Abstract group 288.344: $C_2^4:C_{18}$

• Label: 288.344
• Order: \( 2^{5} \cdot 3^{2} \)
• Exponent: \( 2^{2} \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 3^{2} \)
• $\card{Z(G)}$: \( 2 \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{6} \cdot 3^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \cdot 3 \)
• Perm deg.: $17$
• Trans deg.: $36$
• Rank: $2$

Group information

Description:	$C_2^4:C_{18}$
Order:	\(288\)\(\medspace = 2^{5} \cdot 3^{2} \)
Exponent:	\(36\)\(\medspace = 2^{2} \cdot 3^{2} \)
Automorphism group:	$\GL(2,\mathbb{Z}/4):C_6$, of order \(576\)\(\medspace = 2^{6} \cdot 3^{2} \)
Composition factors:	$C_2$ x 5, $C_3$ x 2
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	2	8	46	24	16	120	48	288
Conjugacy classes	1	7	2	2	14	6	4	18	6	60
Divisions	1	7	1	2	7	1	2	3	1	25
Autjugacy classes	1	5	1	2	5	1	2	2	1	20

Dimension	1	2	3	4	6	12
Irr. complex chars.	36	9	12	0	3	0	60
Irr. rational chars.	4	5	4	1	9	2	25

Minimal presentations

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

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	6	12	12
Arbitrary	5	7	8

Constructions

Presentation:	$\langle a, b, c, d \mid a^{2}=b^{36}=c^{2}=d^{2}=[a,c]=[a,d]=[c,d]=1, b^{a}=b^{19}, c^{b}=d, d^{b}=cd \rangle$
Permutation group:	Degree $17$ $\langle(15,17), (14,15)(16,17), (2,3,4)(5,6,8,7,9,11,10,12,13), (14,16)(15,17), (5,7,10)(6,9,12)(8,11,13), (1,2)(3,4), (1,3)(2,4)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 35 & 16 \\ 28 & 3 \end{array}\right), \left(\begin{array}{rr} 75 & 0 \\ 38 & 75 \end{array}\right), \left(\begin{array}{rr} 37 & 0 \\ 0 & 37 \end{array}\right), \left(\begin{array}{rr} 49 & 0 \\ 0 & 49 \end{array}\right), \left(\begin{array}{rr} 25 & 19 \\ 19 & 6 \end{array}\right), \left(\begin{array}{rr} 25 & 8 \\ 12 & 13 \end{array}\right), \left(\begin{array}{rr} 37 & 38 \\ 38 & 37 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/76\Z)$
Transitive group:	36T311				more information
Direct product:	$D_4$ $\, \times\, $ $(C_2^2:C_9)$
Semidirect product:	$C_2^4$ $\,\rtimes\,$ $C_{18}$ (2)	$(C_2^2\times D_4)$ $\,\rtimes\,$ $C_9$	$C_2^2$ $\,\rtimes\,$ $(D_4\times C_9)$	$(C_2^3:C_{18})$ $\,\rtimes\,$ $C_2$ (2)	all 8
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_3\times D_4)$ . $A_4$	$C_3$ . $(D_4\times A_4)$	$C_{12}$ . $(C_2\times A_4)$	$C_6$ . $(C_2^2\times A_4)$	all 13

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

Homology

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

Subgroups

There are 366 subgroups in 116 conjugacy classes, 30 normal (20 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_6$	$G/Z \simeq$ $C_2^2\times A_4$
Commutator:	$G' \simeq$ $C_2^3$	$G/G' \simeq$ $C_2\times C_{18}$
Frattini:	$\Phi \simeq$ $C_6$	$G/\Phi \simeq$ $C_2^2\times A_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_{12}:C_2^3$	$G/\operatorname{Fit} \simeq$ $C_3$
Radical:	$R \simeq$ $C_2^4:C_{18}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^2\times C_6$	$G/\operatorname{soc} \simeq$ $C_2\times C_6$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2\times D_4$
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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Subgroup information Click on a subgroup in the diagram to see information about it.

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Classes of subgroups up to conjugation

Order 288: $C_2^4:C_{18}$

Order 144: $C_2^3:C_{18}$ x 2, $C_2^2:C_{36}$

Order 96: $C_{12}:C_2^3$

Order 72: $C_2^2:C_{18}$ x 3, $D_4\times C_9$

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

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

Order 32: $C_2^2\times D_4$

Order 24: $C_2^2\times C_6$ x 9, $C_3\times D_4$ x 6, $C_2\times C_{12}$ x 2

Order 18: $C_{18}$ 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 15, $C_{12}$ x 2

Order 9: $C_9$

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

Order 6: $C_6$ x 7

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

Order 3: $C_3$

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 288: $C_2^4:C_{18}$

Order 144: $C_2^2:C_{36}$, $C_2^3:C_{18}$

Order 96: $C_{12}:C_2^3$

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

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

Order 36: $C_2\times C_{18}$, $C_2^2:C_9$, $C_{36}$

Order 32: $C_2^2\times D_4$

Order 24: $C_2^2\times C_6$ x 5, $C_3\times D_4$ x 4, $C_2\times C_{12}$ x 2

Order 18: $C_{18}$ 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 9, $C_{12}$ x 2

Order 9: $C_9$

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

Order 6: $C_6$ x 5

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

Order 3: $C_3$

Order 2: $C_2$ x 5

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 288: $C_2^4:C_{18}$ ($C_1$)

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

Order 96: $C_{12}:C_2^3$ ($C_3$)

Order 72: $C_2^2:C_{18}$ ($C_2^2$)

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

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

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

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

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

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

Order 8: $C_2^3$ ($C_2\times C_{18}$), $D_4$ ($C_2^2:C_9$)

Order 6: $C_6$ ($C_2^2\times A_4$)

Order 4: $C_2^2$ ($C_2^2:C_{18}$) x 2, $C_2^2$ ($D_4\times C_9$), $C_4$ ($C_2^2:C_{18}$)

Order 3: $C_3$ ($D_4\times A_4$)

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

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

Normal subgroups up to automorphism (quotient in parentheses)

Order 288: $C_2^4:C_{18}$ ($C_1$)

Order 144: $C_2^2:C_{36}$ ($C_2$), $C_2^3:C_{18}$ ($C_2$)

Order 96: $C_{12}:C_2^3$ ($C_3$)

Order 72: $C_2^2:C_{18}$ ($C_2^2$)

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

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

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

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

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

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

Order 8: $C_2^3$ ($C_2\times C_{18}$), $D_4$ ($C_2^2:C_9$)

Order 6: $C_6$ ($C_2^2\times A_4$)

Order 4: $C_2^2$ ($C_2^2:C_{18}$), $C_2^2$ ($D_4\times C_9$), $C_4$ ($C_2^2:C_{18}$)

Order 3: $C_3$ ($D_4\times A_4$)

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

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

Series

Derived series

$C_2^4:C_{18}$

$\rhd$

$C_2^3$

$\rhd$

$C_1$

Chief series

$C_2^4:C_{18}$

$\rhd$

$C_2^3:C_{18}$

$\rhd$

$C_2^2:C_{18}$

$\rhd$

$C_2^2:C_9$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Lower central series

$C_2^4:C_{18}$

$\rhd$

$C_2^3$

$\rhd$

$C_2^2$

Upper central series

$C_1$

$\lhd$

$C_6$

$\lhd$

$C_3\times D_4$

Supergroups

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

This group is a maximal quotient of 23 larger groups in the database.

Character theory

Complex character table

See the $60 \times 60$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $25 \times 25$ rational character table.