Abstract group 440.48: $C_{10}\times D_{22}$

• Label: 440.48
• Order: \( 2^{3} \cdot 5 \cdot 11 \)
• Exponent: \( 2 \cdot 5 \cdot 11 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 5 \)
• $\card{Z(G)}$: \( 2^{2} \cdot 5 \)
• $\card{\Aut(G)}$: \( 2^{6} \cdot 3 \cdot 5 \cdot 11 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \cdot 3 \cdot 5 \)
• Perm deg.: $20$
• Trans deg.: $220$
• Rank: $3$

Group information

Description:	$C_{10}\times D_{22}$
Order:	\(440\)\(\medspace = 2^{3} \cdot 5 \cdot 11 \)
Exponent:	\(110\)\(\medspace = 2 \cdot 5 \cdot 11 \)
Automorphism group:	$(C_{11}\times A_4).C_{20}.C_2^2$, of order \(10560\)\(\medspace = 2^{6} \cdot 3 \cdot 5 \cdot 11 \)
Composition factors:	$C_2$ x 3, $C_5$, $C_{11}$
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	5	10	11	22	55	110
Elements	1	47	4	188	10	30	40	120	440
Conjugacy classes	1	7	4	28	5	15	20	60	140
Divisions	1	7	1	7	1	3	1	3	24
Autjugacy classes	1	2	1	2	1	1	1	1	10

Dimension	1	2	4	10	40
Irr. complex chars.	40	100	0	0	0	140
Irr. rational chars.	8	0	8	4	4	24

Minimal presentations

Permutation degree:	$20$
Transitive degree:	$220$
Rank:	$3$
Inequivalent generating triples:	$2604$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	3	4	14

Constructions

Presentation:	$\langle a, b, c \mid a^{2}=b^{2}=c^{110}=[a,b]=[a,c]=1, c^{b}=c^{21} \rangle$
Permutation group:	Degree $20$ $\langle(2,3)(4,5)(6,7)(8,9)(10,11), (17,18)(19,20), (17,19)(18,20), (12,13,14,15,16), (1,2,4,6,8,10,11,9,7,5,3)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrr} 7 & 3 & 4 \\ 1 & 2 & 9 \\ 3 & 5 & 10 \end{array}\right), \left(\begin{array}{rrr} 9 & 5 & 0 \\ 7 & 4 & 0 \\ 7 & 3 & 1 \end{array}\right), \left(\begin{array}{rrr} 8 & 2 & 1 \\ 2 & 0 & 5 \\ 8 & 2 & 4 \end{array}\right), \left(\begin{array}{rrr} 10 & 7 & 0 \\ 0 & 1 & 0 \\ 5 & 10 & 1 \end{array}\right), \left(\begin{array}{rrr} 10 & 0 & 0 \\ 0 & 10 & 0 \\ 0 & 0 & 10 \end{array}\right) \right\rangle \subseteq \GL_{3}(\F_{11})$
Direct product:	$C_2$ ${}^2$ $\, \times\, $ $C_5$ $\, \times\, $ $D_{11}$
Semidirect product:	$C_{55}$ $\,\rtimes\,$ $C_2^3$	$C_{110}$ $\,\rtimes\,$ $C_2^2$ (3)	$(C_2\times C_{110})$ $\,\rtimes\,$ $C_2$	$(C_2\times C_{22})$ $\,\rtimes\,$ $C_{10}$	all 6
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product

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

Homology

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

Subgroups

There are 284 subgroups in 64 conjugacy classes, 42 normal (10 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2\times C_{10}$	$G/Z \simeq$ $D_{11}$
Commutator:	$G' \simeq$ $C_{11}$	$G/G' \simeq$ $C_2^2\times C_{10}$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_{10}\times D_{22}$
Fitting:	$\operatorname{Fit} \simeq$ $C_2\times C_{110}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_{10}\times D_{22}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_{110}$	$G/\operatorname{soc} \simeq$ $C_2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^3$
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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Subgroup information

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Classes of subgroups up to conjugation

Order 440: $C_{10}\times D_{22}$

Order 220: $C_5\times D_{22}$ x 6, $C_2\times C_{110}$

Order 110: $C_5\times D_{11}$ x 4, $C_{110}$ x 3

Order 88: $C_2\times D_{22}$

Order 55: $C_{55}$

Order 44: $D_{22}$ x 6, $C_2\times C_{22}$

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

Order 22: $D_{11}$ x 4, $C_{22}$ x 3

Order 20: $C_2\times C_{10}$ x 7

Order 11: $C_{11}$

Order 10: $C_{10}$ x 7

Order 8: $C_2^3$

Order 5: $C_5$

Order 4: $C_2^2$ x 7

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 440: $C_{10}\times D_{22}$

Order 220: $C_2\times C_{110}$, $C_5\times D_{22}$

Order 110: $C_{110}$, $C_5\times D_{11}$

Order 88: $C_2\times D_{22}$

Order 55: $C_{55}$

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

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

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

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

Order 11: $C_{11}$

Order 10: $C_{10}$ x 2

Order 8: $C_2^3$

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 440: $C_{10}\times D_{22}$ ($C_1$)

Order 220: $C_5\times D_{22}$ ($C_2$) x 6, $C_2\times C_{110}$ ($C_2$)

Order 110: $C_5\times D_{11}$ ($C_2^2$) x 4, $C_{110}$ ($C_2^2$) x 3

Order 88: $C_2\times D_{22}$ ($C_5$)

Order 55: $C_{55}$ ($C_2^3$)

Order 44: $D_{22}$ ($C_{10}$) x 6, $C_2\times C_{22}$ ($C_{10}$)

Order 22: $D_{11}$ ($C_2\times C_{10}$) x 4, $C_{22}$ ($C_2\times C_{10}$) x 3

Order 20: $C_2\times C_{10}$ ($D_{11}$)

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

Order 10: $C_{10}$ ($D_{22}$) x 3

Order 5: $C_5$ ($C_2\times D_{22}$)

Order 4: $C_2^2$ ($C_5\times D_{11}$)

Order 2: $C_2$ ($C_5\times D_{22}$) x 3

Order 1: $C_1$ ($C_{10}\times D_{22}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 440: $C_{10}\times D_{22}$ ($C_1$)

Order 220: $C_2\times C_{110}$ ($C_2$), $C_5\times D_{22}$ ($C_2$)

Order 110: $C_{110}$ ($C_2^2$), $C_5\times D_{11}$ ($C_2^2$)

Order 88: $C_2\times D_{22}$ ($C_5$)

Order 55: $C_{55}$ ($C_2^3$)

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

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

Order 20: $C_2\times C_{10}$ ($D_{11}$)

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

Order 10: $C_{10}$ ($D_{22}$)

Order 5: $C_5$ ($C_2\times D_{22}$)

Order 4: $C_2^2$ ($C_5\times D_{11}$)

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

Order 1: $C_1$ ($C_{10}\times D_{22}$)

Series

Derived series

$C_{10}\times D_{22}$

$\rhd$

$C_{11}$

$\rhd$

$C_1$

Chief series

$C_{10}\times D_{22}$

$\rhd$

$C_5\times D_{22}$

$\rhd$

$C_{110}$

$\rhd$

$C_{55}$

$\rhd$

$C_{11}$

$\rhd$

$C_1$

Lower central series

$C_{10}\times D_{22}$

$\rhd$

$C_{11}$

Upper central series

$C_1$

$\lhd$

$C_2\times C_{10}$

Supergroups

This group is a maximal subgroup of 24 larger groups in the database.

This group is a maximal quotient of 21 larger groups in the database.

Character theory

Complex character table

See the $140 \times 140$ 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 $24 \times 24$ rational character table.