4:D_9$ x 2, $C_2^4:C_{18}$, $C_2^2:D_{36}$, $C_{12}.S_4$

Order 192: $C_2^4:D_6$

Order 144: $C_2^2:D_{18}$ x 7, $C_2^3:C_{18}$ x 2, $D_4\times D_9$, $C_2^2:C_{36}$, $C_6.S_4$

Order 96: $C_2^3:D_6$ x 2, $C_2^4:S_3$ x 2, $C_2^3.D_6$ x 2, $D_6:D_4$ x 2, $C_{12}:C_2^3$, $D_4\times D_6$, $C_{12}:D_4$, $C_{12}:D_4$, $C_2^3.D_6$, $C_{12}:D_4$, $C_{12}:D_4$

Order 72: $C_2^2:D_9$ x 4, $C_2^2:C_{18}$ x 3, $C_9:D_4$ x 2, $C_2\times D_{18}$ x 2, $D_{36}$, $C_4\times D_9$, $D_4\times C_9$

Order 64: $D_4^2$

Order 48: $C_6:D_4$ x 19, $C_6.D_4$ x 5, $C_6\times D_4$ x 4, $S_3\times D_4$ x 4, $D_6:C_4$ x 3, $C_2^3\times C_6$ x 2, $C_2^2\times D_6$ x 2, $C_6.C_2^3$ x 2, $C_2^2\times C_{12}$, $C_2\times D_{12}$, $C_4\times D_6$, $C_{12}:C_4$, $C_6.D_4$, $C_{12}:C_4$

Order 36: $D_{18}$ x 7, $C_2\times C_{18}$ x 2, $C_2^2:C_9$, $C_{36}$, $C_9:C_4$

Order 32: $C_2^2\times D_4$ x 4, $C_4:D_4$ x 4, $C_2^2\wr C_2$ x 4, $C_4\times D_4$ x 2, $C_4:D_4$

Order 24: $C_3:D_4$ x 22, $C_2\times D_6$ x 13, $C_6:C_4$ x 9, $C_2^2\times C_6$ x 9, $C_3\times D_4$ x 5, $C_2\times C_{12}$ x 2, $D_{12}$ x 2, $C_4\times S_3$ x 2

Order 18: $D_9$ x 4, $C_{18}$ x 3

Order 16: $C_2\times D_4$ x 28, $C_2^2:C_4$ x 8, $C_2^4$ x 4, $C_2^2\times C_4$ x 4, $C_4:C_4$ x 2, $C_4^2$

Order 12: $D_6$ x 18, $C_2\times C_6$ x 15, $C_3:C_4$ x 6, $C_{12}$ x 2

Order 9: $C_9$

Order 8: $D_4$ x 29, $C_2^3$ x 22, $C_2\times C_4$ x 13

Order 6: $C_6$ x 7, $S_3$ x 4

Order 4: $C_2^2$ x 33, $C_4$ x 8

Order 3: $C_3$

Order 2: $C_2$ x 11

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 576: $C_2^4:D_{18}$

Order 288: $C_2^3:D_{18}$, $C_2^4:C_{18}$, $C_2^4:D_9$, $C_2^2:D_{36}$, $C_{12}.S_4$

Order 192: $C_2^4:D_6$

Order 144: $C_2^2:D_{18}$ x 3, $D_4\times D_9$, $C_2^2:C_{36}$, $C_6.S_4$, $C_2^3:C_{18}$

Order 96: $C_{12}:C_2^3$, $C_2^3:D_6$, $D_4\times D_6$, $C_2^4:S_3$, $C_{12}:D_4$, $C_2^3.D_6$, $C_{12}:D_4$, $D_6:D_4$, $C_2^3.D_6$, $C_{12}:D_4$, $C_{12}:D_4$

Order 72: $C_2^2:C_{18}$ x 2, $C_2^2:D_9$ x 2, $C_9:D_4$, $D_{36}$, $C_4\times D_9$, $C_2\times D_{18}$, $D_4\times C_9$

Order 64: $D_4^2$

Order 48: $C_6:D_4$ x 8, $C_6\times D_4$ x 3, $S_3\times D_4$ x 3, $C_6.D_4$ x 3, $D_6:C_4$ x 2, $C_2^3\times C_6$, $C_2^2\times D_6$, $C_2^2\times C_{12}$, $C_6.C_2^3$, $C_2\times D_{12}$, $C_4\times D_6$, $C_{12}:C_4$, $C_6.D_4$, $C_{12}:C_4$

Order 36: $D_{18}$ x 3, $C_2\times C_{18}$, $C_2^2:C_9$, $C_{36}$, $C_9:C_4$

Order 32: $C_2^2\times D_4$ x 3, $C_4:D_4$ x 3, $C_2^2\wr C_2$ x 2, $C_4\times D_4$ x 2, $C_4:D_4$

Order 24: $C_3:D_4$ x 9, $C_2\times D_6$ x 6, $C_6:C_4$ x 5, $C_2^2\times C_6$ x 5, $C_3\times D_4$ x 4, $C_2\times C_{12}$ x 2, $D_{12}$ x 2, $C_4\times S_3$ x 2

Order 18: $C_{18}$ x 2, $D_9$ x 2

Order 16: $C_2\times D_4$ x 15, $C_2^2:C_4$ x 5, $C_2^2\times C_4$ x 3, $C_4:C_4$ x 2, $C_2^4$ x 2, $C_4^2$

Order 12: $C_2\times C_6$ x 9, $D_6$ x 8, $C_3:C_4$ x 4, $C_{12}$ x 2

Order 9: $C_9$

Order 8: $D_4$ x 15, $C_2^3$ x 11, $C_2\times C_4$ x 9

Order 6: $C_6$ x 5, $S_3$ x 2

Order 4: $C_2^2$ x 17, $C_4$ x 6

Order 3: $C_3$

Order 2: $C_2$ x 7

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 576: $C_2^4:D_{18}$ ($C_1$)

Order 288: $C_2^3:D_{18}$ ($C_2$) x 2, $C_2^4:D_9$ ($C_2$) x 2, $C_2^4:C_{18}$ ($C_2$), $C_2^2:D_{36}$ ($C_2$), $C_{12}.S_4$ ($C_2$)

Order 144: $C_2^2:D_{18}$ ($C_2^2$) x 3, $C_2^3:C_{18}$ ($C_2^2$) x 2, $C_2^2:C_{36}$ ($C_2^2$), $C_6.S_4$ ($C_2^2$)

Order 96: $C_{12}:C_2^3$ ($S_3$)

Order 72: $C_2^2:D_9$ ($D_4$) x 2, $C_2^2:C_{18}$ ($C_2^3$)

Order 48: $C_2^3\times C_6$ ($D_6$) x 2, $C_2^2\times C_{12}$ ($D_6$)

Order 36: $C_2^2:C_9$ ($C_2\times D_4$)

Order 32: $C_2^2\times D_4$ ($D_9$)

Order 24: $C_2^2\times C_6$ ($C_2\times D_6$), $C_3\times D_4$ ($S_4$)

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

Order 12: $C_2\times C_6$ ($C_2\times S_4$) x 2, $C_2\times C_6$ ($S_3\times D_4$), $C_{12}$ ($C_2\times S_4$)

Order 8: $C_2^3$ ($C_2\times D_{18}$), $D_4$ ($C_2^2:D_9$)

Order 6: $C_6$ ($C_2^2\times S_4$)

Order 4: $C_2^2$ ($C_2^2:D_{18}$) x 2, $C_2^2$ ($D_4\times D_9$), $C_4$ ($C_2^2:D_{18}$)

Order 3: $C_3$ ($D_4\times S_4$)

Order 2: $C_2$ ($C_2^3:D_{18}$)

Order 1: $C_1$ ($C_2^4:D_{18}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 576: $C_2^4:D_{18}$ ($C_1$)

Order 288: $C_2^3:D_{18}$ ($C_2$), $C_2^4:C_{18}$ ($C_2$), $C_2^4:D_9$ ($C_2$), $C_2^2:D_{36}$ ($C_2$), $C_{12}.S_4$ ($C_2$)

Order 144: $C_2^2:D_{18}$ ($C_2^2$) x 2, $C_2^2:C_{36}$ ($C_2^2$), $C_6.S_4$ ($C_2^2$), $C_2^3:C_{18}$ ($C_2^2$)

Order 96: $C_{12}:C_2^3$ ($S_3$)

Order 72: $C_2^2:C_{18}$ ($C_2^3$), $C_2^2:D_9$ ($D_4$)

Order 48: $C_2^3\times C_6$ ($D_6$), $C_2^2\times C_{12}$ ($D_6$)

Order 36: $C_2^2:C_9$ ($C_2\times D_4$)

Order 32: $C_2^2\times D_4$ ($D_9$)

Order 24: $C_2^2\times C_6$ ($C_2\times D_6$), $C_3\times D_4$ ($S_4$)

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

Order 12: $C_2\times C_6$ ($C_2\times S_4$), $C_2\times C_6$ ($S_3\times D_4$), $C_{12}$ ($C_2\times S_4$)

Order 8: $C_2^3$ ($C_2\times D_{18}$), $D_4$ ($C_2^2:D_9$)

Order 6: $C_6$ ($C_2^2\times S_4$)

Order 4: $C_2^2$ ($D_4\times D_9$), $C_2^2$ ($C_2^2:D_{18}$), $C_4$ ($C_2^2:D_{18}$)

Order 3: $C_3$ ($D_4\times S_4$)

Order 2: $C_2$ ($C_2^3:D_{18}$)

Order 1: $C_1$ ($C_2^4:D_{18}$)

Series

Derived series

$C_2^4:D_{18}$

$\rhd$

$C_2^2:C_{18}$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Chief series

$C_2^4:D_{18}$

$\rhd$

$C_2^3:D_{18}$

$\rhd$

$C_2^2:D_{18}$

$\rhd$

$C_2^2:C_{18}$

$\rhd$

$C_2^2:C_9$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Lower central series

$C_2^4:D_{18}$

$\rhd$

$C_2^2:C_{18}$

$\rhd$

$C_2^2:C_9$

Upper central series

$C_1$

$\lhd$

$C_2$

$\lhd$

$D_4$

Supergroups

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

This group is a maximal quotient of 11 larger groups in the database.

Character theory

Complex character table

See the $45 \times 45$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $35 \times 35$ rational character table.