Abstract group 880.30: $C_{88}.C_{10}$

• Label: 880.30
• Order: \( 2^{4} \cdot 5 \cdot 11 \)
• Exponent: \( 2^{3} \cdot 5 \cdot 11 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 5 \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{6} \cdot 5 \cdot 11 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \)
• Perm deg.: $27$
• Trans deg.: $176$
• Rank: $2$

Group information

Description:	$C_{88}.C_{10}$
Order:	\(880\)\(\medspace = 2^{4} \cdot 5 \cdot 11 \)
Exponent:	\(440\)\(\medspace = 2^{3} \cdot 5 \cdot 11 \)
Automorphism group:	$D_8:C_2\times F_{11}$, of order \(3520\)\(\medspace = 2^{6} \cdot 5 \cdot 11 \)
Composition factors:	$C_2$ x 4, $C_5$, $C_{11}$
Derived length:	$2$

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

Group statistics

Order	1	2	4	5	8	10	11	20	22	40	44	88
Elements	1	1	10	44	4	44	10	440	10	176	100	40	880
Conjugacy classes	1	1	3	4	2	4	2	12	2	8	6	4	49
Divisions	1	1	3	1	1	1	1	3	1	1	3	1	18
Autjugacy classes	1	1	2	4	1	4	1	8	1	4	2	1	30

Dimension	1	2	4	5	8	10	16	20	40
Irr. complex chars.	20	15	0	8	0	6	0	0	0	49
Irr. rational chars.	4	1	5	0	1	4	1	1	1	18

Minimal presentations

Permutation degree:	$27$
Transitive degree:	$176$
Rank:	$2$
Inequivalent generating pairs:	$72$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	10	20	40
Arbitrary	7	14	18

Constructions

Presentation:	$\langle a, b \mid b^{88}=1, a^{10}=b^{44}, b^{a}=b^{15} \rangle$
Permutation group:	Degree $27$ $\langle(12,13,16,19)(14,20,22,27)(15,23,25,18)(17,24,26,21), (12,14,16,22)(13,17,19,26) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 33 \\ 55 & 45 \end{array}\right), \left(\begin{array}{rr} 10 & 66 \\ 55 & 32 \end{array}\right), \left(\begin{array}{rr} 15 & 0 \\ 0 & 64 \end{array}\right), \left(\begin{array}{rr} 1 & 7 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 45 & 55 \\ 66 & 67 \end{array}\right), \left(\begin{array}{rr} 34 & 0 \\ 0 & 34 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/77\Z)$
Direct product:	$Q_{16}$ $\, \times\, $ $(C_{11}:C_5)$
Semidirect product:	$(C_{11}\times Q_{16})$ $\,\rtimes\,$ $C_5$	$C_{11}$ $\,\rtimes\,$ $(C_5\times Q_{16})$			more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{88}$ . $C_{10}$	$C_{22}$ . $(C_5\times D_4)$	$(Q_8\times C_{11})$ . $C_{10}$ (2)	$Q_8$ . $(C_{11}:C_{10})$ (2)	all 12

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

Homology

Abelianization:	$C_{2} \times C_{10} \simeq C_{2}^{2} \times C_{5}$
Schur multiplier:	$C_1$
Commutator length:	$1$

Subgroups

There are 154 subgroups in 36 conjugacy classes, 21 normal (15 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_{44}:C_{10}$
Commutator:	$G' \simeq$ $C_{44}$	$G/G' \simeq$ $C_2\times C_{10}$
Frattini:	$\Phi \simeq$ $C_4$	$G/\Phi \simeq$ $C_{22}:C_{10}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{11}\times Q_{16}$	$G/\operatorname{Fit} \simeq$ $C_5$
Radical:	$R \simeq$ $C_{88}.C_{10}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{22}$	$G/\operatorname{soc} \simeq$ $C_5\times D_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $Q_{16}$
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

Classes of subgroups up to automorphism

Normal subgroups

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 880: $C_{88}.C_{10}$

Order 440: $C_{44}.C_{10}$ x 2, $C_{11}:C_{40}$

Order 220: $C_{11}:C_{20}$ x 3

Order 176: $C_{11}\times Q_{16}$

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

Order 88: $Q_8\times C_{11}$ x 2, $C_{88}$

Order 80: $C_5\times Q_{16}$

Order 55: $C_{11}:C_5$

Order 44: $C_{44}$ x 3

Order 40: $C_5\times Q_8$ x 2, $C_{40}$

Order 22: $C_{22}$

Order 20: $C_{20}$ x 3

Order 16: $Q_{16}$

Order 11: $C_{11}$

Order 10: $C_{10}$

Order 8: $Q_8$ x 2, $C_8$

Order 5: $C_5$

Order 4: $C_4$ x 3

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 880: $C_{88}.C_{10}$

Order 440: $C_{11}:C_{40}$, $C_{44}.C_{10}$

Order 220: $C_{11}:C_{20}$ x 2

Order 176: $C_{11}\times Q_{16}$

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

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

Order 80: $C_5\times Q_{16}$

Order 55: $C_{11}:C_5$

Order 44: $C_{44}$ x 2

Order 40: $C_{40}$, $C_5\times Q_8$

Order 22: $C_{22}$

Order 20: $C_{20}$ x 2

Order 16: $Q_{16}$

Order 11: $C_{11}$

Order 10: $C_{10}$

Order 8: $Q_8$, $C_8$

Order 5: $C_5$

Order 4: $C_4$ x 2

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 880: $C_{88}.C_{10}$ ($C_1$)

Order 440: $C_{44}.C_{10}$ ($C_2$) x 2, $C_{11}:C_{40}$ ($C_2$)

Order 220: $C_{11}:C_{20}$ ($C_2^2$)

Order 176: $C_{11}\times Q_{16}$ ($C_5$)

Order 110: $C_{11}:C_{10}$ ($D_4$)

Order 88: $Q_8\times C_{11}$ ($C_{10}$) x 2, $C_{88}$ ($C_{10}$)

Order 55: $C_{11}:C_5$ ($Q_{16}$)

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

Order 22: $C_{22}$ ($C_5\times D_4$)

Order 16: $Q_{16}$ ($C_{11}:C_5$)

Order 11: $C_{11}$ ($C_5\times Q_{16}$)

Order 8: $Q_8$ ($C_{11}:C_{10}$) x 2, $C_8$ ($C_{11}:C_{10}$)

Order 4: $C_4$ ($C_{22}:C_{10}$)

Order 2: $C_2$ ($C_{44}:C_{10}$)

Order 1: $C_1$ ($C_{88}.C_{10}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 880: $C_{88}.C_{10}$ ($C_1$)

Order 440: $C_{11}:C_{40}$ ($C_2$), $C_{44}.C_{10}$ ($C_2$)

Order 220: $C_{11}:C_{20}$ ($C_2^2$)

Order 176: $C_{11}\times Q_{16}$ ($C_5$)

Order 110: $C_{11}:C_{10}$ ($D_4$)

Order 88: $C_{88}$ ($C_{10}$), $Q_8\times C_{11}$ ($C_{10}$)

Order 55: $C_{11}:C_5$ ($Q_{16}$)

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

Order 22: $C_{22}$ ($C_5\times D_4$)

Order 16: $Q_{16}$ ($C_{11}:C_5$)

Order 11: $C_{11}$ ($C_5\times Q_{16}$)

Order 8: $Q_8$ ($C_{11}:C_{10}$), $C_8$ ($C_{11}:C_{10}$)

Order 4: $C_4$ ($C_{22}:C_{10}$)

Order 2: $C_2$ ($C_{44}:C_{10}$)

Order 1: $C_1$ ($C_{88}.C_{10}$)

Series

Derived series

$C_{88}.C_{10}$

$\rhd$

$C_{44}$

$\rhd$

$C_1$

Chief series

$C_{88}.C_{10}$

$\rhd$

$C_{11}:C_{40}$

$\rhd$

$C_{88}$

$\rhd$

$C_{44}$

$\rhd$

$C_{22}$

$\rhd$

$C_{11}$

$\rhd$

$C_1$

Lower central series

$C_{88}.C_{10}$

$\rhd$

$C_{44}$

$\rhd$

$C_{22}$

$\rhd$

$C_{11}$

Upper central series

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_4$

$\lhd$

$Q_{16}$

Supergroups

This group is a maximal subgroup of 10 larger groups in the database.

This group is a maximal quotient of 3 larger groups in the database.

Character theory

Complex character table

See the $49 \times 49$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $18 \times 18$ rational character table.