Abstract group 1584.173: $C_{132}.D_6$

• Label: 1584.173
• Order: \( 2^{4} \cdot 3^{2} \cdot 11 \)
• Exponent: \( 2^{3} \cdot 3 \cdot 11 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 3 \)
• $\card{Z(G)}$: \( 2^{2} \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{6} \cdot 3 \cdot 5 \cdot 11 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \cdot 5 \)
• Perm deg.: $25$
• Trans deg.: $528$
• Rank: $2$

Group information

Description:	$C_{132}.D_6$
Order:	\(1584\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 11 \)
Exponent:	\(264\)\(\medspace = 2^{3} \cdot 3 \cdot 11 \)
Automorphism group:	$C_{33}.(C_2^5\times C_{10})$, of order \(10560\)\(\medspace = 2^{6} \cdot 3 \cdot 5 \cdot 11 \)
Composition factors:	$C_2$ x 4, $C_3$ x 2, $C_{11}$
Derived length:	$2$

This group is nonabelian, supersolvable (hence solvable and monomial), and metabelian.

Group statistics

Order	1	2	3	4	6	8	11	12	22	24	33	44	66	88	132	264
Elements	1	67	8	68	140	56	10	148	10	376	80	20	80	120	160	240	1584
Conjugacy classes	1	2	5	3	7	4	5	12	5	20	25	10	25	20	50	40	234
Divisions	1	2	3	2	4	2	1	4	1	4	3	1	3	1	3	1	36
Autjugacy classes	1	2	3	2	4	2	1	4	1	4	3	1	3	1	3	1	36

Dimension	1	2	4	8	10	16	20	40	80
Irr. complex chars.	24	150	60	0	0	0	0	0	0	234
Irr. rational chars.	4	8	6	3	2	1	5	5	2	36

Minimal presentations

Permutation degree:	$25$
Transitive degree:	$528$
Rank:	$2$
Inequivalent generating pairs:	$48$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	4	8	80
Arbitrary	4	6	18

Constructions

Presentation:	$\langle a, b, c \mid b^{24}=c^{33}=[a,c]=1, a^{2}=b^{6}, b^{a}=b^{5}, c^{b}=c^{10} \rangle$
Permutation group:	Degree $25$ $\langle(2,3)(4,5)(6,7)(8,9)(10,11)(15,16,18,17,19,20,22,21), (15,17,18,20,19,21,22,16) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 109 & 88 \\ 22 & 21 \end{array}\right), \left(\begin{array}{rr} 23 & 0 \\ 0 & 23 \end{array}\right), \left(\begin{array}{rr} 1 & 10 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 89 & 0 \\ 0 & 89 \end{array}\right), \left(\begin{array}{rr} 1 & 66 \\ 22 & 23 \end{array}\right), \left(\begin{array}{rr} 67 & 108 \\ 66 & 43 \end{array}\right), \left(\begin{array}{rr} 1 & 55 \\ 55 & 56 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/110\Z)$
Direct product:	$C_3$ $\, \times\, $ $(C_{33}:\OD_{16})$
Semidirect product:	$(C_3:C_{88})$ $\,\rtimes\,$ $C_6$	$(C_{11}:C_{24})$ $\,\rtimes\,$ $S_3$	$(C_3:C_{24})$ $\,\rtimes\,$ $D_{11}$	$(C_3:C_{264})$ $\,\rtimes\,$ $C_2$	all 14
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$D_{66}$ . $C_{12}$	$C_{132}$ . $D_6$	$C_{66}$ . $(C_4\times S_3)$	$C_{44}$ . $(C_6\times S_3)$	all 22

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 700 subgroups in 88 conjugacy classes, 40 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_{12}$	$G/Z \simeq$ $S_3\times D_{11}$
Commutator:	$G' \simeq$ $C_{66}$	$G/G' \simeq$ $C_2\times C_{12}$
Frattini:	$\Phi \simeq$ $C_4$	$G/\Phi \simeq$ $C_{33}:D_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_3\times C_{132}$	$G/\operatorname{Fit} \simeq$ $C_2^2$
Radical:	$R \simeq$ $C_{132}.D_6$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3\times C_{66}$	$G/\operatorname{soc} \simeq$ $C_2\times C_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $\OD_{16}$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2$
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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Subgroup information

Click on a subgroup in the diagram to see information about it.

Classes of subgroups up to conjugation

Order 1584: $C_{132}.D_6$

Order 792: $C_{12}\times D_{33}$, $C_3:C_{264}$, $C_{33}:C_{24}$

Order 528: $C_{264}:C_2$, $C_{33}:\OD_{16}$

Order 396: $C_3\times D_{66}$, $C_3\times C_{132}$, $C_{33}:C_{12}$

Order 264: $C_{11}:C_{24}$ x 3, $C_{264}$, $C_4\times D_{33}$, $C_{12}\times D_{11}$, $C_3:C_{88}$

Order 198: $C_3\times D_{33}$, $C_3\times C_{66}$

Order 176: $C_{88}:C_2$

Order 144: $C_{24}:C_6$

Order 132: $C_{132}$ x 3, $D_{66}$, $C_3\times D_{22}$, $C_{33}:C_4$, $C_{11}:C_{12}$

Order 99: $C_3\times C_{33}$

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

Order 72: $S_3\times C_{12}$, $C_3\times C_{24}$, $C_3:C_{24}$

Order 66: $C_{66}$ x 3, $D_{33}$, $C_3\times D_{11}$

Order 48: $C_{24}:C_2$, $C_3\times \OD_{16}$

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

Order 36: $C_3\times C_{12}$, $C_3:C_{12}$, $C_6\times S_3$

Order 33: $C_{33}$ x 3

Order 24: $C_{24}$ x 4, $C_2\times C_{12}$, $C_4\times S_3$, $C_3:C_8$

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

Order 18: $C_3\times C_6$, $C_3\times S_3$

Order 16: $\OD_{16}$

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

Order 11: $C_{11}$

Order 9: $C_3^2$

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

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

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

Order 3: $C_3$ x 3

Order 2: $C_2$ x 2

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1584: $C_{132}.D_6$

Order 792: $C_{12}\times D_{33}$, $C_3:C_{264}$, $C_{33}:C_{24}$

Order 528: $C_{264}:C_2$, $C_{33}:\OD_{16}$

Order 396: $C_3\times D_{66}$, $C_3\times C_{132}$, $C_{33}:C_{12}$

Order 264: $C_{11}:C_{24}$ x 3, $C_{264}$, $C_4\times D_{33}$, $C_{12}\times D_{11}$, $C_3:C_{88}$

Order 198: $C_3\times D_{33}$, $C_3\times C_{66}$

Order 176: $C_{88}:C_2$

Order 144: $C_{24}:C_6$

Order 132: $C_{132}$ x 3, $D_{66}$, $C_3\times D_{22}$, $C_{33}:C_4$, $C_{11}:C_{12}$

Order 99: $C_3\times C_{33}$

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

Order 72: $S_3\times C_{12}$, $C_3\times C_{24}$, $C_3:C_{24}$

Order 66: $C_{66}$ x 3, $D_{33}$, $C_3\times D_{11}$

Order 48: $C_{24}:C_2$, $C_3\times \OD_{16}$

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

Order 36: $C_3\times C_{12}$, $C_3:C_{12}$, $C_6\times S_3$

Order 33: $C_{33}$ x 3

Order 24: $C_{24}$ x 4, $C_2\times C_{12}$, $C_4\times S_3$, $C_3:C_8$

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

Order 18: $C_3\times C_6$, $C_3\times S_3$

Order 16: $\OD_{16}$

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

Order 11: $C_{11}$

Order 9: $C_3^2$

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

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

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

Order 3: $C_3$ x 3

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1584: $C_{132}.D_6$ ($C_1$)

Order 792: $C_{12}\times D_{33}$ ($C_2$), $C_3:C_{264}$ ($C_2$), $C_{33}:C_{24}$ ($C_2$)

Order 528: $C_{33}:\OD_{16}$ ($C_3$)

Order 396: $C_3\times D_{66}$ ($C_4$), $C_3\times C_{132}$ ($C_2^2$), $C_{33}:C_{12}$ ($C_4$)

Order 264: $C_4\times D_{33}$ ($C_6$), $C_{11}:C_{24}$ ($C_6$), $C_{11}:C_{24}$ ($S_3$), $C_3:C_{88}$ ($C_6$)

Order 198: $C_3\times C_{66}$ ($C_2\times C_4$)

Order 132: $D_{66}$ ($C_{12}$), $C_{132}$ ($C_2\times C_6$), $C_{132}$ ($D_6$), $C_{33}:C_4$ ($C_{12}$)

Order 99: $C_3\times C_{33}$ ($\OD_{16}$)

Order 88: $C_{11}:C_8$ ($C_3\times S_3$)

Order 72: $C_3:C_{24}$ ($D_{11}$)

Order 66: $C_{66}$ ($C_2\times C_{12}$), $C_{66}$ ($C_4\times S_3$)

Order 44: $C_{44}$ ($C_6\times S_3$)

Order 36: $C_3\times C_{12}$ ($D_{22}$)

Order 33: $C_{33}$ ($C_{24}:C_2$), $C_{33}$ ($C_3\times \OD_{16}$)

Order 24: $C_3:C_8$ ($C_3\times D_{11}$)

Order 22: $C_{22}$ ($S_3\times C_{12}$)

Order 18: $C_3\times C_6$ ($C_4\times D_{11}$)

Order 12: $C_{12}$ ($C_3\times D_{22}$), $C_{12}$ ($S_3\times D_{11}$)

Order 11: $C_{11}$ ($C_{24}:C_6$)

Order 9: $C_3^2$ ($C_{88}:C_2$)

Order 6: $C_6$ ($D_{33}:C_4$), $C_6$ ($C_{12}\times D_{11}$)

Order 4: $C_4$ ($C_{33}:D_6$)

Order 3: $C_3$ ($C_{264}:C_2$), $C_3$ ($C_{33}:\OD_{16}$)

Order 2: $C_2$ ($D_{33}:C_{12}$)

Order 1: $C_1$ ($C_{132}.D_6$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1584: $C_{132}.D_6$ ($C_1$)

Order 792: $C_{12}\times D_{33}$ ($C_2$), $C_3:C_{264}$ ($C_2$), $C_{33}:C_{24}$ ($C_2$)

Order 528: $C_{33}:\OD_{16}$ ($C_3$)

Order 396: $C_3\times D_{66}$ ($C_4$), $C_3\times C_{132}$ ($C_2^2$), $C_{33}:C_{12}$ ($C_4$)

Order 264: $C_4\times D_{33}$ ($C_6$), $C_{11}:C_{24}$ ($C_6$), $C_{11}:C_{24}$ ($S_3$), $C_3:C_{88}$ ($C_6$)

Order 198: $C_3\times C_{66}$ ($C_2\times C_4$)

Order 132: $D_{66}$ ($C_{12}$), $C_{132}$ ($C_2\times C_6$), $C_{132}$ ($D_6$), $C_{33}:C_4$ ($C_{12}$)

Order 99: $C_3\times C_{33}$ ($\OD_{16}$)

Order 88: $C_{11}:C_8$ ($C_3\times S_3$)

Order 72: $C_3:C_{24}$ ($D_{11}$)

Order 66: $C_{66}$ ($C_2\times C_{12}$), $C_{66}$ ($C_4\times S_3$)

Order 44: $C_{44}$ ($C_6\times S_3$)

Order 36: $C_3\times C_{12}$ ($D_{22}$)

Order 33: $C_{33}$ ($C_{24}:C_2$), $C_{33}$ ($C_3\times \OD_{16}$)

Order 24: $C_3:C_8$ ($C_3\times D_{11}$)

Order 22: $C_{22}$ ($S_3\times C_{12}$)

Order 18: $C_3\times C_6$ ($C_4\times D_{11}$)

Order 12: $C_{12}$ ($C_3\times D_{22}$), $C_{12}$ ($S_3\times D_{11}$)

Order 11: $C_{11}$ ($C_{24}:C_6$)

Order 9: $C_3^2$ ($C_{88}:C_2$)

Order 6: $C_6$ ($D_{33}:C_4$), $C_6$ ($C_{12}\times D_{11}$)

Order 4: $C_4$ ($C_{33}:D_6$)

Order 3: $C_3$ ($C_{264}:C_2$), $C_3$ ($C_{33}:\OD_{16}$)

Order 2: $C_2$ ($D_{33}:C_{12}$)

Order 1: $C_1$ ($C_{132}.D_6$)

Series

Derived series

$C_{132}.D_6$

$\rhd$

$C_{66}$

$\rhd$

$C_1$

Chief series

$C_{132}.D_6$

$\rhd$

$C_3:C_{264}$

$\rhd$

$C_3\times C_{132}$

$\rhd$

$C_3\times C_{66}$

$\rhd$

$C_3\times C_{33}$

$\rhd$

$C_{33}$

$\rhd$

$C_{11}$

$\rhd$

$C_1$

Lower central series

$C_{132}.D_6$

$\rhd$

$C_{66}$

$\rhd$

$C_{33}$

Upper central series

$C_1$

$\lhd$

$C_{12}$

Character theory

Complex character table

See the $234 \times 234$ 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 $36 \times 36$ rational character table.