Abstract group 1056.626: $C_{66}:C_4^2$

• Label: 1056.626
• Order: \( 2^{5} \cdot 3 \cdot 11 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 11 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{5} \cdot 3 \)
• $\card{Z(G)}$: \( 2^{4} \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{11} \cdot 5 \cdot 11 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{10} \cdot 5 \)
• Perm deg.: $24$
• Trans deg.: $1056$
• Rank: $3$

Group information

Description:	$C_{66}:C_4^2$
Order:	\(1056\)\(\medspace = 2^{5} \cdot 3 \cdot 11 \)
Exponent:	\(132\)\(\medspace = 2^{2} \cdot 3 \cdot 11 \)
Automorphism group:	$C_{22}.C_2^5.C_5.C_2^5$, of order \(112640\)\(\medspace = 2^{11} \cdot 5 \cdot 11 \)
Composition factors:	$C_2$ x 5, $C_3$, $C_{11}$
Derived length:	$2$

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

Group statistics

Order	1	2	3	4	6	11	12	22	33	44	66	132
Elements	1	7	2	184	14	10	368	70	20	80	140	160	1056
Conjugacy classes	1	7	2	24	14	5	48	35	10	40	70	80	336
Divisions	1	7	1	12	7	1	12	7	1	4	7	4	64
Autjugacy classes	1	3	1	2	3	1	2	3	1	1	3	1	22

Dimension	1	2	4	10	20	40
Irr. complex chars.	96	240	0	0	0	0	336
Irr. rational chars.	8	20	12	8	12	4	64

Minimal presentations

Permutation degree:	$24$
Transitive degree:	$1056$
Rank:	$3$
Inequivalent generating triples:	$3276$

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:	$\langle a, b, c \mid a^{2}=b^{4}=c^{132}=[a,b]=[a,c]=1, c^{b}=c^{109} \rangle$
Permutation group:	Degree $24$ $\langle(2,3)(4,5)(6,7)(8,9)(10,11)(12,13,14,15), (12,14)(13,15)(19,20)(21,22)(23,24) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 89 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 23 & 0 \\ 0 & 23 \end{array}\right), \left(\begin{array}{rr} 1 & 10 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 89 & 0 \\ 0 & 89 \end{array}\right), \left(\begin{array}{rr} 89 & 30 \\ 0 & 87 \end{array}\right), \left(\begin{array}{rr} 109 & 0 \\ 0 & 109 \end{array}\right), \left(\begin{array}{rr} 1 & 55 \\ 55 & 56 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/110\Z)$
Direct product:	$C_2$ $\, \times\, $ $C_4$ $\, \times\, $ $C_3$ $\, \times\, $ $(C_{11}:C_4)$
Semidirect product:	$C_{66}$ $\,\rtimes\,$ $C_4^2$	$(C_2\times C_{132})$ $\,\rtimes\,$ $C_4$	$C_{132}$ $\,\rtimes\,$ $(C_2\times C_4)$	$(C_2\times C_{44})$ $\,\rtimes\,$ $C_{12}$	all 8
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_6$ . $(C_4\times D_{22})$	$(C_2\times C_{12})$ . $D_{22}$	$C_2$ . $(C_{12}\times D_{22})$	$(C_2^2.D_{22})$ . $C_6$	all 32

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

Homology

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

Subgroups

There are 756 subgroups in 216 conjugacy classes, 162 normal (22 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2\times C_{12}$	$G/Z \simeq$ $D_{11}$
Commutator:	$G' \simeq$ $C_{11}$	$G/G' \simeq$ $C_2\times C_4\times C_{12}$
Frattini:	$\Phi \simeq$ $C_2^2$	$G/\Phi \simeq$ $C_6\times D_{22}$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^2\times C_{132}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_{66}:C_4^2$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^2\times C_{66}$	$G/\operatorname{soc} \simeq$ $C_2^2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times C_4^2$
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

No diagram available: subgroups only stored up to automorphism

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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Subgroup information

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Classes of subgroups up to conjugation

Order 1056: $C_{66}:C_4^2$

Order 528: $C_{132}:C_4$ x 4, $C_{66}.C_2^3$ x 2, $C_2^2\times C_{132}$

Order 352: $C_{22}:C_4^2$

Order 264: $C_{22}:C_{12}$ x 12, $C_2\times C_{132}$ x 6, $C_2^2\times C_{66}$

Order 176: $C_{44}:C_4$ x 4, $C_2^2.D_{22}$ x 2, $C_2^2\times C_{44}$

Order 132: $C_{11}:C_{12}$ x 8, $C_2\times C_{66}$ x 7, $C_{132}$ x 4

Order 96: $C_2\times C_4\times C_{12}$

Order 88: $C_{22}:C_4$ x 12, $C_2\times C_{44}$ x 6, $C_2^2\times C_{22}$

Order 66: $C_{66}$ x 7

Order 48: $C_4\times C_{12}$ x 4, $C_2^2\times C_{12}$ x 3

Order 44: $C_{11}:C_4$ x 8, $C_2\times C_{22}$ x 7, $C_{44}$ x 4

Order 33: $C_{33}$

Order 32: $C_2\times C_4^2$

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

Order 22: $C_{22}$ x 7

Order 16: $C_4^2$ x 4, $C_2^2\times C_4$ x 3

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

Order 11: $C_{11}$

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

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 1056: $C_{66}:C_4^2$

Order 528: $C_{132}:C_4$, $C_2^2\times C_{132}$, $C_{66}.C_2^3$

Order 352: $C_{22}:C_4^2$

Order 264: $C_2\times C_{132}$ x 2, $C_{22}:C_{12}$ x 2, $C_2^2\times C_{66}$

Order 176: $C_2^2\times C_{44}$, $C_2^2.D_{22}$, $C_{44}:C_4$

Order 132: $C_2\times C_{66}$ x 3, $C_{132}$, $C_{11}:C_{12}$

Order 96: $C_2\times C_4\times C_{12}$

Order 88: $C_2\times C_{44}$ x 2, $C_{22}:C_4$ x 2, $C_2^2\times C_{22}$

Order 66: $C_{66}$ x 3

Order 48: $C_2^2\times C_{12}$ x 2, $C_4\times C_{12}$

Order 44: $C_2\times C_{22}$ x 3, $C_{44}$, $C_{11}:C_4$

Order 33: $C_{33}$

Order 32: $C_2\times C_4^2$

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

Order 22: $C_{22}$ x 3

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

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

Order 11: $C_{11}$

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

Order 6: $C_6$ x 3

Order 4: $C_2^2$ x 3, $C_4$ x 2

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1056: $C_{66}:C_4^2$ ($C_1$)

Order 528: $C_{132}:C_4$ ($C_2$) x 4, $C_{66}.C_2^3$ ($C_2$) x 2, $C_2^2\times C_{132}$ ($C_2$)

Order 352: $C_{22}:C_4^2$ ($C_3$)

Order 264: $C_{22}:C_{12}$ ($C_4$) x 8, $C_2\times C_{132}$ ($C_4$) x 4, $C_{22}:C_{12}$ ($C_2^2$) x 4, $C_2\times C_{132}$ ($C_2^2$) x 2, $C_2^2\times C_{66}$ ($C_2^2$)

Order 176: $C_{44}:C_4$ ($C_6$) x 4, $C_2^2.D_{22}$ ($C_6$) x 2, $C_2^2\times C_{44}$ ($C_6$)

Order 132: $C_{11}:C_{12}$ ($C_2\times C_4$) x 8, $C_2\times C_{66}$ ($C_2\times C_4$) x 6, $C_{132}$ ($C_2\times C_4$) x 4, $C_2\times C_{66}$ ($C_2^3$)

Order 88: $C_{22}:C_4$ ($C_{12}$) x 8, $C_2\times C_{44}$ ($C_{12}$) x 4, $C_{22}:C_4$ ($C_2\times C_6$) x 4, $C_2\times C_{44}$ ($C_2\times C_6$) x 2, $C_2^2\times C_{22}$ ($C_2\times C_6$)

Order 66: $C_{66}$ ($C_4^2$) x 4, $C_{66}$ ($C_2^2\times C_4$) x 3

Order 48: $C_2^2\times C_{12}$ ($D_{11}$)

Order 44: $C_{11}:C_4$ ($C_2\times C_{12}$) x 8, $C_2\times C_{22}$ ($C_2\times C_{12}$) x 6, $C_{44}$ ($C_2\times C_{12}$) x 4, $C_2\times C_{22}$ ($C_2^2\times C_6$)

Order 33: $C_{33}$ ($C_2\times C_4^2$)

Order 24: $C_2\times C_{12}$ ($C_{11}:C_4$) x 4, $C_2\times C_{12}$ ($D_{22}$) x 2, $C_2^2\times C_6$ ($D_{22}$)

Order 22: $C_{22}$ ($C_4\times C_{12}$) x 4, $C_{22}$ ($C_2^2\times C_{12}$) x 3

Order 16: $C_2^2\times C_4$ ($C_3\times D_{11}$)

Order 12: $C_2\times C_6$ ($C_4\times D_{11}$) x 4, $C_{12}$ ($C_{22}:C_4$) x 4, $C_2\times C_6$ ($C_{22}:C_4$) x 2, $C_2\times C_6$ ($C_2\times D_{22}$)

Order 11: $C_{11}$ ($C_2\times C_4\times C_{12}$)

Order 8: $C_2\times C_4$ ($C_{11}:C_{12}$) x 4, $C_2\times C_4$ ($C_3\times D_{22}$) x 2, $C_2^3$ ($C_3\times D_{22}$)

Order 6: $C_6$ ($C_{44}:C_4$) x 4, $C_6$ ($C_4\times D_{22}$) x 2, $C_6$ ($C_2^2.D_{22}$)

Order 4: $C_2^2$ ($C_{12}\times D_{11}$) x 4, $C_4$ ($C_{22}:C_{12}$) x 4, $C_2^2$ ($C_{22}:C_{12}$) x 2, $C_2^2$ ($C_6\times D_{22}$)

Order 3: $C_3$ ($C_{22}:C_4^2$)

Order 2: $C_2$ ($C_{132}:C_4$) x 4, $C_2$ ($C_{12}\times D_{22}$) x 2, $C_2$ ($C_{66}.C_2^3$)

Order 1: $C_1$ ($C_{66}:C_4^2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1056: $C_{66}:C_4^2$ ($C_1$)

Order 528: $C_{132}:C_4$ ($C_2$), $C_2^2\times C_{132}$ ($C_2$), $C_{66}.C_2^3$ ($C_2$)

Order 352: $C_{22}:C_4^2$ ($C_3$)

Order 264: $C_2^2\times C_{66}$ ($C_2^2$), $C_2\times C_{132}$ ($C_2^2$), $C_2\times C_{132}$ ($C_4$), $C_{22}:C_{12}$ ($C_2^2$), $C_{22}:C_{12}$ ($C_4$)

Order 176: $C_2^2\times C_{44}$ ($C_6$), $C_2^2.D_{22}$ ($C_6$), $C_{44}:C_4$ ($C_6$)

Order 132: $C_2\times C_{66}$ ($C_2\times C_4$) x 2, $C_{132}$ ($C_2\times C_4$), $C_{11}:C_{12}$ ($C_2\times C_4$), $C_2\times C_{66}$ ($C_2^3$)

Order 88: $C_2\times C_{44}$ ($C_2\times C_6$), $C_2\times C_{44}$ ($C_{12}$), $C_{22}:C_4$ ($C_2\times C_6$), $C_{22}:C_4$ ($C_{12}$), $C_2^2\times C_{22}$ ($C_2\times C_6$)

Order 66: $C_{66}$ ($C_2^2\times C_4$) x 2, $C_{66}$ ($C_4^2$)

Order 48: $C_2^2\times C_{12}$ ($D_{11}$)

Order 44: $C_2\times C_{22}$ ($C_2\times C_{12}$) x 2, $C_2\times C_{22}$ ($C_2^2\times C_6$), $C_{44}$ ($C_2\times C_{12}$), $C_{11}:C_4$ ($C_2\times C_{12}$)

Order 33: $C_{33}$ ($C_2\times C_4^2$)

Order 24: $C_2\times C_{12}$ ($D_{22}$), $C_2\times C_{12}$ ($C_{11}:C_4$), $C_2^2\times C_6$ ($D_{22}$)

Order 22: $C_{22}$ ($C_2^2\times C_{12}$) x 2, $C_{22}$ ($C_4\times C_{12}$)

Order 16: $C_2^2\times C_4$ ($C_3\times D_{11}$)

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

Order 11: $C_{11}$ ($C_2\times C_4\times C_{12}$)

Order 8: $C_2^3$ ($C_3\times D_{22}$), $C_2\times C_4$ ($C_3\times D_{22}$), $C_2\times C_4$ ($C_{11}:C_{12}$)

Order 6: $C_6$ ($C_2^2.D_{22}$), $C_6$ ($C_4\times D_{22}$), $C_6$ ($C_{44}:C_4$)

Order 4: $C_2^2$ ($C_6\times D_{22}$), $C_2^2$ ($C_{22}:C_{12}$), $C_2^2$ ($C_{12}\times D_{11}$), $C_4$ ($C_{22}:C_{12}$)

Order 3: $C_3$ ($C_{22}:C_4^2$)

Order 2: $C_2$ ($C_{132}:C_4$), $C_2$ ($C_{66}.C_2^3$), $C_2$ ($C_{12}\times D_{22}$)

Order 1: $C_1$ ($C_{66}:C_4^2$)

Series

Derived series

$C_{66}:C_4^2$

$\rhd$

$C_{11}$

$\rhd$

$C_1$

Chief series

$C_{66}:C_4^2$

$\rhd$

$C_{66}.C_2^3$

$\rhd$

$C_2^2\times C_{66}$

$\rhd$

$C_2\times C_{66}$

$\rhd$

$C_{66}$

$\rhd$

$C_{33}$

$\rhd$

$C_{11}$

$\rhd$

$C_1$

Lower central series

$C_{66}:C_4^2$

$\rhd$

$C_{11}$

Upper central series

$C_1$

$\lhd$

$C_2^2\times C_{12}$

Character theory

Complex character table

See the $336 \times 336$ 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 $64 \times 64$ rational character table.