Abstract group 1464100.g: $\He_{11}.(C_{10}\times F_{11})$

• Label: 1464100.g
• Order: \( 2^{2} \cdot 5^{2} \cdot 11^{4} \)
• Exponent: \( 2 \cdot 5 \cdot 11 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 5^{2} \)
• $\card{Z(G)}$: 2
• $\card{\Aut(G)}$: \( 2^{3} \cdot 5^{2} \cdot 11^{4} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \)
• Perm deg.: $123$
• Trans deg.: $242$
• Rank: $2$

Group information

Description:	$\He_{11}.(C_{10}\times F_{11})$
Order:	\(1464100\)\(\medspace = 2^{2} \cdot 5^{2} \cdot 11^{4} \)
Exponent:	\(110\)\(\medspace = 2 \cdot 5 \cdot 11 \)
Automorphism group:	$C_2\times C_{11}^2.F_{11}^2$, of order \(2928200\)\(\medspace = 2^{3} \cdot 5^{2} \cdot 11^{4} \)
Composition factors:	$C_2$ x 2, $C_5$ x 2, $C_{11}$ x 4
Derived length:	$3$

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

Group statistics

Order	1	2	5	10	11	22	55	110
Elements	1	2663	138424	734712	14640	41260	212960	319440	1464100
Conjugacy classes	1	3	24	72	9	13	20	36	178
Divisions	1	3	6	18	7	9	4	6	54
Autjugacy classes	1	2	24	48	7	8	16	20	126

Dimension	1	4	5	10	40	50	100	110	440	550
Irr. complex chars.	100	0	40	10	0	4	0	20	0	4	178
Irr. rational chars.	4	24	0	6	6	0	2	4	4	4	54

Minimal presentations

Permutation degree:	$123$
Transitive degree:	$242$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	110	110	110
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e \mid a^{10}=b^{110}=c^{11}=d^{11}=e^{11}=[c,d]=[c,e]= \!\cdots\! \rangle}$
Permutation group:	Degree $123$ $\langle(1,2,6,48,114,37,60,66,27,44,9)(3,23,4,32,52,121,55,33,92,14,29)(5,17,51,84,41,106,63,62,75,104,45) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrr} 8 & 8 & 0 & 4 \\ 9 & 5 & 2 & 8 \\ 10 & 2 & 10 & 6 \\ 8 & 1 & 5 & 7 \end{array}\right), \left(\begin{array}{rrrr} 5 & 9 & 8 & 3 \\ 9 & 1 & 5 & 9 \\ 10 & 4 & 0 & 8 \\ 5 & 6 & 7 & 10 \end{array}\right), \left(\begin{array}{rrrr} 8 & 0 & 1 & 1 \\ 8 & 0 & 8 & 1 \\ 8 & 3 & 9 & 0 \\ 10 & 8 & 3 & 1 \end{array}\right), \left(\begin{array}{rrrr} 0 & 8 & 5 & 0 \\ 1 & 4 & 8 & 5 \\ 7 & 8 & 5 & 3 \\ 3 & 7 & 10 & 9 \end{array}\right), \left(\begin{array}{rrrr} 7 & 0 & 0 & 0 \\ 3 & 6 & 0 & 0 \\ 6 & 0 & 6 & 0 \\ 2 & 2 & 10 & 2 \end{array}\right), \left(\begin{array}{rrrr} 6 & 9 & 1 & 4 \\ 4 & 6 & 3 & 1 \\ 8 & 10 & 7 & 2 \\ 2 & 8 & 7 & 7 \end{array}\right), \left(\begin{array}{rrrr} 7 & 6 & 9 & 9 \\ 2 & 10 & 5 & 8 \\ 6 & 0 & 5 & 5 \\ 6 & 9 & 9 & 7 \end{array}\right), \left(\begin{array}{rrrr} 10 & 0 & 0 & 0 \\ 0 & 10 & 0 & 0 \\ 0 & 0 & 10 & 0 \\ 0 & 0 & 0 & 10 \end{array}\right) \right\rangle \subseteq \GL_{4}(\F_{11})$
Direct product:	not computed
Semidirect product:	not isomorphic to a non-trivial semidirect product
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$(\He_{11}:F_{11})$ . $C_{10}$ (2)	$(\He_{11}:F_{11})$ . $C_{10}$ (2)	$(\He_{11}:F_{11})$ . $C_{10}$ (2)	$(\He_{11}:F_{11})$ . $C_{10}$ (2)	all 45

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

Homology

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

Subgroups

There are 712000 subgroups in 532 conjugacy classes, 60 normal (40 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 $C_2$
Commutator:	a subgroup isomorphic to $C_{11}^3:C_{11}$
Frattini:	a subgroup isomorphic to $C_{11}^2$
Fitting:	not computed
Radical:	not computed
Socle:	not computed
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5^2$
11-Sylow subgroup:	$P_{ 11 } \simeq$ $C_{11}^3:C_{11}$

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 1464100 (order at least 1) are shown, as well as all normal subgroups of any index.

Order 1464100: $\He_{11}.(C_{10}\times F_{11})$

Order 732050: $\He_{11}:(C_5\times F_{11})$ x 2, $C_{11}^3:(C_{110}:C_5)$

Order 366025: $C_{11}^3:(C_{55}:C_5)$

Order 292820: $C_2\times \He_{11}:F_{11}$, $C_2\times C_{11}^3:C_{110}$, $C_2\times \He_{11}:F_{11}$, $C_2\times \He_{11}:F_{11}$, $C_2\times \He_{11}:F_{11}$, $\He_{11}:(C_5\times D_{22})$

Order 146410: $\He_{11}:F_{11}$ x 2, $C_{11}^3:C_{110}$ x 2, $\He_{11}:F_{11}$ x 2, $\He_{11}:F_{11}$ x 2, $\He_{11}:(C_5\times D_{11})$ x 2, $\He_{11}:F_{11}$ x 2, $C_{11}^3:(C_{11}:C_{10})$, $\He_{11}:C_{110}$, $C_{11}^3:C_{110}$, $C_{11}^3:C_{11}:C_{10}$, $C_{11}^3:(C_{11}:C_{10})$, $C_{11}^3:(C_{11}:C_{10})$

Order 133100: $\He_{11}:C_{10}^2$, $C_{11}^3:C_{10}^2$

Order 73205: $\He_{11}:C_{55}$, $C_{11}^3:(C_{11}:C_5)$, $C_{11}^3:(C_{11}:C_5)$, $C_{11}^3:(C_{11}:C_5)$, $C_{11}^3:C_{55}$, $C_{11}^3:(C_{11}:C_5)$

Order 66550: $C_{11}^2:(C_5\times F_{11})$ x 2, $C_{11}^3:(C_5\times C_{10})$ x 2, $C_2\times \He_{11}:C_5^2$, $C_2\times C_{11}^3:C_5^2$

Order 58564: $\He_{11}:D_{22}$

Order 33275: $\He_{11}:C_5^2$, $C_{11}^3:C_5^2$

Order 29282: $C_{11}^3:C_{22}$ x 2, $C_{11}^3:C_{22}$

Order 26620: $C_2\times C_{11}^3:C_{10}$ x 2, $C_2\times C_{11}^3:C_{10}$ x 2, $C_2\times C_{11}^2:F_{11}$, $C_2\times \He_{11}:C_{10}$, $C_2\times C_{11}^3:C_{10}$, $C_2\times C_{11}^2:F_{11}$, $C_2\times C_{11}^2:F_{11}$, $C_2\times C_{11}^3:C_{10}$, $C_2\times C_{11}^2:F_{11}$, $C_2\times C_{11}^2:C_{110}$

Order 14641: $C_{11}^3:C_{11}$

Order 13310: $C_{11}^3:C_{10}$ x 4, $C_{11}^3:C_{10}$ x 4, $\He_{11}:C_{10}$ x 3, $C_{11}^2:C_{110}$ x 2, $C_{11}^3:C_{10}$ x 2, $C_{11}^2:C_{110}$ x 2, $C_{11}^3:C_{10}$ x 2, $C_{11}^2:F_{11}$ x 2, $C_{11}^2:F_{11}$ x 2, $\He_{11}:C_{10}$ x 2, $C_{11}^2:F_{11}$ x 2, $C_{11}^2:C_{110}$ x 2, $C_{11}^3:C_{10}$ x 2, $C_{11}^2:F_{11}$ x 2, $C_{11}^3:C_{10}$, $C_{11}^2:C_{110}$, $\He_{11}:C_{10}$, $\He_{11}:C_{10}$

Order 12100: $C_{11}^2:C_{10}^2$ x 3, $C_{11}^2:C_{10}^2$

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

Order 6050: $C_{11}:(C_5\times F_{11})$ x 6, $C_2\times C_{11}^2:C_5^2$ x 4, $C_{11}:C_5\times F_{11}$ x 2

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

Order 3025: $C_{11}^2:C_5^2$ x 4

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

Order 2420: $C_{22}:F_{11}$ x 6, $C_{22}:F_{11}$ x 6, $C_{22}:F_{11}$ x 5, $C_{22}:F_{11}$ x 3, $C_{22}:F_{11}$, $C_{22}\times F_{11}$, $C_{22}:F_{11}$, $C_{11}:C_5\times D_{22}$, $C_{22}:F_{11}$, $C_{22}:F_{11}$

Order 1331: $\He_{11}$ x 2, $C_{11}^3$

Order 1210: $C_{11}:F_{11}$ x 12, $C_{11}:F_{11}$ x 12, $C_{11}:F_{11}$ x 10, $C_{11}^2:C_{10}$ x 9, $C_{11}:C_{110}$ x 8, $C_{11}^2:C_{10}$ x 6, $C_{11}:F_{11}$ x 6, $C_{11}^2:C_{10}$ x 4, $C_{11}\times F_{11}$ x 2, $C_{11}^2:C_{10}$ x 2, $C_{11}:F_{11}$ x 2, $C_{11}:F_{11}$ x 2, $C_{11}:F_{11}$ x 2, $C_{11}:F_{11}$ x 2

Order 1100: $C_{10}\times F_{11}$ x 3, $C_{110}:C_{10}$

Order 605: $C_{11}^2:C_5$ x 9, $C_{11}:C_{55}$ x 8, $C_{11}^2:C_5$ x 6, $C_{11}^2:C_5$ x 4

Order 550: $C_5\times F_{11}$ x 6, $C_{110}:C_5$ x 6

Order 484: $C_{11}:D_{22}$ x 5, $C_{11}\times D_{22}$

Order 275: $C_{55}:C_5$ x 4

Order 242: $C_{11}:D_{11}$ x 10, $C_{11}\times C_{22}$ x 7, $C_{11}\times D_{11}$ x 2

Order 220: $C_2\times F_{11}$ x 17, $C_{22}:C_{10}$ x 5, $C_5\times D_{22}$ x 3, $C_2\times C_{110}$

Order 121: $C_{11}^2$ x 7

Order 110: $F_{11}$ x 34, $C_{11}:C_{10}$ x 33, $C_{110}$ x 6, $C_5\times D_{11}$ x 6

Order 100: $C_{10}^2$

Order 55: $C_{11}:C_5$ x 23, $C_{55}$ x 4

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

Order 44: $D_{22}$ x 5, $C_2\times C_{22}$

Order 25: $C_5^2$

Order 22: $D_{11}$ x 10, $C_{22}$ x 9

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

Order 11: $C_{11}$ x 7

Order 10: $C_{10}$ x 18

Order 5: $C_5$ x 6

Order 4: $C_2^2$

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 1464100: $\He_{11}.(C_{10}\times F_{11})$ ($C_1$)

Order 732050: $\He_{11}:(C_5\times F_{11})$ ($C_2$) x 2, $C_{11}^3:(C_{110}:C_5)$ ($C_2$)

Order 366025: $C_{11}^3:(C_{55}:C_5)$ ($C_2^2$)

Order 292820: $C_2\times \He_{11}:F_{11}$ ($C_5$), $C_2\times C_{11}^3:C_{110}$ ($C_5$), $C_2\times \He_{11}:F_{11}$ ($C_5$), $C_2\times \He_{11}:F_{11}$ ($C_5$), $C_2\times \He_{11}:F_{11}$ ($C_5$), $\He_{11}:(C_5\times D_{22})$ ($C_5$)

Order 146410: $\He_{11}:F_{11}$ ($C_{10}$) x 2, $C_{11}^3:C_{110}$ ($C_{10}$) x 2, $\He_{11}:F_{11}$ ($C_{10}$) x 2, $\He_{11}:F_{11}$ ($C_{10}$) x 2, $\He_{11}:(C_5\times D_{11})$ ($C_{10}$) x 2, $\He_{11}:F_{11}$ ($C_{10}$) x 2, $C_{11}^3:(C_{11}:C_{10})$ ($C_{10}$), $\He_{11}:C_{110}$ ($C_{10}$), $C_{11}^3:C_{110}$ ($C_{10}$), $C_{11}^3:C_{11}:C_{10}$ ($C_{10}$), $C_{11}^3:(C_{11}:C_{10})$ ($C_{10}$), $C_{11}^3:(C_{11}:C_{10})$ ($C_{10}$)

Order 73205: $\He_{11}:C_{55}$ ($C_2\times C_{10}$), $C_{11}^3:(C_{11}:C_5)$ ($C_2\times C_{10}$), $C_{11}^3:(C_{11}:C_5)$ ($C_2\times C_{10}$), $C_{11}^3:(C_{11}:C_5)$ ($C_2\times C_{10}$), $C_{11}^3:C_{55}$ ($C_2\times C_{10}$), $C_{11}^3:(C_{11}:C_5)$ ($C_2\times C_{10}$)

Order 58564: $\He_{11}:D_{22}$ ($C_5^2$)

Order 29282: $C_{11}^3:C_{22}$ ($C_5\times C_{10}$) x 2, $C_{11}^3:C_{22}$ ($C_5\times C_{10}$)

Order 26620: $C_2\times C_{11}^3:C_{10}$ ($C_{11}:C_5$)

Order 14641: $C_{11}^3:C_{11}$ ($C_{10}^2$)

Order 13310: $C_{11}^3:C_{10}$ ($C_{11}:C_{10}$) x 2, $C_{11}^3:C_{10}$ ($C_{11}:C_{10}$), $\He_{11}:C_{10}$ ($F_{11}$)

Order 6655: $\He_{11}:C_5$ ($C_2\times F_{11}$), $C_{11}^3:C_5$ ($C_{22}:C_{10}$)

Order 5324: $C_{11}^2:D_{22}$ ($C_{55}:C_5$)

Order 2662: $C_{11}^3:C_2$ ($C_{110}:C_5$) x 2, $C_2\times \He_{11}$ ($C_5\times F_{11}$), $C_{11}^2\times C_{22}$ ($C_{110}:C_5$)

Order 1331: $C_{11}^3$ ($C_{110}:C_{10}$), $\He_{11}$ ($C_{10}\times F_{11}$)

Order 242: $C_{11}\times C_{22}$ ($C_{11}:C_5\times F_{11}$)

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

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

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

Order 2: $C_2$ ($\He_{11}:(C_5\times F_{11})$)

Order 1: $C_1$ ($\He_{11}.(C_{10}\times F_{11})$)

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

Supergroups

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

This group is a maximal quotient of 1 larger groups in the database.

Character theory

Complex character table

See the $178 \times 178$ 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 $54 \times 54$ rational character table.