Abstract group 2200.u: $C_{220}.C_{10}$

• Label: 2200.u
• Order: \( 2^{3} \cdot 5^{2} \cdot 11 \)
• Exponent: \( 2^{2} \cdot 5 \cdot 11 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 5^{2} \)
• $\card{Z(G)}$: \( 2 \cdot 5 \)
• $\card{\Aut(G)}$: \( 2^{6} \cdot 5^{2} \cdot 11 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \cdot 5 \)
• Perm deg.: $24$
• Trans deg.: $440$
• Rank: $2$

Group information

Description:	$C_{220}.C_{10}$
Order:	\(2200\)\(\medspace = 2^{3} \cdot 5^{2} \cdot 11 \)
Exponent:	\(220\)\(\medspace = 2^{2} \cdot 5 \cdot 11 \)
Automorphism group:	$C_{110}.C_{10}.C_2^4$, of order \(17600\)\(\medspace = 2^{6} \cdot 5^{2} \cdot 11 \)
Composition factors:	$C_2$ x 3, $C_5$ x 2, $C_{11}$
Derived length:	$2$

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

Group statistics

Order	1	2	4	5	10	11	20	22	44	55	110	220
Elements	1	1	46	224	224	10	1504	10	20	40	40	80	2200
Conjugacy classes	1	1	3	24	24	1	72	1	2	4	4	8	145
Divisions	1	1	3	6	6	1	18	1	1	1	1	1	41
Autjugacy classes	1	1	2	5	5	1	10	1	1	1	1	1	30

Dimension	1	2	4	8	10	20	40	80
Irr. complex chars.	100	25	0	0	20	0	0	0	145
Irr. rational chars.	4	1	24	6	2	1	2	1	41

Minimal presentations

Permutation degree:	$24$
Transitive degree:	$440$
Rank:	$2$
Inequivalent generating pairs:	$72$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	10	20	80
Arbitrary	10	14	18

Constructions

Presentation:	$\langle a, b \mid b^{220}=1, a^{10}=b^{110}, b^{a}=b^{171} \rangle$
Permutation group:	Degree $24$ $\langle(20,21,22,23,24), (1,2,5,9,11,6,10,4,3,7,8)(20,21,22,23,24), (2,6,9,11,7) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrr} 3 & 0 & 0 & 0 \\ 0 & 3 & 0 & 0 \\ 0 & 0 & 3 & 0 \\ 0 & 0 & 0 & 3 \end{array}\right), \left(\begin{array}{rrrr} 4 & 10 & 10 & 9 \\ 8 & 10 & 0 & 10 \\ 1 & 2 & 8 & 1 \\ 1 & 1 & 3 & 3 \end{array}\right), \left(\begin{array}{rrrr} 8 & 0 & 5 & 6 \\ 2 & 10 & 5 & 5 \\ 5 & 5 & 7 & 0 \\ 10 & 5 & 9 & 9 \end{array}\right), \left(\begin{array}{rrrr} 8 & 0 & 0 & 0 \\ 0 & 8 & 0 & 0 \\ 0 & 0 & 8 & 0 \\ 0 & 0 & 0 & 8 \end{array}\right), \left(\begin{array}{rrrr} 10 & 0 & 2 & 1 \\ 0 & 8 & 4 & 6 \\ 9 & 1 & 3 & 0 \\ 1 & 5 & 6 & 1 \end{array}\right), \left(\begin{array}{rrrr} 1 & 7 & 0 & 7 \\ 4 & 8 & 7 & 1 \\ 5 & 4 & 4 & 7 \\ 7 & 7 & 8 & 9 \end{array}\right) \right\rangle \subseteq \GL_{4}(\F_{11})$
Direct product:	$C_5$ $\, \times\, $ $(C_{44}.C_{10})$
Semidirect product:	$(C_{55}:Q_8)$ $\,\rtimes\,$ $C_5$	$(C_{55}:C_5)$ $\,\rtimes\,$ $Q_8$	$C_{55}$ $\,\rtimes\,$ $(C_5\times Q_8)$	$(C_{11}:Q_8)$ $\,\rtimes\,$ $C_5^2$	all 6
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{20}$ . $F_{11}$	$C_{220}$ . $C_{10}$	$C_{22}$ . $C_{10}^2$	$C_4$ . $(C_5\times F_{11})$	all 16
Aut. group:	$\Aut((C_6\times D_{12}):D_6)$	$\Aut(C_3^6.(S_3\times C_{18}))$	$\Aut(C_3^5.C_3\wr C_4)$

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

Homology

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

Subgroups

There are 516 subgroups in 96 conjugacy classes, 54 normal (18 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_{10}$	$G/Z \simeq$ $C_2\times F_{11}$
Commutator:	$G' \simeq$ $C_{22}$	$G/G' \simeq$ $C_{10}^2$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_{10}\times F_{11}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{220}$	$G/\operatorname{Fit} \simeq$ $C_{10}$
Radical:	$R \simeq$ $C_{220}.C_{10}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{110}$	$G/\operatorname{soc} \simeq$ $C_2\times C_{10}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $Q_8$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5^2$
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 2200: $C_{220}.C_{10}$

Order 1100: $C_{55}:C_{20}$ x 2, $C_{220}:C_5$

Order 550: $C_{110}:C_5$

Order 440: $C_{44}.C_{10}$ x 5, $C_{55}:Q_8$

Order 275: $C_{55}:C_5$

Order 220: $C_{11}:C_{20}$ x 10, $C_{11}:C_{20}$ x 5, $C_{11}:C_{20}$ x 2, $C_{220}$

Order 200: $Q_8\times C_5^2$

Order 110: $C_{11}:C_{10}$ x 5, $C_{110}$

Order 100: $C_5\times C_{20}$ x 3

Order 88: $C_{11}:Q_8$

Order 55: $C_{11}:C_5$ x 5, $C_{55}$

Order 50: $C_5\times C_{10}$

Order 44: $C_{11}:C_4$ x 2, $C_{44}$

Order 40: $C_5\times Q_8$ x 6

Order 25: $C_5^2$

Order 22: $C_{22}$

Order 20: $C_{20}$ x 18

Order 11: $C_{11}$

Order 10: $C_{10}$ x 6

Order 8: $Q_8$

Order 5: $C_5$ x 6

Order 4: $C_4$ x 3

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 2200: $C_{220}.C_{10}$

Order 1100: $C_{220}:C_5$, $C_{55}:C_{20}$

Order 550: $C_{110}:C_5$

Order 440: $C_{44}.C_{10}$, $C_{55}:Q_8$

Order 275: $C_{55}:C_5$

Order 220: $C_{220}$, $C_{11}:C_{20}$, $C_{11}:C_{20}$, $C_{11}:C_{20}$

Order 200: $Q_8\times C_5^2$

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

Order 100: $C_5\times C_{20}$ x 2

Order 88: $C_{11}:Q_8$

Order 55: $C_{55}$, $C_{11}:C_5$

Order 50: $C_5\times C_{10}$

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

Order 40: $C_5\times Q_8$ x 2

Order 25: $C_5^2$

Order 22: $C_{22}$

Order 20: $C_{20}$ x 4

Order 11: $C_{11}$

Order 10: $C_{10}$ x 2

Order 8: $Q_8$

Order 5: $C_5$ x 2

Order 4: $C_4$ x 2

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 2200: $C_{220}.C_{10}$ ($C_1$)

Order 1100: $C_{55}:C_{20}$ ($C_2$) x 2, $C_{220}:C_5$ ($C_2$)

Order 550: $C_{110}:C_5$ ($C_2^2$)

Order 440: $C_{44}.C_{10}$ ($C_5$) x 5, $C_{55}:Q_8$ ($C_5$)

Order 275: $C_{55}:C_5$ ($Q_8$)

Order 220: $C_{11}:C_{20}$ ($C_{10}$) x 10, $C_{11}:C_{20}$ ($C_{10}$) x 5, $C_{11}:C_{20}$ ($C_{10}$) x 2, $C_{220}$ ($C_{10}$)

Order 110: $C_{11}:C_{10}$ ($C_2\times C_{10}$) x 5, $C_{110}$ ($C_2\times C_{10}$)

Order 88: $C_{11}:Q_8$ ($C_5^2$)

Order 55: $C_{11}:C_5$ ($C_5\times Q_8$) x 5, $C_{55}$ ($C_5\times Q_8$)

Order 44: $C_{11}:C_4$ ($C_5\times C_{10}$) x 2, $C_{44}$ ($C_5\times C_{10}$)

Order 22: $C_{22}$ ($C_{10}^2$)

Order 20: $C_{20}$ ($F_{11}$)

Order 11: $C_{11}$ ($Q_8\times C_5^2$)

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

Order 5: $C_5$ ($C_{44}.C_{10}$)

Order 4: $C_4$ ($C_5\times F_{11}$)

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

Order 1: $C_1$ ($C_{220}.C_{10}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 2200: $C_{220}.C_{10}$ ($C_1$)

Order 1100: $C_{220}:C_5$ ($C_2$), $C_{55}:C_{20}$ ($C_2$)

Order 550: $C_{110}:C_5$ ($C_2^2$)

Order 440: $C_{44}.C_{10}$ ($C_5$), $C_{55}:Q_8$ ($C_5$)

Order 275: $C_{55}:C_5$ ($Q_8$)

Order 220: $C_{220}$ ($C_{10}$), $C_{11}:C_{20}$ ($C_{10}$), $C_{11}:C_{20}$ ($C_{10}$), $C_{11}:C_{20}$ ($C_{10}$)

Order 110: $C_{110}$ ($C_2\times C_{10}$), $C_{11}:C_{10}$ ($C_2\times C_{10}$)

Order 88: $C_{11}:Q_8$ ($C_5^2$)

Order 55: $C_{55}$ ($C_5\times Q_8$), $C_{11}:C_5$ ($C_5\times Q_8$)

Order 44: $C_{44}$ ($C_5\times C_{10}$), $C_{11}:C_4$ ($C_5\times C_{10}$)

Order 22: $C_{22}$ ($C_{10}^2$)

Order 20: $C_{20}$ ($F_{11}$)

Order 11: $C_{11}$ ($Q_8\times C_5^2$)

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

Order 5: $C_5$ ($C_{44}.C_{10}$)

Order 4: $C_4$ ($C_5\times F_{11}$)

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

Order 1: $C_1$ ($C_{220}.C_{10}$)

Series

Derived series

$C_{220}.C_{10}$

$\rhd$

$C_{22}$

$\rhd$

$C_1$

Chief series

$C_{220}.C_{10}$

$\rhd$

$C_{220}:C_5$

$\rhd$

$C_{220}$

$\rhd$

$C_{110}$

$\rhd$

$C_{55}$

$\rhd$

$C_{11}$

$\rhd$

$C_1$

Lower central series

$C_{220}.C_{10}$

$\rhd$

$C_{22}$

$\rhd$

$C_{11}$

Upper central series

$C_1$

$\lhd$

$C_{10}$

$\lhd$

$C_{20}$

Supergroups

This group is a maximal subgroup of 7 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 $145 \times 145$ 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 $41 \times 41$ rational character table.