Abstract group 880.210: $C_2^3\times F_{11}$

• Label: 880.210
• Order: \( 2^{4} \cdot 5 \cdot 11 \)
• Exponent: \( 2 \cdot 5 \cdot 11 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \cdot 5 \)
• $\card{Z(G)}$: \( 2^{3} \)
• $\card{\Aut(G)}$: \( 2^{7} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \cdot 3 \cdot 7 \)
• Perm deg.: $17$
• Trans deg.: $88$
• Rank: $4$

Group information

Description:	$C_2^3\times F_{11}$
Order:	\(880\)\(\medspace = 2^{4} \cdot 5 \cdot 11 \)
Exponent:	\(110\)\(\medspace = 2 \cdot 5 \cdot 11 \)
Automorphism group:	$F_{11}\times C_2^3:\GL(3,2)$, of order \(147840\)\(\medspace = 2^{7} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \)
Composition factors:	$C_2$ x 4, $C_5$, $C_{11}$
Derived length:	$2$

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

Group statistics

Order	1	2	5	10	11	22
Elements	1	95	44	660	10	70	880
Conjugacy classes	1	15	4	60	1	7	88
Divisions	1	15	1	15	1	7	40
Autjugacy classes	1	2	4	8	1	1	17

Dimension	1	4	10
Irr. complex chars.	80	0	8	88
Irr. rational chars.	16	16	8	40

Minimal presentations

Permutation degree:	$17$
Transitive degree:	$88$
Rank:	$4$
Inequivalent generating quadruples:	$1244880$

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:	$\langle a, b, c, d \mid a^{2}=b^{2}=c^{10}=d^{22}=[a,b]=[a,c]=[a,d]=[b,c]=[b,d]=1, d^{c}=d^{7} \rangle$
Permutation group:	Degree $17$ $\langle(2,3)(4,6)(5,8)(7,11)(9,10)(12,13)(14,15)(16,17), (12,14)(13,15)(16,17) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 25 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 23 & 0 \\ 0 & 23 \end{array}\right), \left(\begin{array}{rr} 10 & 0 \\ 0 & 10 \end{array}\right), \left(\begin{array}{rr} 1 & 3 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 23 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 32 & 12 \\ 0 & 23 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/33\Z)$
Direct product:	$C_2$ ${}^3$ $\, \times\, $ $F_{11}$
Semidirect product:	$(C_2\times D_{22})$ $\,\rtimes\,$ $C_{10}$	$D_{22}$ $\,\rtimes\,$ $(C_2\times C_{10})$	$(C_2^2\times D_{22})$ $\,\rtimes\,$ $C_5$	$(C_{11}:C_5)$ $\,\rtimes\,$ $C_2^4$	all 12
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Aut. group:	$\Aut(C_8\times D_{11})$	$\Aut(C_{88}:C_2)$	$\Aut(C_{11}:C_{24})$	$\Aut(C_{12}\times D_{11})$	all 11

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

Homology

Abelianization:	$C_{2}^{3} \times C_{10} \simeq C_{2}^{4} \times C_{5}$
Schur multiplier:	$C_{2}^{6}$
Commutator length:	$1$

Subgroups

There are 1448 subgroups in 268 conjugacy classes, 150 normal (8 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^3$	$G/Z \simeq$ $F_{11}$
Commutator:	$G' \simeq$ $C_{11}$	$G/G' \simeq$ $C_2^3\times C_{10}$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_2^3\times F_{11}$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^2\times C_{22}$	$G/\operatorname{Fit} \simeq$ $C_{10}$
Radical:	$R \simeq$ $C_2^3\times F_{11}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^2\times C_{22}$	$G/\operatorname{soc} \simeq$ $C_{10}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^4$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5$
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

For the default diagram, subgroups are sorted vertically by the number of prime divisors (counted with multiplicity) in their orders.
To see subgroups sorted vertically by order instead, check this box.

Sorry, your browser does not support the subgroup diagram.

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

See a full page version of the diagram

See a full page version of the diagram

See a full page version of the diagram

See a full page version of the diagram

Classes of subgroups up to conjugation

Order 880: $C_2^3\times F_{11}$

Order 440: $C_2^2\times F_{11}$ x 14, $C_2\times C_{22}:C_{10}$

Order 220: $C_2\times F_{11}$ x 28, $C_{22}:C_{10}$ x 7

Order 176: $C_2^2\times D_{22}$

Order 110: $F_{11}$ x 8, $C_{11}:C_{10}$ x 7

Order 88: $C_2\times D_{22}$ x 14, $C_2^2\times C_{22}$

Order 80: $C_2^3\times C_{10}$

Order 55: $C_{11}:C_5$

Order 44: $D_{22}$ x 28, $C_2\times C_{22}$ x 7

Order 40: $C_2^2\times C_{10}$ x 15

Order 22: $D_{11}$ x 8, $C_{22}$ x 7

Order 20: $C_2\times C_{10}$ x 35

Order 16: $C_2^4$

Order 11: $C_{11}$

Order 10: $C_{10}$ x 15

Order 8: $C_2^3$ x 15

Order 5: $C_5$

Order 4: $C_2^2$ x 35

Order 2: $C_2$ x 15

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 880: $C_2^3\times F_{11}$

Order 440: $C_2\times C_{22}:C_{10}$, $C_2^2\times F_{11}$

Order 220: $C_{22}:C_{10}$, $C_2\times F_{11}$

Order 176: $C_2^2\times D_{22}$

Order 110: $C_{11}:C_{10}$, $F_{11}$

Order 88: $C_2^2\times C_{22}$, $C_2\times D_{22}$

Order 80: $C_2^3\times C_{10}$

Order 55: $C_{11}:C_5$

Order 44: $C_2\times C_{22}$, $D_{22}$

Order 40: $C_2^2\times C_{10}$ x 2

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

Order 20: $C_2\times C_{10}$ x 2

Order 16: $C_2^4$

Order 11: $C_{11}$

Order 10: $C_{10}$ x 2

Order 8: $C_2^3$ x 2

Order 5: $C_5$

Order 4: $C_2^2$ x 2

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 880: $C_2^3\times F_{11}$ ($C_1$)

Order 440: $C_2^2\times F_{11}$ ($C_2$) x 14, $C_2\times C_{22}:C_{10}$ ($C_2$)

Order 220: $C_2\times F_{11}$ ($C_2^2$) x 28, $C_{22}:C_{10}$ ($C_2^2$) x 7

Order 176: $C_2^2\times D_{22}$ ($C_5$)

Order 110: $F_{11}$ ($C_2^3$) x 8, $C_{11}:C_{10}$ ($C_2^3$) x 7

Order 88: $C_2\times D_{22}$ ($C_{10}$) x 14, $C_2^2\times C_{22}$ ($C_{10}$)

Order 55: $C_{11}:C_5$ ($C_2^4$)

Order 44: $D_{22}$ ($C_2\times C_{10}$) x 28, $C_2\times C_{22}$ ($C_2\times C_{10}$) x 7

Order 22: $D_{11}$ ($C_2^2\times C_{10}$) x 8, $C_{22}$ ($C_2^2\times C_{10}$) x 7

Order 11: $C_{11}$ ($C_2^3\times C_{10}$)

Order 8: $C_2^3$ ($F_{11}$)

Order 4: $C_2^2$ ($C_2\times F_{11}$) x 7

Order 2: $C_2$ ($C_2^2\times F_{11}$) x 7

Order 1: $C_1$ ($C_2^3\times F_{11}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 880: $C_2^3\times F_{11}$ ($C_1$)

Order 440: $C_2\times C_{22}:C_{10}$ ($C_2$), $C_2^2\times F_{11}$ ($C_2$)

Order 220: $C_{22}:C_{10}$ ($C_2^2$), $C_2\times F_{11}$ ($C_2^2$)

Order 176: $C_2^2\times D_{22}$ ($C_5$)

Order 110: $C_{11}:C_{10}$ ($C_2^3$), $F_{11}$ ($C_2^3$)

Order 88: $C_2^2\times C_{22}$ ($C_{10}$), $C_2\times D_{22}$ ($C_{10}$)

Order 55: $C_{11}:C_5$ ($C_2^4$)

Order 44: $C_2\times C_{22}$ ($C_2\times C_{10}$), $D_{22}$ ($C_2\times C_{10}$)

Order 22: $C_{22}$ ($C_2^2\times C_{10}$), $D_{11}$ ($C_2^2\times C_{10}$)

Order 11: $C_{11}$ ($C_2^3\times C_{10}$)

Order 8: $C_2^3$ ($F_{11}$)

Order 4: $C_2^2$ ($C_2\times F_{11}$)

Order 2: $C_2$ ($C_2^2\times F_{11}$)

Order 1: $C_1$ ($C_2^3\times F_{11}$)

Series

Derived series

$C_2^3\times F_{11}$

$\rhd$

$C_{11}$

$\rhd$

$C_1$

Chief series

$C_2^3\times F_{11}$

$\rhd$

$C_2\times C_{22}:C_{10}$

$\rhd$

$C_{22}:C_{10}$

$\rhd$

$C_{11}:C_{10}$

$\rhd$

$C_{22}$

$\rhd$

$C_{11}$

$\rhd$

$C_1$

Lower central series

$C_2^3\times F_{11}$

$\rhd$

$C_{11}$

Upper central series

$C_1$

$\lhd$

$C_2^3$

Supergroups

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

This group is a maximal quotient of 16 larger groups in the database.

Character theory

Complex character table

See the $88 \times 88$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $40 \times 40$ rational character table.