Abstract group 528.44: $C_{33}:\SD_{16}$

• Label: 528.44
• Order: \( 2^{4} \cdot 3 \cdot 11 \)
• Exponent: \( 2^{3} \cdot 3 \cdot 11 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 3 \)
• $\card{Z(G)}$: \( 2 \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{6} \cdot 5 \cdot 11 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \cdot 5 \)
• Perm deg.: $22$
• Trans deg.: $264$
• Rank: $2$

Group information

Description:	$C_{33}:\SD_{16}$
Order:	\(528\)\(\medspace = 2^{4} \cdot 3 \cdot 11 \)
Exponent:	\(264\)\(\medspace = 2^{3} \cdot 3 \cdot 11 \)
Automorphism group:	$C_{22}.(C_2^4\times C_{10})$, of order \(3520\)\(\medspace = 2^{6} \cdot 5 \cdot 11 \)
Composition factors:	$C_2$ x 4, $C_3$, $C_{11}$
Derived length:	$2$

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

Group statistics

Order	1	2	3	4	6	8	11	12	22	24	33	44	66	132
Elements	1	45	2	6	90	44	10	12	10	88	20	60	20	120	528
Conjugacy classes	1	2	2	2	4	2	5	4	5	4	10	15	10	30	96
Divisions	1	2	1	2	2	1	1	2	1	1	1	2	1	2	20
Autjugacy classes	1	2	1	2	2	1	1	2	1	1	1	2	1	2	20

Dimension	1	2	4	8	10	20	40
Irr. complex chars.	12	69	15	0	0	0	0	96
Irr. rational chars.	4	5	2	1	2	4	2	20

Minimal presentations

Permutation degree:	$22$
Transitive degree:	$264$
Rank:	$2$
Inequivalent generating pairs:	$24$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	4	8	40
Arbitrary	4	6	16

Constructions

Presentation:	$\langle a, b, c \mid b^{8}=c^{33}=[a,c]=1, a^{2}=b^{4}, b^{a}=b^{3}, c^{b}=c^{10} \rangle$
Permutation group:	Degree $22$ $\langle(2,5)(3,7)(6,8)(13,14)(15,16)(17,18)(19,20)(21,22), (1,2,4,6)(3,8,7,5), (9,10,11) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 65 & 42 \\ 44 & 1 \end{array}\right), \left(\begin{array}{rr} 23 & 0 \\ 0 & 23 \end{array}\right), \left(\begin{array}{rr} 34 & 33 \\ 33 & 1 \end{array}\right), \left(\begin{array}{rr} 65 & 22 \\ 22 & 43 \end{array}\right), \left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 23 & 44 \\ 44 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/66\Z)$
Direct product:	$C_3$ $\, \times\, $ $(Q_8:D_{11})$
Semidirect product:	$C_{33}$ $\,\rtimes\,$ $\SD_{16}$	$(C_3\times Q_8)$ $\,\rtimes\,$ $D_{11}$	$Q_8$ $\,\rtimes\,$ $(C_3\times D_{11})$	$(Q_8\times C_{33})$ $\,\rtimes\,$ $C_2$	all 8
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$D_{44}$ . $C_6$	$C_{66}$ . $D_4$	$C_{12}$ . $D_{22}$	$C_{132}$ . $C_2^2$	all 10

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

Homology

Abelianization:	$C_{2} \times C_{6} \simeq C_{2}^{2} \times C_{3}$
Schur multiplier:	$C_1$
Commutator length:	$1$

Subgroups

There are 240 subgroups in 40 conjugacy classes, 22 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_6$	$G/Z \simeq$ $C_{11}:D_4$
Commutator:	$G' \simeq$ $C_{44}$	$G/G' \simeq$ $C_2\times C_6$
Frattini:	$\Phi \simeq$ $C_4$	$G/\Phi \simeq$ $C_3\times D_{22}$
Fitting:	$\operatorname{Fit} \simeq$ $Q_8\times C_{33}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_{33}:\SD_{16}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{66}$	$G/\operatorname{soc} \simeq$ $D_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $\SD_{16}$
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: $C_{33}:\SD_{16}$

Order 264: $Q_8\times C_{33}$, $C_{11}:C_{24}$, $C_3\times D_{44}$

Order 176: $Q_8:D_{11}$

Order 132: $C_{132}$ x 2, $C_3\times D_{22}$

Order 88: $D_{44}$, $Q_8\times C_{11}$, $C_{11}:C_8$

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

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

Order 44: $C_{44}$ x 2, $D_{22}$

Order 33: $C_{33}$

Order 24: $C_{24}$, $C_3\times Q_8$, $C_3\times D_4$

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

Order 16: $\SD_{16}$

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

Order 11: $C_{11}$

Order 8: $Q_8$, $D_4$, $C_8$

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: $C_{33}:\SD_{16}$

Order 264: $Q_8\times C_{33}$, $C_{11}:C_{24}$, $C_3\times D_{44}$

Order 176: $Q_8:D_{11}$

Order 132: $C_{132}$ x 2, $C_3\times D_{22}$

Order 88: $D_{44}$, $Q_8\times C_{11}$, $C_{11}:C_8$

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

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

Order 44: $C_{44}$ x 2, $D_{22}$

Order 33: $C_{33}$

Order 24: $C_{24}$, $C_3\times Q_8$, $C_3\times D_4$

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

Order 16: $\SD_{16}$

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

Order 11: $C_{11}$

Order 8: $Q_8$, $D_4$, $C_8$

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: $C_{33}:\SD_{16}$ ($C_1$)

Order 264: $Q_8\times C_{33}$ ($C_2$), $C_{11}:C_{24}$ ($C_2$), $C_3\times D_{44}$ ($C_2$)

Order 176: $Q_8:D_{11}$ ($C_3$)

Order 132: $C_{132}$ ($C_2^2$)

Order 88: $D_{44}$ ($C_6$), $Q_8\times C_{11}$ ($C_6$), $C_{11}:C_8$ ($C_6$)

Order 66: $C_{66}$ ($D_4$)

Order 44: $C_{44}$ ($C_2\times C_6$)

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

Order 24: $C_3\times Q_8$ ($D_{11}$)

Order 22: $C_{22}$ ($C_3\times D_4$)

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

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

Order 8: $Q_8$ ($C_3\times D_{11}$)

Order 6: $C_6$ ($C_{11}:D_4$)

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

Order 3: $C_3$ ($Q_8:D_{11}$)

Order 2: $C_2$ ($C_{33}:D_4$)

Order 1: $C_1$ ($C_{33}:\SD_{16}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 528: $C_{33}:\SD_{16}$ ($C_1$)

Order 264: $Q_8\times C_{33}$ ($C_2$), $C_{11}:C_{24}$ ($C_2$), $C_3\times D_{44}$ ($C_2$)

Order 176: $Q_8:D_{11}$ ($C_3$)

Order 132: $C_{132}$ ($C_2^2$)

Order 88: $D_{44}$ ($C_6$), $Q_8\times C_{11}$ ($C_6$), $C_{11}:C_8$ ($C_6$)

Order 66: $C_{66}$ ($D_4$)

Order 44: $C_{44}$ ($C_2\times C_6$)

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

Order 24: $C_3\times Q_8$ ($D_{11}$)

Order 22: $C_{22}$ ($C_3\times D_4$)

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

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

Order 8: $Q_8$ ($C_3\times D_{11}$)

Order 6: $C_6$ ($C_{11}:D_4$)

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

Order 3: $C_3$ ($Q_8:D_{11}$)

Order 2: $C_2$ ($C_{33}:D_4$)

Order 1: $C_1$ ($C_{33}:\SD_{16}$)

Series

Derived series

$C_{33}:\SD_{16}$

$\rhd$

$C_{44}$

$\rhd$

$C_1$

Chief series

$C_{33}:\SD_{16}$

$\rhd$

$Q_8\times C_{33}$

$\rhd$

$C_{132}$

$\rhd$

$C_{66}$

$\rhd$

$C_{33}$

$\rhd$

$C_{11}$

$\rhd$

$C_1$

Lower central series

$C_{33}:\SD_{16}$

$\rhd$

$C_{44}$

$\rhd$

$C_{22}$

$\rhd$

$C_{11}$

Upper central series

$C_1$

$\lhd$

$C_6$

$\lhd$

$C_{12}$

$\lhd$

$C_3\times Q_8$

Supergroups

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

This group is a maximal quotient of 9 larger groups in the database.

Character theory

Complex character table

See the $96 \times 96$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $20 \times 20$ rational character table.