Abstract group 11000.b: $C_{110}.C_{10}^2$

• Label: 11000.b
• Order: \( 2^{3} \cdot 5^{3} \cdot 11 \)
• Exponent: \( 2^{2} \cdot 5 \cdot 11 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 5^{2} \)
• $\card{Z(G)}$: \( 2 \cdot 5 \)
• $\card{\Aut(G)}$: \( 2^{7} \cdot 5^{3} \cdot 11 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \cdot 5 \)
• Perm deg.: $25$
• Trans deg.: $220$
• Rank: $2$

Group information

Description:	$C_{110}.C_{10}^2$
Order:	\(11000\)\(\medspace = 2^{3} \cdot 5^{3} \cdot 11 \)
Exponent:	\(220\)\(\medspace = 2^{2} \cdot 5 \cdot 11 \)
Automorphism group:	$C_{55}.(C_{10}\times D_5).C_2^5$, of order \(176000\)\(\medspace = 2^{7} \cdot 5^{3} \cdot 11 \)
Composition factors:	$C_2$ x 3, $C_5$ x 3, $C_{11}$
Derived length:	$2$

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

Group statistics

Order	1	2	4	5	10	11	20	22	44	55	110	220
Elements	1	133	10	1124	6492	10	2240	10	100	240	240	400	11000
Conjugacy classes	1	3	1	74	222	1	24	1	2	14	14	8	365
Divisions	1	3	1	19	56	1	6	1	1	4	4	1	98
Autjugacy classes	1	3	1	11	27	1	5	1	1	3	3	1	58

Dimension	1	2	4	8	10	20	40	80
Irr. complex chars.	100	225	0	0	20	20	0	0	365
Irr. rational chars.	4	1	26	55	2	1	4	5	98

Minimal presentations

Permutation degree:	$25$
Transitive degree:	$220$
Rank:	$2$
Inequivalent generating pairs:	$144$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	20	40	80
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	$\langle a, b, c \mid a^{10}=b^{10}=c^{110}=1, b^{a}=bc^{85}, c^{a}=c^{61}, c^{b}=c^{39} \rangle$
Permutation group:	Degree $25$ $\langle(10,11,14,13,12), (6,7)(8,9)(10,12,13,14,11)(16,19,21,18,24,25,22,17,23,20) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrr} 3 & 0 & 0 & 0 \\ 0 & 3 & 0 & 0 \\ 0 & 0 & 3 & 0 \\ 0 & 0 & 0 & 3 \end{array}\right), \left(\begin{array}{rrrr} 6 & 4 & 10 & 5 \\ 9 & 1 & 6 & 7 \\ 3 & 5 & 5 & 2 \\ 2 & 9 & 8 & 3 \end{array}\right), \left(\begin{array}{rrrr} 1 & 2 & 3 & 4 \\ 4 & 5 & 0 & 9 \\ 1 & 9 & 6 & 8 \\ 10 & 0 & 6 & 4 \end{array}\right), \left(\begin{array}{rrrr} 10 & 6 & 3 & 0 \\ 2 & 5 & 7 & 5 \\ 2 & 4 & 4 & 1 \\ 0 & 9 & 5 & 8 \end{array}\right), \left(\begin{array}{rrrr} 6 & 3 & 3 & 9 \\ 3 & 0 & 9 & 0 \\ 5 & 5 & 3 & 3 \\ 5 & 0 & 3 & 0 \end{array}\right), \left(\begin{array}{rrrr} 6 & 10 & 10 & 4 \\ 4 & 2 & 5 & 5 \\ 3 & 6 & 5 & 0 \\ 3 & 2 & 6 & 6 \end{array}\right), \left(\begin{array}{rrrr} 10 & 4 & 9 & 3 \\ 3 & 10 & 5 & 9 \\ 6 & 2 & 8 & 7 \\ 7 & 6 & 8 & 8 \end{array}\right) \right\rangle \subseteq \GL_{4}(\F_{11})$
Direct product:	$C_5$ $\, \times\, $ $(D_{110}:C_{10})$
Semidirect product:	$(C_{10}\times F_{11})$ $\,\rtimes\,$ $D_5$	$(C_5:C_{20})$ $\,\rtimes\,$ $F_{11}$	$D_{22}$ $\,\rtimes\,$ $(D_5\times C_5^2)$	$(D_{110}:C_{10})$ $\,\rtimes\,$ $C_5$	all 30
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{110}$ . $C_{10}^2$	$C_{10}$ . $(D_5\times F_{11})$	$(C_{110}:C_5)$ . $D_{10}$	$C_{110}$ . $(C_5\times D_{10})$	all 12

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:	$C_{10}$
Commutator length:	$1$

Subgroups

There are 6440 subgroups in 496 conjugacy classes, 82 normal (37 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_{10}$	$G/Z \simeq$ $D_5\times F_{11}$
Commutator:	$G' \simeq$ $C_{110}$	$G/G' \simeq$ $C_{10}^2$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_{55}:C_{10}^2$
Fitting:	$\operatorname{Fit} \simeq$ $C_5\times C_{110}$	$G/\operatorname{Fit} \simeq$ $C_2\times C_{10}$
Radical:	$R \simeq$ $C_{110}.C_{10}^2$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_5\times C_{110}$	$G/\operatorname{soc} \simeq$ $C_2\times C_{10}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_4$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5^3$
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

No diagram available

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

No diagram available

Classes of subgroups up to conjugation

Order 11000: $C_{110}.C_{10}^2$

Order 5500: $C_{55}:C_{10}^2$, $C_{55}:C_{10}^2$, $C_5\times C_{55}:C_{20}$

Order 2750: $C_5^2:F_{11}$, $C_{110}:C_5^2$, $C_5^2\times F_{11}$

Order 2200: $D_{110}:C_{10}$ x 5, $C_{220}:C_{10}$, $D_{110}:C_{10}$

Order 1375: $C_{11}:C_5^3$

Order 1100: $C_{10}\times F_{11}$ x 19, $C_{110}:C_{10}$ x 5, $C_{55}:C_{20}$ x 5, $C_5\times D_{110}$, $C_5^2\times D_{22}$, $C_5:C_{220}$, $C_{220}:C_5$

Order 1000: $C_{10}^2:C_{10}$

Order 550: $C_5\times F_{11}$ x 31, $C_{110}:C_5$ x 18, $C_{55}:C_{10}$ x 5, $C_5\times C_{110}$, $C_5\times D_{55}$, $D_{11}\times C_5^2$

Order 500: $C_5\times C_{10}^2$, $D_{10}\times C_5^2$, $C_5^2:C_{20}$

Order 440: $C_{44}:C_{10}$ x 5, $C_5\times D_{44}$, $C_5:D_{44}$

Order 275: $C_{55}:C_5$ x 18, $C_5\times C_{55}$

Order 250: $C_5^2\times C_{10}$ x 2, $D_5\times C_5^2$

Order 220: $C_2\times F_{11}$ x 20, $C_{11}:C_{20}$ x 5, $C_5\times D_{22}$ x 5, $C_{220}$, $C_5:C_{44}$, $D_{110}$

Order 200: $C_{10}\wr C_2$ x 6, $D_4\times C_5^2$

Order 125: $C_5^3$

Order 110: $F_{11}$ x 30, $C_{11}:C_{10}$ x 15, $C_5\times D_{11}$ x 7, $C_{110}$ x 4, $D_{55}$

Order 100: $C_{10}^2$ x 20, $C_5:C_{20}$ x 6, $C_5\times D_{10}$ x 6, $C_5\times C_{20}$

Order 88: $D_{44}$

Order 55: $C_{11}:C_5$ x 15, $C_{55}$ x 4

Order 50: $C_5\times C_{10}$ x 51, $C_5\times D_5$ x 6

Order 44: $D_{22}$ x 2, $C_{44}$

Order 40: $C_5\times D_4$ x 6, $C_5:D_4$

Order 25: $C_5^2$ x 19

Order 22: $D_{11}$ x 2, $C_{22}$

Order 20: $C_2\times C_{10}$ x 25, $C_{20}$ x 6, $D_{10}$, $C_5:C_4$

Order 11: $C_{11}$

Order 10: $C_{10}$ x 56, $D_5$

Order 8: $D_4$

Order 5: $C_5$ x 19

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

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 11000: $C_{110}.C_{10}^2$

Order 5500: $C_{55}:C_{10}^2$, $C_{55}:C_{10}^2$, $C_5\times C_{55}:C_{20}$

Order 2750: $C_5^2:F_{11}$, $C_{110}:C_5^2$, $C_5^2\times F_{11}$

Order 2200: $C_{220}:C_{10}$, $D_{110}:C_{10}$, $D_{110}:C_{10}$

Order 1375: $C_{11}:C_5^3$

Order 1100: $C_{10}\times F_{11}$ x 5, $C_5\times D_{110}$, $C_5^2\times D_{22}$, $C_{110}:C_{10}$, $C_5:C_{220}$, $C_{220}:C_5$, $C_{55}:C_{20}$

Order 1000: $C_{10}^2:C_{10}$

Order 550: $C_5\times F_{11}$ x 5, $C_{110}:C_5$ x 4, $C_{55}:C_{10}$, $C_5\times C_{110}$, $C_5\times D_{55}$, $D_{11}\times C_5^2$

Order 500: $C_5\times C_{10}^2$, $D_{10}\times C_5^2$, $C_5^2:C_{20}$

Order 440: $C_{44}:C_{10}$, $C_5\times D_{44}$, $C_5:D_{44}$

Order 275: $C_{55}:C_5$ x 4, $C_5\times C_{55}$

Order 250: $C_5^2\times C_{10}$ x 2, $D_5\times C_5^2$

Order 220: $C_5\times D_{22}$ x 4, $C_2\times F_{11}$ x 3, $C_{220}$, $C_5:C_{44}$, $C_{11}:C_{20}$, $D_{110}$

Order 200: $C_{10}\wr C_2$ x 2, $D_4\times C_5^2$

Order 125: $C_5^3$

Order 110: $C_5\times D_{11}$ x 4, $C_{110}$ x 3, $F_{11}$ x 3, $C_{11}:C_{10}$ x 2, $D_{55}$

Order 100: $C_{10}^2$ x 6, $C_5:C_{20}$ x 2, $C_5\times D_{10}$ x 2, $C_5\times C_{20}$

Order 88: $D_{44}$

Order 55: $C_{55}$ x 3, $C_{11}:C_5$ x 2

Order 50: $C_5\times C_{10}$ x 11, $C_5\times D_5$ x 2

Order 44: $D_{22}$ x 2, $C_{44}$

Order 40: $C_5\times D_4$ x 2, $C_5:D_4$

Order 25: $C_5^2$ x 5

Order 22: $D_{11}$ x 2, $C_{22}$

Order 20: $C_2\times C_{10}$ x 7, $C_{20}$ x 2, $D_{10}$, $C_5:C_4$

Order 11: $C_{11}$

Order 10: $C_{10}$ x 12, $D_5$

Order 8: $D_4$

Order 5: $C_5$ x 5

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

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 11000: $C_{110}.C_{10}^2$ ($C_1$)

Order 5500: $C_{55}:C_{10}^2$ ($C_2$), $C_{55}:C_{10}^2$ ($C_2$), $C_5\times C_{55}:C_{20}$ ($C_2$)

Order 2750: $C_{110}:C_5^2$ ($C_2^2$)

Order 2200: $D_{110}:C_{10}$ ($C_5$) x 5, $D_{110}:C_{10}$ ($C_5$)

Order 1375: $C_{11}:C_5^3$ ($D_4$)

Order 1100: $C_{110}:C_{10}$ ($C_{10}$) x 5, $C_{10}\times F_{11}$ ($C_{10}$) x 5, $C_{55}:C_{20}$ ($C_{10}$) x 5, $C_5\times D_{110}$ ($C_{10}$), $C_5^2\times D_{22}$ ($C_{10}$), $C_{10}\times F_{11}$ ($D_5$), $C_5:C_{220}$ ($C_{10}$)

Order 550: $C_{110}:C_5$ ($C_2\times C_{10}$) x 5, $C_5\times C_{110}$ ($C_2\times C_{10}$), $C_{110}:C_5$ ($D_{10}$)

Order 440: $C_5:D_{44}$ ($C_5^2$)

Order 275: $C_{55}:C_5$ ($C_5\times D_4$) x 5, $C_5\times C_{55}$ ($C_5\times D_4$), $C_{55}:C_5$ ($C_5:D_4$)

Order 220: $C_2\times F_{11}$ ($C_5\times D_5$) x 5, $C_5:C_{44}$ ($C_5\times C_{10}$), $D_{110}$ ($C_5\times C_{10}$), $C_5\times D_{22}$ ($C_5\times C_{10}$), $C_5\times D_{22}$ ($C_5\times D_5$)

Order 110: $C_{11}:C_{10}$ ($C_5\times D_{10}$) x 5, $C_{110}$ ($C_{10}^2$), $C_{110}$ ($C_5\times D_{10}$)

Order 100: $C_5:C_{20}$ ($F_{11}$)

Order 55: $C_{11}:C_5$ ($C_{10}\wr C_2$) x 5, $C_{55}$ ($D_4\times C_5^2$), $C_{55}$ ($C_{10}\wr C_2$)

Order 50: $C_5\times C_{10}$ ($C_2\times F_{11}$)

Order 44: $D_{22}$ ($D_5\times C_5^2$)

Order 25: $C_5^2$ ($C_{44}:C_{10}$)

Order 22: $C_{22}$ ($D_{10}\times C_5^2$)

Order 20: $C_5:C_4$ ($C_5\times F_{11}$)

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

Order 10: $C_{10}$ ($C_{10}\times F_{11}$), $C_{10}$ ($D_5\times F_{11}$)

Order 5: $C_5$ ($C_{220}:C_{10}$), $C_5$ ($D_{110}:C_{10}$)

Order 2: $C_2$ ($C_{55}:C_{10}^2$)

Order 1: $C_1$ ($C_{110}.C_{10}^2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 11000: $C_{110}.C_{10}^2$ ($C_1$)

Order 5500: $C_{55}:C_{10}^2$ ($C_2$), $C_{55}:C_{10}^2$ ($C_2$), $C_5\times C_{55}:C_{20}$ ($C_2$)

Order 2750: $C_{110}:C_5^2$ ($C_2^2$)

Order 2200: $D_{110}:C_{10}$ ($C_5$), $D_{110}:C_{10}$ ($C_5$)

Order 1375: $C_{11}:C_5^3$ ($D_4$)

Order 1100: $C_5\times D_{110}$ ($C_{10}$), $C_5^2\times D_{22}$ ($C_{10}$), $C_{110}:C_{10}$ ($C_{10}$), $C_{10}\times F_{11}$ ($C_{10}$), $C_{10}\times F_{11}$ ($D_5$), $C_5:C_{220}$ ($C_{10}$), $C_{55}:C_{20}$ ($C_{10}$)

Order 550: $C_5\times C_{110}$ ($C_2\times C_{10}$), $C_{110}:C_5$ ($C_2\times C_{10}$), $C_{110}:C_5$ ($D_{10}$)

Order 440: $C_5:D_{44}$ ($C_5^2$)

Order 275: $C_5\times C_{55}$ ($C_5\times D_4$), $C_{55}:C_5$ ($C_5:D_4$), $C_{55}:C_5$ ($C_5\times D_4$)

Order 220: $C_2\times F_{11}$ ($C_5\times D_5$), $C_5:C_{44}$ ($C_5\times C_{10}$), $D_{110}$ ($C_5\times C_{10}$), $C_5\times D_{22}$ ($C_5\times C_{10}$), $C_5\times D_{22}$ ($C_5\times D_5$)

Order 110: $C_{110}$ ($C_{10}^2$), $C_{110}$ ($C_5\times D_{10}$), $C_{11}:C_{10}$ ($C_5\times D_{10}$)

Order 100: $C_5:C_{20}$ ($F_{11}$)

Order 55: $C_{55}$ ($D_4\times C_5^2$), $C_{55}$ ($C_{10}\wr C_2$), $C_{11}:C_5$ ($C_{10}\wr C_2$)

Order 50: $C_5\times C_{10}$ ($C_2\times F_{11}$)

Order 44: $D_{22}$ ($D_5\times C_5^2$)

Order 25: $C_5^2$ ($C_{44}:C_{10}$)

Order 22: $C_{22}$ ($D_{10}\times C_5^2$)

Order 20: $C_5:C_4$ ($C_5\times F_{11}$)

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

Order 10: $C_{10}$ ($C_{10}\times F_{11}$), $C_{10}$ ($D_5\times F_{11}$)

Order 5: $C_5$ ($C_{220}:C_{10}$), $C_5$ ($D_{110}:C_{10}$)

Order 2: $C_2$ ($C_{55}:C_{10}^2$)

Order 1: $C_1$ ($C_{110}.C_{10}^2$)

Series

Derived series

$C_{110}.C_{10}^2$

$\rhd$

$C_{110}.C_{10}^2$

$\rhd$

$C_{110}$

$\rhd$

$C_{110}$

$\rhd$

$C_1$

$\rhd$

$C_1$

Chief series

$C_{110}.C_{10}^2$

$\rhd$

$C_{110}.C_{10}^2$

$\rhd$

$C_{55}:C_{10}^2$

$\rhd$

$C_{55}:C_{10}^2$

$\rhd$

$C_{110}:C_{10}$

$\rhd$

$C_{110}:C_{10}$

$\rhd$

$C_{110}:C_5$

$\rhd$

$C_{110}:C_5$

$\rhd$

$C_{110}$

$\rhd$

$C_{110}$

$\rhd$

$C_{55}$

$\rhd$

$C_{55}$

$\rhd$

$C_{11}$

$\rhd$

$C_{11}$

$\rhd$

$C_1$

$\rhd$

$C_1$

Lower central series

$C_{110}.C_{10}^2$

$\rhd$

$C_{110}.C_{10}^2$

$\rhd$

$C_{110}$

$\rhd$

$C_{110}$

$\rhd$

$C_{55}$

$\rhd$

$C_{55}$

Upper central series

$C_1$

$\lhd$

$C_1$

$\lhd$

$C_{10}$

$\lhd$

$C_{10}$

Supergroups

This group is a maximal subgroup of 5 larger groups in the database.

This group is a maximal quotient of 2 larger groups in the database.

Character theory

Complex character table

See the $365 \times 365$ 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 $98 \times 98$ rational character table.