Abstract group 288.369: $C_{36}.C_2^3$

• Label: 288.369
• Order: \( 2^{5} \cdot 3^{2} \)
• Exponent: \( 2^{2} \cdot 3^{2} \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \cdot 3^{2} \)
• $\card{Z(G)}$: \( 2^{3} \cdot 3^{2} \)
• $\card{\Aut(G)}$: \( 2^{11} \cdot 3^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{9} \cdot 3^{3} \)
• Perm deg.: $21$
• Trans deg.: $288$
• Rank: $4$

Group information

Description:	$C_{36}.C_2^3$
Order:	\(288\)\(\medspace = 2^{5} \cdot 3^{2} \)
Exponent:	\(36\)\(\medspace = 2^{2} \cdot 3^{2} \)
Automorphism group:	$C_2^7.C_6^2:D_6$, of order \(55296\)\(\medspace = 2^{11} \cdot 3^{3} \)
Composition factors:	$C_2$ x 5, $C_3$ x 2
Nilpotency class:	$2$
Derived length:	$2$

This group is nonabelian, elementary for $p = 2$ (hence nilpotent, solvable, supersolvable, monomial, and hyperelementary), and metabelian.

Group statistics

Order	1	2	3	4	6	9	12	18	36
Elements	1	7	2	24	14	6	48	42	144	288
Conjugacy classes	1	7	2	12	14	6	24	42	72	180
Divisions	1	7	1	12	7	1	12	7	12	60
Autjugacy classes	1	2	1	1	2	1	1	2	1	12

Dimension	1	2	4	6	12
Irr. complex chars.	144	36	0	0	0	180
Irr. rational chars.	16	20	4	16	4	60

Minimal presentations

Permutation degree:	$21$
Transitive degree:	$288$
Rank:	$4$
Inequivalent generating quadruples:	$37800$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	$\langle a, b, c, d \mid a^{2}=c^{2}=d^{36}=[a,b]=[a,c]=[a,d]=[b,c]=[c,d]=1, b^{2}=d^{18}, d^{b}=d^{19} \rangle$
Permutation group:	Degree $21$ $\langle(1,2,4,6)(3,7,8,5), (1,3,4,8)(2,5,6,7)(9,10)(11,12), (1,4)(2,6)(3,8)(5,7) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 20 & 19 \\ 19 & 1 \end{array}\right), \left(\begin{array}{rr} 25 & 0 \\ 0 & 7 \end{array}\right), \left(\begin{array}{rr} 7 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 37 & 0 \\ 0 & 37 \end{array}\right), \left(\begin{array}{rr} 20 & 0 \\ 0 & 20 \end{array}\right), \left(\begin{array}{rr} 20 & 38 \\ 38 & 1 \end{array}\right), \left(\begin{array}{rr} 20 & 0 \\ 0 & 56 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/57\Z)$
Direct product:	$C_2$ ${}^2$ $\, \times\, $ $C_9$ $\, \times\, $ $Q_8$
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{36}$ . $C_2^3$	$C_{18}$ . $C_2^4$	$(C_6\times Q_8)$ . $C_6$	$C_6$ . $(C_6\times Q_8)$	all 24

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

Homology

Abelianization:	$C_{2}^{3} \times C_{18} \simeq C_{2}^{4} \times C_{9}$
Schur multiplier:	$C_{2}^{5}$
Commutator length:	$1$

Subgroups

There are 234 subgroups, all normal (12 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2\times C_{18}$	$G/Z \simeq$ $C_2^2$
Commutator:	$G' \simeq$ $C_2$	$G/G' \simeq$ $C_2^3\times C_{18}$
Frattini:	$\Phi \simeq$ $C_6$	$G/\Phi \simeq$ $C_2^3\times C_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_{36}.C_2^3$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $C_{36}.C_2^3$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^2\times C_6$	$G/\operatorname{soc} \simeq$ $C_2\times C_6$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2\times Q_8$
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

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 288: $C_{36}.C_2^3$

Order 144: $Q_8\times C_{18}$ x 12, $C_2^2\times C_{36}$ x 3

Order 96: $C_6.C_2^4$

Order 72: $C_2\times C_{36}$ x 18, $Q_8\times C_9$ x 16, $C_2^2\times C_{18}$

Order 48: $C_6\times Q_8$ x 12, $C_2^2\times C_{12}$ x 3

Order 36: $C_{36}$ x 12, $C_2\times C_{18}$ x 7

Order 32: $C_2^2\times Q_8$

Order 24: $C_2\times C_{12}$ x 18, $C_3\times Q_8$ x 16, $C_2^2\times C_6$

Order 18: $C_{18}$ x 7

Order 16: $C_2\times Q_8$ x 12, $C_2^2\times C_4$ x 3

Order 12: $C_{12}$ x 12, $C_2\times C_6$ x 7

Order 9: $C_9$

Order 8: $C_2\times C_4$ x 18, $Q_8$ x 16, $C_2^3$

Order 6: $C_6$ x 7

Order 4: $C_4$ x 12, $C_2^2$ x 7

Order 3: $C_3$

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 288: $C_{36}.C_2^3$

Order 144: $Q_8\times C_{18}$, $C_2^2\times C_{36}$

Order 96: $C_6.C_2^4$

Order 72: $C_2\times C_{36}$, $C_2^2\times C_{18}$, $Q_8\times C_9$

Order 48: $C_6\times Q_8$, $C_2^2\times C_{12}$

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

Order 32: $C_2^2\times Q_8$

Order 24: $C_2\times C_{12}$, $C_2^2\times C_6$, $C_3\times Q_8$

Order 18: $C_{18}$ x 2

Order 16: $C_2\times Q_8$, $C_2^2\times C_4$

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

Order 9: $C_9$

Order 8: $C_2^3$, $Q_8$, $C_2\times C_4$

Order 6: $C_6$ x 2

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

Order 3: $C_3$

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 288: $C_{36}.C_2^3$ ($C_1$)

Order 144: $Q_8\times C_{18}$ ($C_2$) x 12, $C_2^2\times C_{36}$ ($C_2$) x 3

Order 96: $C_6.C_2^4$ ($C_3$)

Order 72: $C_2\times C_{36}$ ($C_2^2$) x 18, $Q_8\times C_9$ ($C_2^2$) x 16, $C_2^2\times C_{18}$ ($C_2^2$)

Order 48: $C_6\times Q_8$ ($C_6$) x 12, $C_2^2\times C_{12}$ ($C_6$) x 3

Order 36: $C_{36}$ ($C_2^3$) x 12, $C_2\times C_{18}$ ($Q_8$) x 4, $C_2\times C_{18}$ ($C_2^3$) x 3

Order 32: $C_2^2\times Q_8$ ($C_9$)

Order 24: $C_2\times C_{12}$ ($C_2\times C_6$) x 18, $C_3\times Q_8$ ($C_2\times C_6$) x 16, $C_2^2\times C_6$ ($C_2\times C_6$)

Order 18: $C_{18}$ ($C_2\times Q_8$) x 6, $C_{18}$ ($C_2^4$)

Order 16: $C_2\times Q_8$ ($C_{18}$) x 12, $C_2^2\times C_4$ ($C_{18}$) x 3

Order 12: $C_{12}$ ($C_2^2\times C_6$) x 12, $C_2\times C_6$ ($C_3\times Q_8$) x 4, $C_2\times C_6$ ($C_2^2\times C_6$) x 3

Order 9: $C_9$ ($C_2^2\times Q_8$)

Order 8: $C_2\times C_4$ ($C_2\times C_{18}$) x 18, $Q_8$ ($C_2\times C_{18}$) x 16, $C_2^3$ ($C_2\times C_{18}$)

Order 6: $C_6$ ($C_6\times Q_8$) x 6, $C_6$ ($C_2^3\times C_6$)

Order 4: $C_4$ ($C_2^2\times C_{18}$) x 12, $C_2^2$ ($Q_8\times C_9$) x 4, $C_2^2$ ($C_2^2\times C_{18}$) x 3

Order 3: $C_3$ ($C_6.C_2^4$)

Order 2: $C_2$ ($Q_8\times C_{18}$) x 6, $C_2$ ($C_2^3\times C_{18}$)

Order 1: $C_1$ ($C_{36}.C_2^3$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 288: $C_{36}.C_2^3$ ($C_1$)

Order 144: $Q_8\times C_{18}$ ($C_2$), $C_2^2\times C_{36}$ ($C_2$)

Order 96: $C_6.C_2^4$ ($C_3$)

Order 72: $C_2\times C_{36}$ ($C_2^2$), $C_2^2\times C_{18}$ ($C_2^2$), $Q_8\times C_9$ ($C_2^2$)

Order 48: $C_6\times Q_8$ ($C_6$), $C_2^2\times C_{12}$ ($C_6$)

Order 36: $C_2\times C_{18}$ ($C_2^3$), $C_2\times C_{18}$ ($Q_8$), $C_{36}$ ($C_2^3$)

Order 32: $C_2^2\times Q_8$ ($C_9$)

Order 24: $C_2\times C_{12}$ ($C_2\times C_6$), $C_2^2\times C_6$ ($C_2\times C_6$), $C_3\times Q_8$ ($C_2\times C_6$)

Order 18: $C_{18}$ ($C_2^4$), $C_{18}$ ($C_2\times Q_8$)

Order 16: $C_2\times Q_8$ ($C_{18}$), $C_2^2\times C_4$ ($C_{18}$)

Order 12: $C_2\times C_6$ ($C_2^2\times C_6$), $C_2\times C_6$ ($C_3\times Q_8$), $C_{12}$ ($C_2^2\times C_6$)

Order 9: $C_9$ ($C_2^2\times Q_8$)

Order 8: $C_2^3$ ($C_2\times C_{18}$), $Q_8$ ($C_2\times C_{18}$), $C_2\times C_4$ ($C_2\times C_{18}$)

Order 6: $C_6$ ($C_2^3\times C_6$), $C_6$ ($C_6\times Q_8$)

Order 4: $C_2^2$ ($C_2^2\times C_{18}$), $C_2^2$ ($Q_8\times C_9$), $C_4$ ($C_2^2\times C_{18}$)

Order 3: $C_3$ ($C_6.C_2^4$)

Order 2: $C_2$ ($Q_8\times C_{18}$), $C_2$ ($C_2^3\times C_{18}$)

Order 1: $C_1$ ($C_{36}.C_2^3$)

Series

Derived series

$C_{36}.C_2^3$

$\rhd$

$C_2$

$\rhd$

$C_1$

Chief series

$C_{36}.C_2^3$

$\rhd$

$C_2^2\times C_{36}$

$\rhd$

$C_2\times C_{36}$

$\rhd$

$C_2\times C_{18}$

$\rhd$

$C_{18}$

$\rhd$

$C_9$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_{36}.C_2^3$

$\rhd$

$C_2$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_2^2\times C_{18}$

$\lhd$

$C_{36}.C_2^3$

Supergroups

This group is a maximal subgroup of 61 larger groups in the database.

This group is a maximal quotient of 20 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 $60 \times 60$ rational character table.