Abstract group 2662000.a: $C_{22}:F_{11}^2.C_{10}$

• Label: 2662000.a
• Order: \( 2^{4} \cdot 5^{3} \cdot 11^{3} \)
• Exponent: \( 2^{2} \cdot 5 \cdot 11 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 5^{2} \)
• $\card{Z(G)}$: 10
• $\card{\Aut(G)}$: \( 2^{8} \cdot 5^{4} \cdot 11^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \cdot 5^{2} \)
• Perm deg.: $42$
• Trans deg.: not computed
• Rank: $3$

Group information

Description:	$C_{22}:F_{11}^2.C_{10}$
Order:	\(2662000\)\(\medspace = 2^{4} \cdot 5^{3} \cdot 11^{3} \)
Exponent:	\(220\)\(\medspace = 2^{2} \cdot 5 \cdot 11 \)
Automorphism group:	Group of order \(212960000\)\(\medspace = 2^{8} \cdot 5^{4} \cdot 11^{3} \)
Composition factors:	$C_2$ x 4, $C_5$ x 3, $C_{11}$ x 3
Derived length:	$3$

This group is nonabelian and solvable. Whether it is monomial has not been computed.

Group statistics

Order	1	2	4	5	10	11	20	22	44	55	110	220
Elements	1	4247	1320	87124	1449628	1330	537680	31690	25300	77920	344560	101200	2662000
Conjugacy classes	1	7	2	74	518	15	48	33	13	90	222	52	1075
Divisions	1	7	2	19	131	15	12	33	8	23	56	8	315
Autjugacy classes	1	6	2	17	78	7	10	15	4	15	31	4	190

Minimal presentations

Permutation degree:	$42$
Transitive degree:	not computed
Rank:	$3$
Inequivalent generating triples:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	100	not computed	not computed
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e \mid a^{10}=b^{10}=c^{110}=d^{11}=e^{22}=[c,e]=[d,e]= \!\cdots\! \rangle}$
Permutation group:	Degree $42$ $\langle(38,39,40,42,41), (1,2,6,9,4,14,5,3,8,12,11)(7,15,20,10,22,18,17,16,19,21,13) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrr} 2 & 0 & 0 & 0 \\ 5 & 7 & 4 & 0 \\ 10 & 10 & 10 & 0 \\ 8 & 10 & 6 & 4 \end{array}\right), \left(\begin{array}{rrrr} 4 & 0 & 0 & 0 \\ 5 & 9 & 4 & 0 \\ 2 & 2 & 10 & 0 \\ 2 & 2 & 6 & 4 \end{array}\right), \left(\begin{array}{rrrr} 3 & 2 & 9 & 8 \\ 10 & 6 & 4 & 9 \\ 6 & 4 & 1 & 9 \\ 0 & 6 & 1 & 4 \end{array}\right), \left(\begin{array}{rrrr} 9 & 0 & 0 & 0 \\ 7 & 5 & 10 & 0 \\ 6 & 6 & 5 & 0 \\ 3 & 6 & 4 & 1 \end{array}\right), \left(\begin{array}{rrrr} 7 & 2 & 7 & 7 \\ 3 & 9 & 3 & 8 \\ 1 & 2 & 5 & 0 \\ 8 & 9 & 0 & 1 \end{array}\right), \left(\begin{array}{rrrr} 3 & 0 & 0 & 0 \\ 0 & 3 & 0 & 0 \\ 0 & 0 & 3 & 0 \\ 0 & 0 & 0 & 3 \end{array}\right), \left(\begin{array}{rrrr} 8 & 0 & 0 & 0 \\ 0 & 8 & 0 & 0 \\ 0 & 0 & 8 & 0 \\ 0 & 0 & 0 & 8 \end{array}\right), \left(\begin{array}{rrrr} 6 & 4 & 3 & 2 \\ 5 & 7 & 3 & 5 \\ 0 & 4 & 6 & 9 \\ 1 & 1 & 1 & 9 \end{array}\right), \left(\begin{array}{rrrr} 7 & 0 & 9 & 0 \\ 9 & 9 & 7 & 10 \\ 0 & 4 & 3 & 2 \\ 0 & 9 & 4 & 3 \end{array}\right), \left(\begin{array}{rrrr} 0 & 10 & 3 & 5 \\ 10 & 8 & 3 & 3 \\ 6 & 7 & 4 & 1 \\ 0 & 6 & 1 & 1 \end{array}\right) \right\rangle \subseteq \GL_{4}(\F_{11})$
Direct product:	$C_5$ $\, \times\, $ $(C_{22}:F_{11}^2.C_2)$
Semidirect product:	$C_{11}^3$ $\,\rtimes\,$ $(D_{10}:C_{10}^2)$	$(C_{11}^3:C_5^3)$ $\,\rtimes\,$ $(C_2\times D_4)$			more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$(C_{22}:F_{11}^2)$ . $C_{10}$ (5)	$(C_{11}^3:C_{10}^3)$ . $C_2$	$(C_{11}^3:C_{10}^2)$ . $D_{10}$	$(C_{11}^3:C_{10}^2)$ . $D_{10}$	all 124

Elements of the group are displayed as matrices in $\GL_{4}(\F_{11})$.

Homology

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

Subgroups

There are 244 normal subgroups (97 characteristic).

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

Special subgroups

Center:	a subgroup isomorphic to $C_{10}$
Commutator:	a subgroup isomorphic to $C_{11}^2:C_{110}$
Frattini:	a subgroup isomorphic to $C_2$
Fitting:	not computed
Radical:	not computed
Socle:	not computed
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times D_4$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5^3$
11-Sylow subgroup:	$P_{ 11 } \simeq$ $C_{11}^3$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: inclusions not computed

Classes of subgroups up to automorphism

No diagram available: inclusions not computed

Normal subgroups

No diagram available: inclusions not computed

Normal subgroups up to automorphism

No diagram available: inclusions not computed

Classes of subgroups up to conjugation

All subgroups of index up to 200 (order at least 13310) are shown, as well as all normal subgroups of any index.

Order 2662000: $C_{22}:F_{11}^2.C_{10}$

Order 1331000: $C_{11}^3:C_{10}^3$, $C_{11}^2:C_{110}.C_{10}^2$, $C_{11}^3:C_5^3:D_4$, $C_{11}^3:(D_5\times C_{10}^2)$, $C_{11}^3:C_5^3:D_4$, $C_{11}^3.C_{10}.C_{10}^2$, $C_{11}^3.C_{10}.C_{10}^2$

Order 665500: $C_{55}:F_{11}^2$ x 2, $C_{11}^3:(D_{10}\times C_5^2)$ x 2, $C_5\times C_{11}^3:C_{10}^2$, $C_5\times C_{11}^3:C_{10}^2$, $C_{11}^3:C_5^2:C_{20}$, $C_{11}^3:C_5^2:C_{20}$, $C_{11}^3:(D_{10}\times C_5^2)$, $C_{11}^3:(D_{10}\times C_5^2)$, $C_5\times C_{11}^3:C_{10}^2$

Order 532400: $C_{22}:F_{11}^2.C_2$ x 5, $C_{110}:F_{11}.D_{22}$, $(D_{22}\times C_{11}^2):C_{10}^2$

Order 332750: $C_{11}^2:(C_5^2\times F_{11})$ x 2, $C_{10}\times C_{11}^3:C_5^2$, $C_{11}^2:(C_5^2\times F_{11})$, $C_{11}^3:(D_5\times C_5^2)$, $C_{11}^3:(D_5\times C_5^2)$, $C_{11}^2:(C_5^2\times F_{11})$

Order 266200: $C_{22}:F_{11}^2$ x 15, $C_{11}^3:C_{10}\wr C_2$ x 5, $C_{11}^3:(C_{10}:C_{20})$ x 5, $C_{11}^3:(C_{10}\times D_{10})$ x 5, $C_{11}^3:C_{10}\wr C_2$ x 5, $C_{11}^3:C_{10}\wr C_2$ x 5, $C_{11}^3:C_{10}\wr C_2$ x 5, $C_2\times C_{11}^3:C_{10}^2$ x 2, $(D_{11}\times C_{110}):F_{11}$ x 2, $C_{11}^3:(C_2\times C_{10}^2)$, $(C_{11}^2\times C_{110}).D_{10}$, $C_{11}^3:(D_4\times C_5^2)$, $C_{11}^3:(D_4\times C_5^2)$, $C_{11}^3:C_{10}\wr C_2$, $C_{11}^3:(C_{10}\times C_{20})$, $C_{11}^3:C_{10}\wr C_2$, $C_{11}^3:(C_{10}\times D_{10})$, $D_{11}\times C_{110}:F_{11}$, $C_2\times C_{11}^3:C_{10}^2$, $C_{11}^3:C_{10}\wr C_2$, $C_{11}^3.C_5.C_{20}.C_2$

Order 242000: $C_{11}^2.C_{10}.C_{10}^2.C_2$

Order 166375: $C_{11}^3:C_5^3$

Order 133100: $C_{11}:F_{11}^2$ x 50, $C_{11}^3:C_{10}^2$ x 15, $C_{11}^3:C_{10}^2$ x 15, $C_{11}^3:C_{10}^2$ x 15, $C_{11}^3:(C_5\times D_{10})$ x 10, $C_{11}^2:C_{55}:C_{20}$ x 5, $C_{11}^2:C_{55}:C_{20}$ x 5, $C_{11}^3:(C_5\times D_{10})$ x 5, $C_{11}^3:(C_5\times D_{10})$ x 5, $C_{11}^3:C_{10}^2$ x 4, $C_{11}^3:C_{10}^2$ x 4, $C_{11}^3:C_{10}^2$ x 4, $C_5\times C_{11}^3:D_{10}$ x 2, $C_{11}^3:C_{10}^2$ x 2, $C_{11}^3:C_{10}^2$ x 2, $C_{11}^3:C_{10}^2$ x 2, $C_5\times C_{11}^3:C_{20}$, $C_5\times C_{11}^3:C_{20}$, $C_{11}^3:C_{10}^2$, $C_{110}\times C_{11}^2:D_5$, $(C_{11}^2\times C_{110}).D_5$, $C_{11}^3:C_{10}^2$, $C_{11}^3:C_{10}^2$, $C_{11}^3:C_{10}^2$, $C_{11}^3:C_{10}^2$, $C_{11}^3:C_{10}^2$, $(C_{11}\times C_{110}).D_{55}$, $C_{11}^3:C_{10}^2$, $C_{11}^3:C_{10}^2$, $C_{11}^3:C_{10}^2$, $C_{11}^3:C_{10}^2$, $C_5\times C_{11}^2:D_{110}$

Order 121000: $C_{110}:F_{11}:C_{10}$ x 2, $C_{22}^2:(D_5\times C_5^2)$ x 2, $C_{11}^2:C_{10}^3$, $C_{55}:F_{11}:C_{20}$, $C_{10}\times F_{11}^2$, $C_{11}^2:(D_5\times C_{10}^2)$

Order 106480: $C_{11}^3:(D_4\times C_{10})$ x 5, $(D_{22}\times C_{11}^2).D_{10}$, $C_5\times D_{22}:D_{11}^2$

Order 66550: $C_{11}^3:(C_5\times C_{10})$ x 30, $C_{11}^3:(C_5\times C_{10})$ x 25, $C_{11}^3:(C_5\times C_{10})$ x 25, $C_2\times C_{11}^3:C_5^2$ x 15, $C_{11}^3:(C_5\times D_5)$ x 5, $C_{11}^3:(C_5\times D_5)$ x 5, $C_5\times C_{11}^3:C_{10}$ x 2, $C_{11}\times C_{55}:F_{11}$ x 2, $C_5\times C_{11}^3:C_{10}$ x 2, $C_5\times C_{11}^3:C_{10}$ x 2, $C_5\times C_{11}^3:C_{10}$ x 2, $C_5\times C_{11}^3:C_{10}$ x 2, $C_5\times C_{11}^3:C_{10}$ x 2, $C_5\times C_{11}^3:C_{10}$ x 2, $C_5\times C_{11}^3:C_{10}$ x 2, $C_5\times C_{11}^3:C_{10}$ x 2, $C_{11}^2:D_5\times C_{55}$, $C_5\times C_{11}^2:C_{110}$, $C_{11}\times C_{55}:F_{11}$, $C_5\times C_{11}^2:C_{110}$, $C_5\times C_{11}^3:C_{10}$, $C_5\times C_{11}^3:C_{10}$, $C_5\times C_{11}^3:C_{10}$, $C_5\times C_{11}^2:D_{55}$

Order 60500: $C_5\times C_{11}^2:C_{10}^2$ x 5, $C_5\times F_{11}^2$ x 4, $C_{11}^2:C_5^2:C_{20}$ x 2, $C_{110}:C_5\times F_{11}$ x 2, $C_{11}^2:(D_{10}\times C_5^2)$ x 2, $C_{11}^2:(D_{10}\times C_5^2)$ x 2, $C_{22}^2:C_5^3$

Order 53240: $(C_{11}\times D_{22}):F_{11}$ x 10, $(C_{11}\times D_{22}):F_{11}$ x 10, $C_{11}^3:(C_5\times D_4)$ x 5, $(C_{11}\times D_{22}):F_{11}$ x 5, $C_{11}^3:(C_2\times C_{20})$ x 5, $C_{11}:(D_{22}\times F_{11})$ x 5, $(C_{11}\times D_{22}):F_{11}$ x 5, $D_{11}\times C_{22}:F_{11}$ x 3, $C_{110}.D_{11}^2$ x 2, $C_{110}:D_{11}^2$ x 2, $C_{110}.D_{11}^2$, $(C_{11}\times C_{22}).D_{110}$, $(C_{11}^2\times C_{22}).D_{10}$, $C_{110}.D_{11}^2$, $C_{110}.D_{11}^2$, $(D_{22}\times C_{11}^2):D_5$, $C_2\times C_{11}^3:D_{10}$, $(C_{11}^2\times C_{22}).D_{10}$, $(C_{11}^2\times C_{22}).D_{10}$

Order 48400: $C_{22}^2:(C_5\times D_{10})$ x 5, $C_{22}^2:C_{10}^2$ x 2, $C_5\times D_{44}:F_{11}$, $C_5\times C_{22}^2:D_{10}$

Order 33275: $C_{11}^3:C_5^2$ x 15, $C_{11}^3:C_5^2$, $C_{11}^3:C_5^2$, $C_{11}^3:C_5^2$, $C_{11}^3:C_5^2$

Order 30250: $C_{55}:(C_5\times F_{11})$ x 5, $C_{55}:C_5\times F_{11}$ x 4, $C_2\times C_{11}^2:C_5^3$ x 3, $C_{11}^2:(D_5\times C_5^2)$ x 2

Order 26620: $C_{11}:(D_{11}\times F_{11})$ x 20, $C_{11}^3:(C_2\times C_{10})$ x 20, $C_{11}^3:(C_2\times C_{10})$ x 20, $C_2\times C_{11}^3:C_{10}$ x 10, $C_2\times C_{11}^3:C_{10}$ x 10, $D_{11}\times C_{11}:F_{11}$ x 10, $C_2\times C_{11}^3:C_{10}$ x 5, $C_2\times C_{11}^3:C_{10}$ x 5, $C_2\times C_{11}^3:C_{10}$ x 5, $C_2\times C_{11}^3:C_{10}$ x 5, $C_2\times C_{11}^3:C_{10}$ x 5, $C_{11}^3:C_{20}$ x 5, $C_{11}^3:C_{20}$ x 5, $C_{11}\times C_{22}:F_{11}$ x 5, $C_2\times C_{11}^3:C_{10}$ x 5, $C_{55}:D_{11}^2$ x 4, $C_2\times C_{11}^3:C_{10}$ x 3, $C_2\times C_{11}^3:C_{10}$ x 3, $C_{11}\times C_{22}:F_{11}$ x 3, $C_{55}.C_{22}^2$ x 2, $C_{11}:D_{11}\times C_{110}$ x 2, $C_{11}^3:D_{10}$ x 2, $(C_{11}\times C_{22}).D_{55}$, $C_{11}^2:D_{110}$, $C_{11}^3:D_{10}$, $(C_{11}^2\times C_{22}).D_5$, $C_{11}^2:C_{220}$, $C_2\times C_{11}^3:C_{10}$, $C_{11}^2:C_{220}$

Order 24200: $C_2\times F_{11}^2$ x 25, $C_2\times C_{11}^2:C_{10}^2$ x 15, $C_{11}^2:C_{10}\wr C_2$ x 10, $C_{22}^2:(C_5\times D_5)$ x 10, $C_5\times D_{22}:F_{11}$ x 10, $C_{11}:(C_{20}\times F_{11})$ x 7, $C_{11}:F_{11}:C_{20}$ x 5, $C_{11}^2:(C_{10}\times D_{10})$ x 5, $C_5\times C_{22}^2:C_{10}$ x 4, $C_5\times D_{22}:F_{11}$ x 4, $C_5\times D_{22}\times F_{11}$ x 2, $C_{220}:F_{11}$ x 2, $C_5\times C_{22}^2:D_5$ x 2, $C_5\times D_{22}:F_{11}$ x 2, $(C_{11}\times C_{110}).D_{10}$ x 2, $C_2\times C_{110}:F_{11}$ x 2, $C_5\times D_{22}:F_{11}$ x 2, $(C_{11}\times C_{110}).D_{10}$, $C_{10}\times C_{11}^2:D_{10}$, $C_2\times C_{110}:F_{11}$, $C_{220}:F_{11}$, $C_5\times D_{22}:F_{11}$, $C_2\times C_{110}:F_{11}$, $C_2\times C_{110}:F_{11}$

Order 22000: $D_{110}:C_{10}^2$

Order 21296: $D_{22}:D_{11}^2$

Order 15125: $C_{11}^2:C_5^3$ x 2

Order 13310: $C_{11}^3:C_{10}$ x 10, $C_{11}^3:C_{10}$ x 10, $C_{11}^3:C_{10}$ x 10, $C_{11}^3:C_{10}$ x 10, $C_{11}^3:C_{10}$ x 10, $C_{11}^3:C_{10}$ x 10, $C_{11}^3:C_{10}$ x 10, $C_{11}^2:C_{110}$ x 10, $C_{11}^3:C_{10}$ x 10, $C_{11}^3:C_{10}$ x 6, $C_{11}^2:C_{110}$ x 5, $C_{11}^3:C_{10}$ x 5, $C_{11}^2:C_{110}$ x 5, $C_{11}^3:C_{10}$ x 5, $C_{11}^3:C_{10}$ x 5, $C_{11}^2:C_{110}$ x 3, $C_{11}^2:C_{110}$ x 2, $C_{11}^3:C_{10}$ x 2, $C_{11}^2:C_{110}$ x 2, $C_{11}^2:D_{55}$, $C_{11}^3:D_5$, $C_{11}^2\times C_{110}$

Order 12100: $C_{110}:F_{11}$, $C_5\times C_{11}^2:D_{10}$, $(C_{11}\times C_{110}).D_5$

Order 10648: $C_{22}:D_{11}^2$

Order 6655: $C_{11}^3:C_5$ x 5, $C_{11}^2\times C_{55}$, $C_{11}^2:C_{55}$

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

Order 5324: $C_{11}^2:D_{22}$, $C_{11}^2:D_{22}$, $D_{22}\times C_{11}^2$

Order 4840: $(C_{11}\times C_{22}).D_{10}$

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

Order 2662: $C_{11}^3:C_2$ x 2, $C_{11}^2\times C_{22}$

Order 2420: $C_{22}:F_{11}$, $C_{11}^2:D_{10}$, $(C_{11}\times C_{22}).D_5$, $C_{110}:D_{11}$

Order 1331: $C_{11}^3$

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

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

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

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

Order 220: $C_5\times D_{22}$

Order 125: $C_5^3$

Order 121: $C_{11}^2$

Order 110: $C_{110}$

Order 55: $C_{55}$

Order 44: $D_{22}$

Order 22: $C_{22}$

Order 16: $C_2\times D_4$

Order 11: $C_{11}$

Order 10: $C_{10}$

Order 5: $C_5$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 2662000: $C_{22}:F_{11}^2.C_{10}$ ($C_1$)

Order 1331000: $C_{11}^3:C_{10}^3$ ($C_2$), $C_{11}^2:C_{110}.C_{10}^2$ ($C_2$), $C_{11}^3:C_5^3:D_4$ ($C_2$), $C_{11}^3:(D_5\times C_{10}^2)$ ($C_2$), $C_{11}^3:C_5^3:D_4$ ($C_2$), $C_{11}^3.C_{10}.C_{10}^2$ ($C_2$), $C_{11}^3.C_{10}.C_{10}^2$ ($C_2$)

Order 665500: $C_5\times C_{11}^3:C_{10}^2$ ($C_2^2$), $C_5\times C_{11}^3:C_{10}^2$ ($C_2^2$), $C_{11}^3:C_5^2:C_{20}$ ($C_2^2$), $C_{11}^3:C_5^2:C_{20}$ ($C_2^2$), $C_{11}^3:(D_{10}\times C_5^2)$ ($C_2^2$), $C_{11}^3:(D_{10}\times C_5^2)$ ($C_2^2$), $C_5\times C_{11}^3:C_{10}^2$ ($C_2^2$)

Order 532400: $C_{22}:F_{11}^2.C_2$ ($C_5$) x 5, $C_{110}:F_{11}.D_{22}$ ($C_5$)

Order 332750: $C_{11}^2:(C_5^2\times F_{11})$ ($D_4$) x 2, $C_{10}\times C_{11}^3:C_5^2$ ($C_2^3$)

Order 266200: $C_{22}:F_{11}^2$ ($C_{10}$) x 5, $C_{11}^3:C_{10}\wr C_2$ ($C_{10}$) x 5, $C_{11}^3:(C_{10}:C_{20})$ ($C_{10}$) x 5, $C_{11}^3:(C_{10}\times D_{10})$ ($C_{10}$) x 5, $C_{11}^3:C_{10}\wr C_2$ ($C_{10}$) x 5, $C_{11}^3:C_{10}\wr C_2$ ($C_{10}$) x 5, $C_{11}^3:C_{10}\wr C_2$ ($C_{10}$) x 5, $C_2\times C_{11}^3:C_{10}^2$ ($D_5$), $(C_{11}^2\times C_{110}).D_{10}$ ($C_{10}$), $C_{11}^3:C_{10}\wr C_2$ ($C_{10}$), $C_{11}^3:C_{10}\wr C_2$ ($C_{10}$), $C_{11}^3:(C_{10}\times D_{10})$ ($C_{10}$), $D_{11}\times C_{110}:F_{11}$ ($C_{10}$), $C_{11}^3:C_{10}\wr C_2$ ($C_{10}$), $C_{11}^3.C_5.C_{20}.C_2$ ($C_{10}$)

Order 166375: $C_{11}^3:C_5^3$ ($C_2\times D_4$)

Order 133100: $C_{11}^2:C_{55}:C_{20}$ ($C_2\times C_{10}$) x 5, $C_{11}^3:C_{10}^2$ ($C_2\times C_{10}$) x 5, $C_{11}^2:C_{55}:C_{20}$ ($C_2\times C_{10}$) x 5, $C_{11}^3:C_{10}^2$ ($C_2\times C_{10}$) x 5, $C_{11}^3:(C_5\times D_{10})$ ($C_2\times C_{10}$) x 5, $C_{11}^3:(C_5\times D_{10})$ ($C_2\times C_{10}$) x 5, $C_{11}^3:C_{10}^2$ ($C_2\times C_{10}$) x 5, $C_{11}^3:C_{10}^2$ ($C_2\times C_{10}$), $C_{110}\times C_{11}^2:D_5$ ($C_2\times C_{10}$), $(C_{11}^2\times C_{110}).D_5$ ($C_2\times C_{10}$), $C_{11}^3:C_{10}^2$ ($C_2\times C_{10}$), $C_{11}^3:C_{10}^2$ ($D_{10}$), $(C_{11}\times C_{110}).D_{55}$ ($C_2\times C_{10}$), $C_{11}^3:C_{10}^2$ ($D_{10}$), $C_{11}^3:C_{10}^2$ ($C_2\times C_{10}$), $C_{11}^3:C_{10}^2$ ($D_{10}$), $C_5\times C_{11}^2:D_{110}$ ($C_2\times C_{10}$)

Order 106480: $(D_{22}\times C_{11}^2).D_{10}$ ($C_5^2$)

Order 66550: $C_{11}^3:(C_5\times C_{10})$ ($C_5\times D_4$) x 10, $C_2\times C_{11}^3:C_5^2$ ($C_2^2\times C_{10}$) x 5, $C_5\times C_{11}^3:C_{10}$ ($C_5:D_4$) x 2, $C_5\times C_{11}^3:C_{10}$ ($C_5\times D_4$) x 2, $C_5\times C_{11}^2:C_{110}$ ($C_2^2\times C_{10}$), $C_5\times C_{11}^3:C_{10}$ ($C_2\times D_{10}$)

Order 53240: $(C_{11}\times D_{22}):F_{11}$ ($C_5\times D_5$) x 5, $(C_{11}\times C_{22}).D_{110}$ ($C_5\times C_{10}$), $(C_{11}^2\times C_{22}).D_{10}$ ($C_5\times C_{10}$), $C_{110}:D_{11}^2$ ($C_5\times D_5$), $(D_{22}\times C_{11}^2):D_5$ ($C_5\times C_{10}$), $C_2\times C_{11}^3:D_{10}$ ($C_5\times C_{10}$), $(C_{11}^2\times C_{22}).D_{10}$ ($C_5\times C_{10}$), $(C_{11}^2\times C_{22}).D_{10}$ ($C_5\times C_{10}$), $D_{11}\times C_{22}:F_{11}$ ($C_5\times C_{10}$)

Order 33275: $C_{11}^3:C_5^2$ ($D_4\times C_{10}$) x 5, $C_{11}^3:C_5^2$ ($C_{10}:D_4$), $C_{11}^3:C_5^2$ ($D_4\times C_{10}$)

Order 26620: $C_2\times C_{11}^3:C_{10}$ ($C_5\times D_{10}$) x 5, $C_2\times C_{11}^3:C_{10}$ ($C_5\times D_{10}$) x 5, $C_2\times C_{11}^3:C_{10}$ ($C_5\times D_{10}$) x 5, $C_2\times C_{11}^3:C_{10}$ ($C_{10}^2$), $C_2\times C_{11}^3:C_{10}$ ($C_{10}^2$), $(C_{11}\times C_{22}).D_{55}$ ($C_{10}^2$), $C_{11}^2:D_{110}$ ($C_{10}^2$), $C_{11}^3:D_{10}$ ($C_{10}^2$), $(C_{11}^2\times C_{22}).D_5$ ($C_{10}^2$), $C_{11}\times C_{22}:F_{11}$ ($C_{10}^2$), $C_{55}.C_{22}^2$ ($C_5\times D_{10}$), $C_2\times C_{11}^3:C_{10}$ ($C_5\times D_{10}$), $C_{11}:D_{11}\times C_{110}$ ($C_5\times D_{10}$)

Order 24200: $(C_{11}\times C_{110}).D_{10}$ ($F_{11}$)

Order 13310: $C_{11}^3:C_{10}$ ($C_{10}\wr C_2$) x 10, $C_{11}^3:C_{10}$ ($C_{10}\times D_{10}$) x 5, $C_{11}^3:C_{10}$ ($D_4\times C_5^2$) x 2, $C_{11}^3:C_{10}$ ($C_{10}\wr C_2$) x 2, $C_{11}^2:C_{110}$ ($C_2\times C_{10}^2$), $C_{11}^2\times C_{110}$ ($C_{10}\times D_{10}$)

Order 12100: $C_{110}:F_{11}$ ($C_2\times F_{11}$), $C_5\times C_{11}^2:D_{10}$ ($C_2\times F_{11}$), $(C_{11}\times C_{110}).D_5$ ($C_2\times F_{11}$)

Order 10648: $C_{22}:D_{11}^2$ ($D_5\times C_5^2$)

Order 6655: $C_{11}^3:C_5$ ($C_{10}^2:C_2^2$) x 5, $C_{11}^2\times C_{55}$ ($C_{10}^2:C_2^2$), $C_{11}^2:C_{55}$ ($C_4:C_{10}^2$)

Order 6050: $C_5\times C_{11}^2:C_{10}$ ($C_2^2\times F_{11}$)

Order 5324: $C_{11}^2:D_{22}$ ($D_{10}\times C_5^2$), $C_{11}^2:D_{22}$ ($D_{10}\times C_5^2$), $D_{22}\times C_{11}^2$ ($D_{10}\times C_5^2$)

Order 4840: $(C_{11}\times C_{22}).D_{10}$ ($C_5\times F_{11}$)

Order 3025: $C_{11}^2:C_5^2$ ($D_4\times F_{11}$)

Order 2662: $C_{11}^3:C_2$ ($C_{10}^2:C_{10}$) x 2, $C_{11}^2\times C_{22}$ ($D_5\times C_{10}^2$)

Order 2420: $C_{22}:F_{11}$ ($C_{10}\times F_{11}$), $C_{11}^2:D_{10}$ ($C_{10}\times F_{11}$), $(C_{11}\times C_{22}).D_5$ ($C_{10}\times F_{11}$), $C_{110}:D_{11}$ ($D_5\times F_{11}$)

Order 1331: $C_{11}^3$ ($D_{10}:C_{10}^2$)

Order 1210: $C_{11}\times C_{110}$ ($D_{10}\times F_{11}$), $C_{11}^2:C_{10}$ ($C_{22}:C_{10}^2$)

Order 605: $C_{11}\times C_{55}$ ($C_5:D_4\times F_{11}$), $C_{11}^2:C_5$ ($C_{44}:C_{10}^2$)

Order 484: $C_{11}:D_{22}$ ($C_{55}:C_{10}^2$)

Order 242: $C_{11}\times C_{22}$ ($C_{110}:C_{10}^2$)

Order 220: $C_5\times D_{22}$ ($C_{11}^2:(C_5\times D_{10})$)

Order 121: $C_{11}^2$ ($D_{110}:C_{10}^2$)

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

Order 55: $C_{55}$ ($C_{22}^2:(C_5\times D_{10})$)

Order 44: $D_{22}$ ($C_{11}^2:(D_{10}\times C_5^2)$)

Order 22: $C_{22}$ ($C_{11}^2:(D_5\times C_{10}^2)$)

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

Order 10: $C_{10}$ ($C_{11}^3.C_5^2.C_2^3$)

Order 5: $C_5$ ($C_{22}:F_{11}^2.C_2$)

Order 2: $C_2$ ($C_{11}^3.C_5^3.C_2^3$)

Order 1: $C_1$ ($C_{22}:F_{11}^2.C_{10}$)

Normal subgroups up to automorphism (quotient in parentheses)

not computed

Series

Derived series	not computed
Chief series	not computed
Lower central series	not computed
Upper central series	not computed

Character theory

Complex character table

The $1075 \times 1075$ character table is not available for this group.

Rational character table

The $315 \times 315$ rational character table is not available for this group.