Abstract group 14641000.t: $C_{11}^4:(D_5^2:C_{10})$

• Label: 14641000.t
• Order: \( 2^{3} \cdot 5^{3} \cdot 11^{4} \)
• Exponent: \( 2^{2} \cdot 5 \cdot 11 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 5 \)
• $\card{Z(G)}$: 1
• $\card{\Aut(G)}$: \( 2^{4} \cdot 5^{3} \cdot 11^{4} \)
• $\card{\mathrm{Out}(G)}$: \( 2 \)
• Perm deg.: $44$
• Trans deg.: $44$
• Rank: $2$

Group information

Description:	$C_{11}^4:(D_5^2:C_{10})$
Order:	\(14641000\)\(\medspace = 2^{3} \cdot 5^{3} \cdot 11^{4} \)
Exponent:	\(220\)\(\medspace = 2^{2} \cdot 5 \cdot 11 \)
Automorphism group:	$C_{11}^4.C_5^2.C_{20}.C_2^2$, of order \(29282000\)\(\medspace = 2^{4} \cdot 5^{3} \cdot 11^{4} \)
Composition factors:	$C_2$ x 3, $C_5$ x 3, $C_{11}$ x 4
Derived length:	$4$

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
Elements	1	17545	66550	828124	6894580	14640	2928200	641300	665500	987360	1597200	14641000
Conjugacy classes	1	3	1	29	32	24	4	13	2	42	12	163
Divisions	1	3	1	9	9	22	1	11	1	8	2	68
Autjugacy classes	1	3	1	29	32	22	4	11	1	26	6	136

Dimension	1	2	4	8	16	32	40	100	160	200	250	320	400	500	800	1000	2000
Irr. complex chars.	20	5	40	5	0	0	25	32	0	7	8	0	0	10	0	11	0	163
Irr. rational chars.	4	1	4	6	8	1	1	4	2	3	4	2	3	12	3	9	1	68

Minimal presentations

Permutation degree:	$44$
Transitive degree:	$44$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of faithful linear representations

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

Constructions

Presentation:	${\langle a, b, c, d, e, f, g \mid a^{10}=c^{5}=d^{11}=e^{55}=f^{11}=g^{11}= \!\cdots\! \rangle}$
Permutation group:	Degree $44$ $\langle(1,42,4,40,9,44,10,36,8,41)(2,34)(3,37,11,39,6,35,5,43,7,38)(12,30,15,29,20,31,21,27,19,24) \!\cdots\! \rangle$
Transitive group:	44T608				more information
Direct product:	not computed
Semidirect product:	$C_{11}^4$ $\,\rtimes\,$ $(D_5^2:C_{10})$				more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$(C_{11}^4.D_5^2)$ . $C_{10}$	$(C_{11}^4:D_5^2)$ . $C_{10}$	$(C_{11}^4:D_5\wr C_2)$ . $C_5$	$(C_{11}^4:(C_5:F_5))$ . $C_{10}$	all 11

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

Homology

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

Subgroups

There are 15 normal subgroups, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	a subgroup isomorphic to $C_1$
Commutator:	not computed
Frattini:	a subgroup isomorphic to $C_1$
Fitting:	not computed
Radical:	not computed
Socle:	not computed
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_4$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5^3$
11-Sylow subgroup:	$P_{ 11 } \simeq$ $C_{11}^4$

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 117128 (order at least 125) are shown, as well as all normal subgroups of any index.

Order 14641000: $C_{11}^4:(D_5^2:C_{10})$

Order 7320500: $C_{11}^4.C_5^3.C_2^2$, $C_{11}^4:(C_5\times D_5^2)$, $C_{11}^4:(C_5^2:C_{20})$

Order 3660250: $C_{11}^4.C_5^3.C_2$ x 3

Order 2928200: $C_{11}^4:D_5\wr C_2$

Order 1830125: $C_{11}^4:C_5^3$

Order 1464100: $C_{11}^4.D_5^2$, $C_{11}^4.(C_{10}\times D_5)$, $C_{11}^4:D_5^2$, $C_{11}^4:(C_5\times D_{10})$, $C_{11}^4:(C_5\times F_5)$, $C_{11}^4:(C_5:F_5)$

Order 732050: $C_{11}^4:(C_5\times D_5)$ x 11, $C_{11}^4:(C_5\times C_{10})$ x 2, $C_{11}^4:(C_5:D_5)$

Order 585640: $C_{11}^4:(C_5\times D_4)$

Order 366025: $C_{11}^4:C_5^2$ x 9

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

Order 292820: $C_{11}^4:D_{10}$, $C_{11}^4:(C_2\times C_{10})$, $C_{11}^4:D_{10}$, $C_{11}^4:(C_2\times C_{10})$, $C_{11}^4:F_5$, $C_{11}^4:C_{20}$

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

Order 146410: $C_{11}^4:C_{10}$ x 9, $C_{11}^4:D_5$ x 4, $C_{11}^3:D_{55}$

Order 133100: $C_{11}^3:(C_{10}\times D_5)$, $C_{11}^3:(C_5\times D_{10})$

Order 117128: $C_{11}^3:D_{44}$

Order 73205: $C_{11}^4:C_5$ x 7, $C_{11}\times C_{11}^3:C_5$, $C_{11}^3:C_{55}$

Order 66550: $C_{11}^3:(C_5\times D_5)$ x 5, $C_{11}^3:(C_5\times D_5)$ x 4, $C_{11}^3:(C_5\times C_{10})$ x 3, $C_{11}^3:(C_5\times D_5)$, $C_{11}^3:(C_5\times C_{10})$, $C_{11}^3:(C_5\times C_{10})$

Order 60500: $C_{11}^2:(C_5\times D_5^2)$

Order 58564: $C_{11}^4:C_2^2$, $C_{11}\wr C_2^2$, $C_{11}\wr C_4$

Order 53240: $C_{11}^3:(C_5\times D_4)$, $D_{11}^2:F_{11}$

Order 33275: $C_{11}^3:C_5^2$ x 17, $C_{11}^3:C_5^2$ x 4, $C_{11}^2:C_5\times C_{11}:C_5$, $C_{11}\times C_{11}^2:C_5^2$

Order 30250: $C_{11}^2:(D_5\times C_5^2)$ x 2, $C_{11}^2:(C_5^2\times D_5)$, $C_{11}^2:(C_5\times C_5:D_5)$

Order 29282: $C_{11}^3:C_{22}$ x 2, $C_{11}^4:C_2$

Order 26620: $C_{11}^3:(C_2\times C_{10})$ x 8, $C_{11}^3:(C_2\times C_{10})$ x 2, $C_{11}^3:D_{10}$, $C_{11}^3:C_{20}$, $C_{11}^3:C_{20}$, $C_{11}^3:D_{10}$, $C_{11}^3:(C_2\times C_{10})$

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

Order 14641: $C_{11}^4$

Order 13310: $C_{11}^3:C_{10}$ x 31, $C_{11}^3:C_{10}$ x 13, $C_{11}^3:C_{10}$ x 7, $C_{11}^2:D_{55}$ x 4, $C_{11}\times D_{11}\times C_{11}:C_5$, $C_{11}\times C_{11}^2:C_{10}$, $C_{11}^3:C_{10}$, $C_{11}^3:C_{10}$, $C_{11}^3:C_{10}$, $C_{11}^3:C_{10}$, $C_{11}^3:D_5$, $C_{11}^2:C_{110}$, $C_{11}^3:C_{10}$, $C_{11}^3:C_{10}$

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

Order 10648: $C_{11}^3:D_4$, $C_{11}^2:D_{44}$

Order 6655: $C_{11}^3:C_5$ x 22, $C_{11}^3:C_5$ x 5, $C_{11}^2:C_{55}$ x 5, $C_{11}^3:C_5$ x 4, $C_{11}^3:C_5$ x 4, $C_{11}^3:C_5$ x 4, $C_{11}^2:C_{55}$ x 4, $C_{11}\times C_{11}^2:C_5$ x 4, $C_{11}^2:C_{55}$ x 4

Order 6050: $C_{11}^2:(C_5\times D_5)$ x 19, $C_{11}^2:(C_5\times D_5)$ x 5, $C_{11}:(C_5\times F_{11})$ x 3, $C_{55}:F_{11}$ x 2, $C_{11}:C_5\times F_{11}$ x 2, $C_5\times C_{11}^2:D_5$ x 2, $C_{11}^2:(C_5:D_5)$, $C_2\times C_{11}^2:C_5^2$

Order 5324: $C_{11}:D_{11}^2$ x 8, $C_{11}\times D_{11}^2$ x 2, $C_{11}^3:C_4$, $C_{11}^2:C_{44}$, $C_{11}:D_{11}^2$

Order 4840: $D_{11}^2:C_{10}$, $D_{22}:F_{11}$

Order 3025: $C_{11}^2:C_5^2$ x 37, $C_{11}^2:C_5^2$ x 2, $C_{11}^2:C_5^2$ x 2, $C_{11}^2:C_5^2$ x 2, $C_{55}:C_{55}$ x 2

Order 2750: $C_5^2:F_{11}$

Order 2662: $C_{11}^2:C_{22}$ x 31, $D_{11}\times C_{11}^2$ x 7, $C_{11}^3:C_2$

Order 2420: $D_{11}:F_{11}$ x 15, $C_{22}:F_{11}$ x 2, $C_{11}^2:D_{10}$ x 2, $C_{11}^2:D_{10}$, $C_{11}^2:D_{10}$, $C_{22}:F_{11}$, $C_{11}^2:C_{20}$, $C_{11}^2:C_{20}$

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

Order 1331: $C_{11}^3$ x 22

Order 1210: $C_{11}:F_{11}$ x 88, $C_{11}:F_{11}$ x 23, $C_{11}^2:D_5$ x 13, $C_{11}:F_{11}$ x 4, $C_{11}:F_{11}$ x 4, $C_{11}:F_{11}$ x 3, $C_{11}\times F_{11}$ x 2, $D_{11}\times C_{55}$ x 2, $C_{11}^2:C_{10}$ 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, $C_{11}:D_{55}$, $C_{11}\times D_{55}$, $C_{11}^2:C_{10}$, $C_{11}^2:C_{10}$, $C_{11}:C_{110}$

Order 1100: $C_{110}:C_{10}$, $D_5\times F_{11}$

Order 1000: $D_5^2:C_{10}$

Order 968: $D_{11}\wr C_2$, $C_{11}:D_{44}$

Order 605: $C_{11}^2:C_5$ x 136, $C_{11}^2:C_5$ x 28, $C_{11}:C_{55}$ x 28, $C_{11}^2:C_5$ x 19, $C_{11}\times C_{55}$ x 2

Order 550: $C_{55}:C_{10}$ x 8, $C_5\times F_{11}$ x 3, $C_{110}:C_5$ x 2, $C_{55}:C_{10}$, $C_5\times D_{55}$

Order 500: $C_5\times D_5^2$ x 2, $C_5^2:C_{20}$

Order 484: $D_{11}^2$ x 15, $C_{11}\times D_{22}$ x 2, $C_{11}^2:C_4$, $C_{11}:C_{44}$, $C_{11}:D_{22}$

Order 440: $C_{44}:C_{10}$, $D_{22}:C_{10}$

Order 275: $C_{55}:C_5$ x 22, $C_5\times C_{55}$

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

Order 242: $C_{11}\times D_{11}$ x 82, $C_{11}:D_{11}$ x 23, $C_{11}\times C_{22}$ x 2

Order 220: $C_2\times F_{11}$ x 10, $C_{22}:C_{10}$, $C_{11}:C_{20}$, $D_{110}$, $D_5\times D_{11}$, $C_{11}:C_{20}$

Order 200: $D_5\wr C_2$

Order 125: $C_5^3$

Order 8: $D_4$

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 14641000: $C_{11}^4:(D_5^2:C_{10})$ ($C_1$)

Order 7320500: $C_{11}^4.C_5^3.C_2^2$ ($C_2$), $C_{11}^4:(C_5\times D_5^2)$ ($C_2$), $C_{11}^4:(C_5^2:C_{20})$ ($C_2$)

Order 3660250: $C_{11}^4.C_5^3.C_2$ ($C_2^2$)

Order 2928200: $C_{11}^4:D_5\wr C_2$ ($C_5$)

Order 1830125: $C_{11}^4:C_5^3$ ($D_4$)

Order 1464100: $C_{11}^4.D_5^2$ ($C_{10}$), $C_{11}^4:D_5^2$ ($C_{10}$), $C_{11}^4:(C_5:F_5)$ ($C_{10}$)

Order 732050: $C_{11}^4:(C_5:D_5)$ ($C_2\times C_{10}$)

Order 366025: $C_{11}^4:C_5^2$ ($C_5\times D_4$)

Order 73205: $C_{11}^4:C_5$ ($D_5\wr C_2$)

Order 14641: $C_{11}^4$ ($D_5^2:C_{10}$)

Order 1: $C_1$ ($C_{11}^4:(D_5^2:C_{10})$)

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

See the $163 \times 163$ 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 $68 \times 68$ rational character table.