Abstract group 36300.o: $C_{11}^2:(D_6\times C_5^2)$

• Label: 36300.o
• Order: \( 2^{2} \cdot 3 \cdot 5^{2} \cdot 11^{2} \)
• Exponent: \( 2 \cdot 3 \cdot 5 \cdot 11 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 5^{2} \)
• $\card{Z(G)}$: \( 2 \cdot 5 \)
• $\card{\Aut(G)}$: \( 2^{5} \cdot 3 \cdot 5^{2} \cdot 11^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \cdot 5 \)
• Perm deg.: $40$
• Trans deg.: $330$
• Rank: $2$

Group information

Description:	$C_{11}^2:(D_6\times C_5^2)$
Order:	\(36300\)\(\medspace = 2^{2} \cdot 3 \cdot 5^{2} \cdot 11^{2} \)
Exponent:	\(330\)\(\medspace = 2 \cdot 3 \cdot 5 \cdot 11 \)
Automorphism group:	$C_{11}^2.C_{15}.C_{10}.C_2^4$, of order \(290400\)\(\medspace = 2^{5} \cdot 3 \cdot 5^{2} \cdot 11^{2} \)
Composition factors:	$C_2$ x 2, $C_3$, $C_5$ x 2, $C_{11}$ x 2
Derived length:	$3$

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

Group statistics

Order	1	2	3	5	6	10	11	15	22	30	55	110
Elements	1	67	242	2424	242	17208	120	5808	780	5808	480	3120	36300
Conjugacy classes	1	3	1	24	1	72	5	24	9	24	20	36	220
Divisions	1	3	1	6	1	18	3	6	5	6	3	5	58
Autjugacy classes	1	2	1	5	1	10	3	5	4	5	3	4	44

Dimension	1	2	4	8	15	30	60	120	240
Irr. complex chars.	100	50	0	0	40	30	0	0	0	220
Irr. rational chars.	4	2	24	12	0	6	2	6	2	58

Minimal presentations

Permutation degree:	$40$
Transitive degree:	$330$
Rank:	$2$
Inequivalent generating pairs:	$864$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	15	30	120
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d \mid a^{10}=b^{3}=c^{11}=d^{110}=[c,d]=1, b^{a}=b^{2}c^{2}d^{40}, c^{a}= \!\cdots\! \rangle}$
Permutation group:	Degree $40$ $\langle(34,35,38,36,37), (34,35,38,36,37)(39,40), (1,2,6,3,10,19,18,11,7,5,9)(4,13,23,25,12,8,21,20,30,14,16) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrr} 3 & 0 & 0 & 0 \\ 0 & 3 & 0 & 0 \\ 0 & 0 & 3 & 0 \\ 0 & 0 & 0 & 3 \end{array}\right), \left(\begin{array}{rrrr} 10 & 2 & 8 & 3 \\ 9 & 5 & 6 & 0 \\ 9 & 5 & 7 & 2 \\ 1 & 9 & 1 & 3 \end{array}\right), \left(\begin{array}{rrrr} 0 & 8 & 9 & 4 \\ 2 & 7 & 2 & 9 \\ 9 & 1 & 7 & 3 \\ 2 & 9 & 9 & 3 \end{array}\right), \left(\begin{array}{rrrr} 9 & 7 & 4 & 6 \\ 7 & 8 & 7 & 4 \\ 8 & 1 & 7 & 4 \\ 10 & 8 & 4 & 6 \end{array}\right), \left(\begin{array}{rrrr} 9 & 6 & 5 & 2 \\ 9 & 8 & 4 & 5 \\ 3 & 8 & 2 & 5 \\ 7 & 3 & 2 & 1 \end{array}\right), \left(\begin{array}{rrrr} 8 & 0 & 0 & 0 \\ 0 & 8 & 0 & 0 \\ 0 & 0 & 8 & 0 \\ 0 & 0 & 0 & 8 \end{array}\right), \left(\begin{array}{rrrr} 2 & 8 & 5 & 0 \\ 1 & 5 & 8 & 8 \\ 3 & 2 & 5 & 7 \\ 6 & 3 & 8 & 10 \end{array}\right) \right\rangle \subseteq \GL_{4}(\F_{11})$
Direct product:	$C_2$ $\, \times\, $ $C_5$ $\, \times\, $ $(C_{11}^2:(C_5\times S_3))$
Semidirect product:	$(C_{11}^2:C_5^2)$ $\,\rtimes\,$ $D_6$	$(C_{11}^2:D_6)$ $\,\rtimes\,$ $C_5^2$	$C_{11}^2$ $\,\rtimes\,$ $(D_6\times C_5^2)$	$(C_{11}^2:C_3)$ $\,\rtimes\,$ $C_{10}^2$	all 13
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:	$C_{10}^{2} \simeq C_{2}^{2} \times C_{5}^{2}$
Schur multiplier:	$C_{10}$
Commutator length:	$1$

Subgroups

There are 16192 subgroups in 256 conjugacy classes, 60 normal (19 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_{10}$	$G/Z \simeq$ $C_{11}^2:(C_5\times S_3)$
Commutator:	$G' \simeq$ $C_{11}^2:C_3$	$G/G' \simeq$ $C_{10}^2$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_{11}^2:(D_6\times C_5^2)$
Fitting:	$\operatorname{Fit} \simeq$ $C_{11}\times C_{110}$	$G/\operatorname{Fit} \simeq$ $C_5\times S_3$
Radical:	$R \simeq$ $C_{11}^2:(D_6\times C_5^2)$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{11}\times C_{110}$	$G/\operatorname{soc} \simeq$ $C_5\times S_3$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5^2$
11-Sylow subgroup:	$P_{ 11 } \simeq$ $C_{11}^2$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

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Classes of subgroups up to automorphism

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Normal subgroups

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Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 36300: $C_{11}^2:(D_6\times C_5^2)$

Order 18150: $C_5\times C_{11}^2:(C_5\times S_3)$ x 2, $C_{10}\times C_{11}^2:C_{15}$

Order 12100: $C_{10}\times C_{11}^2:C_{10}$

Order 9075: $C_5\times C_{11}^2:C_{15}$

Order 7260: $C_2\times C_{11}^2:(C_5\times S_3)$ x 5, $C_{10}\times C_{11}^2:S_3$

Order 6050: $C_5\times C_{11}^2:C_{10}$ x 2, $C_{10}\times C_{11}^2:C_5$

Order 3630: $C_{11}^2:(C_5\times S_3)$ x 10, $C_2\times C_{11}^2:C_{15}$ x 5, $C_5\times C_{11}^2:S_3$ x 2, $C_{10}\times C_{11}^2:C_3$

Order 3025: $C_{11}^2:C_5^2$

Order 2420: $C_2\times C_{11}^2:C_{10}$ x 5, $C_{110}\times D_{11}$

Order 1815: $C_{11}^2:C_{15}$ x 5, $C_{11}^2:C_{15}$

Order 1452: $C_{11}^2:D_6$

Order 1210: $C_{11}:F_{11}$ x 10, $C_{11}^2:C_{10}$ x 5, $D_{11}\times C_{55}$ x 2, $C_{11}\times C_{110}$

Order 1100: $C_{110}:C_{10}$, $C_{10}\times F_{11}$

Order 726: $C_{11}^2:S_3$ x 2, $C_{11}^2:C_6$

Order 605: $C_{11}^2:C_5$ x 5, $C_{11}\times C_{55}$

Order 550: $C_{110}:C_5$ x 5, $C_5\times F_{11}$ x 2

Order 484: $C_{11}\times D_{22}$

Order 363: $C_{11}^2:C_3$

Order 300: $D_6\times C_5^2$

Order 275: $C_{55}:C_5$ x 3

Order 242: $C_{11}\times D_{11}$ x 2, $C_{11}\times C_{22}$

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

Order 150: $S_3\times C_5^2$ x 2, $C_5\times C_{30}$

Order 121: $C_{11}^2$

Order 110: $C_{11}:C_{10}$ x 25, $F_{11}$ x 10, $C_{110}$ x 5, $C_5\times D_{11}$ x 2

Order 100: $C_{10}^2$

Order 75: $C_5\times C_{15}$

Order 60: $S_3\times C_{10}$ x 6

Order 55: $C_{11}:C_5$ x 15, $C_{55}$ x 3

Order 50: $C_5\times C_{10}$ x 3

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

Order 30: $C_5\times S_3$ x 12, $C_{30}$ x 6

Order 25: $C_5^2$

Order 22: $C_{22}$ x 5, $D_{11}$ x 2

Order 20: $C_2\times C_{10}$ x 6

Order 15: $C_{15}$ x 6

Order 12: $D_6$

Order 11: $C_{11}$ x 3

Order 10: $C_{10}$ x 18

Order 6: $S_3$ x 2, $C_6$

Order 5: $C_5$ x 6

Order 4: $C_2^2$

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 36300: $C_{11}^2:(D_6\times C_5^2)$

Order 18150: $C_5\times C_{11}^2:(C_5\times S_3)$, $C_{10}\times C_{11}^2:C_{15}$

Order 12100: $C_{10}\times C_{11}^2:C_{10}$

Order 9075: $C_5\times C_{11}^2:C_{15}$

Order 7260: $C_2\times C_{11}^2:(C_5\times S_3)$, $C_{10}\times C_{11}^2:S_3$

Order 6050: $C_5\times C_{11}^2:C_{10}$, $C_{10}\times C_{11}^2:C_5$

Order 3630: $C_2\times C_{11}^2:C_{15}$, $C_{11}^2:(C_5\times S_3)$, $C_5\times C_{11}^2:S_3$, $C_{10}\times C_{11}^2:C_3$

Order 3025: $C_{11}^2:C_5^2$

Order 2420: $C_{110}\times D_{11}$, $C_2\times C_{11}^2:C_{10}$

Order 1815: $C_{11}^2:C_{15}$, $C_{11}^2:C_{15}$

Order 1452: $C_{11}^2:D_6$

Order 1210: $C_{11}\times C_{110}$, $D_{11}\times C_{55}$, $C_{11}^2:C_{10}$, $C_{11}:F_{11}$

Order 1100: $C_{110}:C_{10}$, $C_{10}\times F_{11}$

Order 726: $C_{11}^2:C_6$, $C_{11}^2:S_3$

Order 605: $C_{11}\times C_{55}$, $C_{11}^2:C_5$

Order 550: $C_{110}:C_5$ x 4, $C_5\times F_{11}$

Order 484: $C_{11}\times D_{22}$

Order 363: $C_{11}^2:C_3$

Order 300: $D_6\times C_5^2$

Order 275: $C_{55}:C_5$ x 3

Order 242: $C_{11}\times C_{22}$, $C_{11}\times D_{11}$

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

Order 150: $C_5\times C_{30}$, $S_3\times C_5^2$

Order 121: $C_{11}^2$

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

Order 100: $C_{10}^2$

Order 75: $C_5\times C_{15}$

Order 60: $S_3\times C_{10}$ x 2

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

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

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

Order 30: $C_{30}$ x 2, $C_5\times S_3$ x 2

Order 25: $C_5^2$

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

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

Order 15: $C_{15}$ x 2

Order 12: $D_6$

Order 11: $C_{11}$ x 3

Order 10: $C_{10}$ x 4

Order 6: $C_6$, $S_3$

Order 5: $C_5$ x 2

Order 4: $C_2^2$

Order 3: $C_3$

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 36300: $C_{11}^2:(D_6\times C_5^2)$ ($C_1$)

Order 18150: $C_5\times C_{11}^2:(C_5\times S_3)$ ($C_2$) x 2, $C_{10}\times C_{11}^2:C_{15}$ ($C_2$)

Order 9075: $C_5\times C_{11}^2:C_{15}$ ($C_2^2$)

Order 7260: $C_2\times C_{11}^2:(C_5\times S_3)$ ($C_5$) x 5, $C_{10}\times C_{11}^2:S_3$ ($C_5$)

Order 6050: $C_{10}\times C_{11}^2:C_5$ ($S_3$)

Order 3630: $C_{11}^2:(C_5\times S_3)$ ($C_{10}$) x 10, $C_2\times C_{11}^2:C_{15}$ ($C_{10}$) x 5, $C_5\times C_{11}^2:S_3$ ($C_{10}$) x 2, $C_{10}\times C_{11}^2:C_3$ ($C_{10}$)

Order 3025: $C_{11}^2:C_5^2$ ($D_6$)

Order 1815: $C_{11}^2:C_{15}$ ($C_2\times C_{10}$) x 5, $C_{11}^2:C_{15}$ ($C_2\times C_{10}$)

Order 1452: $C_{11}^2:D_6$ ($C_5^2$)

Order 1210: $C_{11}^2:C_{10}$ ($C_5\times S_3$) x 5, $C_{11}\times C_{110}$ ($C_5\times S_3$)

Order 726: $C_{11}^2:S_3$ ($C_5\times C_{10}$) x 2, $C_{11}^2:C_6$ ($C_5\times C_{10}$)

Order 605: $C_{11}^2:C_5$ ($S_3\times C_{10}$) x 5, $C_{11}\times C_{55}$ ($S_3\times C_{10}$)

Order 363: $C_{11}^2:C_3$ ($C_{10}^2$)

Order 242: $C_{11}\times C_{22}$ ($S_3\times C_5^2$)

Order 121: $C_{11}^2$ ($D_6\times C_5^2$)

Order 10: $C_{10}$ ($C_{11}^2:(C_5\times S_3)$)

Order 5: $C_5$ ($C_2\times C_{11}^2:(C_5\times S_3)$)

Order 2: $C_2$ ($C_5\times C_{11}^2:(C_5\times S_3)$)

Order 1: $C_1$ ($C_{11}^2:(D_6\times C_5^2)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 36300: $C_{11}^2:(D_6\times C_5^2)$ ($C_1$)

Order 18150: $C_5\times C_{11}^2:(C_5\times S_3)$ ($C_2$), $C_{10}\times C_{11}^2:C_{15}$ ($C_2$)

Order 9075: $C_5\times C_{11}^2:C_{15}$ ($C_2^2$)

Order 7260: $C_2\times C_{11}^2:(C_5\times S_3)$ ($C_5$), $C_{10}\times C_{11}^2:S_3$ ($C_5$)

Order 6050: $C_{10}\times C_{11}^2:C_5$ ($S_3$)

Order 3630: $C_2\times C_{11}^2:C_{15}$ ($C_{10}$), $C_{11}^2:(C_5\times S_3)$ ($C_{10}$), $C_5\times C_{11}^2:S_3$ ($C_{10}$), $C_{10}\times C_{11}^2:C_3$ ($C_{10}$)

Order 3025: $C_{11}^2:C_5^2$ ($D_6$)

Order 1815: $C_{11}^2:C_{15}$ ($C_2\times C_{10}$), $C_{11}^2:C_{15}$ ($C_2\times C_{10}$)

Order 1452: $C_{11}^2:D_6$ ($C_5^2$)

Order 1210: $C_{11}\times C_{110}$ ($C_5\times S_3$), $C_{11}^2:C_{10}$ ($C_5\times S_3$)

Order 726: $C_{11}^2:C_6$ ($C_5\times C_{10}$), $C_{11}^2:S_3$ ($C_5\times C_{10}$)

Order 605: $C_{11}\times C_{55}$ ($S_3\times C_{10}$), $C_{11}^2:C_5$ ($S_3\times C_{10}$)

Order 363: $C_{11}^2:C_3$ ($C_{10}^2$)

Order 242: $C_{11}\times C_{22}$ ($S_3\times C_5^2$)

Order 121: $C_{11}^2$ ($D_6\times C_5^2$)

Order 10: $C_{10}$ ($C_{11}^2:(C_5\times S_3)$)

Order 5: $C_5$ ($C_2\times C_{11}^2:(C_5\times S_3)$)

Order 2: $C_2$ ($C_5\times C_{11}^2:(C_5\times S_3)$)

Order 1: $C_1$ ($C_{11}^2:(D_6\times C_5^2)$)

Series

Derived series

$C_{11}^2:(D_6\times C_5^2)$

$\rhd$

$C_{11}^2:(D_6\times C_5^2)$

$\rhd$

$C_{11}^2:C_3$

$\rhd$

$C_{11}^2:C_3$

$\rhd$

$C_{11}^2$

$\rhd$

$C_{11}^2$

$\rhd$

$C_1$

$\rhd$

$C_1$

Chief series

$C_{11}^2:(D_6\times C_5^2)$

$\rhd$

$C_{11}^2:(D_6\times C_5^2)$

$\rhd$

$C_5\times C_{11}^2:(C_5\times S_3)$

$\rhd$

$C_5\times C_{11}^2:(C_5\times S_3)$

$\rhd$

$C_5\times C_{11}^2:C_{15}$

$\rhd$

$C_5\times C_{11}^2:C_{15}$

$\rhd$

$C_{11}^2:C_{15}$

$\rhd$

$C_{11}^2:C_{15}$

$\rhd$

$C_{11}^2:C_3$

$\rhd$

$C_{11}^2:C_3$

$\rhd$

$C_{11}^2$

$\rhd$

$C_{11}^2$

$\rhd$

$C_1$

$\rhd$

$C_1$

Lower central series

$C_{11}^2:(D_6\times C_5^2)$

$\rhd$

$C_{11}^2:(D_6\times C_5^2)$

$\rhd$

$C_{11}^2:C_3$

$\rhd$

$C_{11}^2:C_3$

Upper central series

$C_1$

$\lhd$

$C_1$

$\lhd$

$C_{10}$

$\lhd$

$C_{10}$

Supergroups

This group is a maximal subgroup of 4 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 $220 \times 220$ 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 $58 \times 58$ rational character table.