Abstract group 528.32: $C_{264}:C_2$

• Label: 528.32
• Order: \( 2^{4} \cdot 3 \cdot 11 \)
• Exponent: \( 2^{3} \cdot 3 \cdot 11 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 3 \)
• $\card{Z(G)}$: \( 2^{2} \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{5} \cdot 5 \cdot 11 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \cdot 5 \)
• Perm deg.: $22$
• Trans deg.: $264$
• Rank: $2$

Group information

Description:	$C_{264}:C_2$
Order:	\(528\)\(\medspace = 2^{4} \cdot 3 \cdot 11 \)
Exponent:	\(264\)\(\medspace = 2^{3} \cdot 3 \cdot 11 \)
Automorphism group:	$C_2^4\times F_{11}$, of order \(1760\)\(\medspace = 2^{5} \cdot 5 \cdot 11 \)
Composition factors:	$C_2$ x 4, $C_3$, $C_{11}$
Derived length:	$2$

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

Group statistics

Order	1	2	3	4	6	8	11	12	22	24	33	44	66	88	132	264
Elements	1	23	2	24	46	48	10	48	10	96	20	20	20	40	40	80	528
Conjugacy classes	1	2	2	3	4	4	5	6	5	8	10	10	10	20	20	40	150
Divisions	1	2	1	2	2	2	1	2	1	2	1	1	1	1	1	1	22
Autjugacy classes	1	2	1	2	2	2	1	2	1	2	1	1	1	1	1	1	22

Dimension	1	2	4	8	10	20	40	80
Irr. complex chars.	24	126	0	0	0	0	0	0	150
Irr. rational chars.	4	6	3	1	2	3	2	1	22

Minimal presentations

Permutation degree:	$22$
Transitive degree:	$264$
Rank:	$2$
Inequivalent generating pairs:	$48$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	2	4	80
Arbitrary	2	4	16

Constructions

Presentation:	$\langle a, b \mid a^{2}=b^{264}=1, b^{a}=b^{109} \rangle$
Permutation group:	Degree $22$ $\langle(2,5)(6,8)(13,14)(15,16)(17,18)(19,20)(21,22), (1,2,3,6,4,5,7,8), (9,10,11) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 69 & 103 \\ 108 & 69 \end{array}\right), \left(\begin{array}{rr} 1 & 0 \\ 0 & 108 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{109})$
Direct product:	$C_3$ $\, \times\, $ $(C_{88}:C_2)$
Semidirect product:	$C_{88}$ $\,\rtimes\,$ $C_6$	$C_{24}$ $\,\rtimes\,$ $D_{11}$	$C_{264}$ $\,\rtimes\,$ $C_2$	$C_{33}$ $\,\rtimes\,$ $\OD_{16}$	all 8
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$D_{22}$ . $C_{12}$	$C_{12}$ . $D_{22}$	$C_{132}$ . $C_2^2$	$(C_3\times D_{22})$ . $C_4$	all 14

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

Homology

Abelianization:	$C_{2} \times C_{12} \simeq C_{2} \times C_{4} \times C_{3}$
Schur multiplier:	$C_1$
Commutator length:	$1$

Subgroups

There are 184 subgroups in 40 conjugacy classes, 26 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_{12}$	$G/Z \simeq$ $D_{22}$
Commutator:	$G' \simeq$ $C_{22}$	$G/G' \simeq$ $C_2\times C_{12}$
Frattini:	$\Phi \simeq$ $C_4$	$G/\Phi \simeq$ $C_3\times D_{22}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{264}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_{264}:C_2$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{66}$	$G/\operatorname{soc} \simeq$ $C_2\times C_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $\OD_{16}$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
11-Sylow subgroup:	$P_{ 11 } \simeq$ $C_{11}$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

Classes of subgroups up to automorphism

Normal subgroups

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 528: $C_{264}:C_2$

Order 264: $C_{264}$, $C_{11}:C_{24}$, $C_{12}\times D_{11}$

Order 176: $C_{88}:C_2$

Order 132: $C_3\times D_{22}$, $C_{132}$, $C_{11}:C_{12}$

Order 88: $C_4\times D_{11}$, $C_{88}$, $C_{11}:C_8$

Order 66: $C_{66}$, $C_3\times D_{11}$

Order 48: $C_3\times \OD_{16}$

Order 44: $D_{22}$, $C_{44}$, $C_{11}:C_4$

Order 33: $C_{33}$

Order 24: $C_{24}$ x 2, $C_2\times C_{12}$

Order 22: $C_{22}$, $D_{11}$

Order 16: $\OD_{16}$

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

Order 11: $C_{11}$

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

Order 6: $C_6$ x 2

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

Order 3: $C_3$

Order 2: $C_2$ x 2

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 528: $C_{264}:C_2$

Order 264: $C_{264}$, $C_{11}:C_{24}$, $C_{12}\times D_{11}$

Order 176: $C_{88}:C_2$

Order 132: $C_3\times D_{22}$, $C_{132}$, $C_{11}:C_{12}$

Order 88: $C_4\times D_{11}$, $C_{88}$, $C_{11}:C_8$

Order 66: $C_{66}$, $C_3\times D_{11}$

Order 48: $C_3\times \OD_{16}$

Order 44: $D_{22}$, $C_{44}$, $C_{11}:C_4$

Order 33: $C_{33}$

Order 24: $C_{24}$ x 2, $C_2\times C_{12}$

Order 22: $C_{22}$, $D_{11}$

Order 16: $\OD_{16}$

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

Order 11: $C_{11}$

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

Order 6: $C_6$ x 2

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

Order 3: $C_3$

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 528: $C_{264}:C_2$ ($C_1$)

Order 264: $C_{264}$ ($C_2$), $C_{11}:C_{24}$ ($C_2$), $C_{12}\times D_{11}$ ($C_2$)

Order 176: $C_{88}:C_2$ ($C_3$)

Order 132: $C_3\times D_{22}$ ($C_4$), $C_{132}$ ($C_2^2$), $C_{11}:C_{12}$ ($C_4$)

Order 88: $C_4\times D_{11}$ ($C_6$), $C_{88}$ ($C_6$), $C_{11}:C_8$ ($C_6$)

Order 66: $C_{66}$ ($C_2\times C_4$)

Order 44: $D_{22}$ ($C_{12}$), $C_{44}$ ($C_2\times C_6$), $C_{11}:C_4$ ($C_{12}$)

Order 33: $C_{33}$ ($\OD_{16}$)

Order 24: $C_{24}$ ($D_{11}$)

Order 22: $C_{22}$ ($C_2\times C_{12}$)

Order 12: $C_{12}$ ($D_{22}$)

Order 11: $C_{11}$ ($C_3\times \OD_{16}$)

Order 8: $C_8$ ($C_3\times D_{11}$)

Order 6: $C_6$ ($C_4\times D_{11}$)

Order 4: $C_4$ ($C_3\times D_{22}$)

Order 3: $C_3$ ($C_{88}:C_2$)

Order 2: $C_2$ ($C_{12}\times D_{11}$)

Order 1: $C_1$ ($C_{264}:C_2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 528: $C_{264}:C_2$ ($C_1$)

Order 264: $C_{264}$ ($C_2$), $C_{11}:C_{24}$ ($C_2$), $C_{12}\times D_{11}$ ($C_2$)

Order 176: $C_{88}:C_2$ ($C_3$)

Order 132: $C_3\times D_{22}$ ($C_4$), $C_{132}$ ($C_2^2$), $C_{11}:C_{12}$ ($C_4$)

Order 88: $C_4\times D_{11}$ ($C_6$), $C_{88}$ ($C_6$), $C_{11}:C_8$ ($C_6$)

Order 66: $C_{66}$ ($C_2\times C_4$)

Order 44: $D_{22}$ ($C_{12}$), $C_{44}$ ($C_2\times C_6$), $C_{11}:C_4$ ($C_{12}$)

Order 33: $C_{33}$ ($\OD_{16}$)

Order 24: $C_{24}$ ($D_{11}$)

Order 22: $C_{22}$ ($C_2\times C_{12}$)

Order 12: $C_{12}$ ($D_{22}$)

Order 11: $C_{11}$ ($C_3\times \OD_{16}$)

Order 8: $C_8$ ($C_3\times D_{11}$)

Order 6: $C_6$ ($C_4\times D_{11}$)

Order 4: $C_4$ ($C_3\times D_{22}$)

Order 3: $C_3$ ($C_{88}:C_2$)

Order 2: $C_2$ ($C_{12}\times D_{11}$)

Order 1: $C_1$ ($C_{264}:C_2$)

Series

Derived series

$C_{264}:C_2$

$\rhd$

$C_{22}$

$\rhd$

$C_1$

Chief series

$C_{264}:C_2$

$\rhd$

$C_{12}\times D_{11}$

$\rhd$

$C_{132}$

$\rhd$

$C_{66}$

$\rhd$

$C_{33}$

$\rhd$

$C_{11}$

$\rhd$

$C_1$

Lower central series

$C_{264}:C_2$

$\rhd$

$C_{22}$

$\rhd$

$C_{11}$

Upper central series

$C_1$

$\lhd$

$C_{12}$

$\lhd$

$C_{24}$

Supergroups

This group is a maximal subgroup of 24 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 $150 \times 150$ 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 $22 \times 22$ rational character table.