Abstract group 52404.d: $C_{1191}:C_{44}$

• Label: 52404.d
• Order: \( 2^{2} \cdot 3 \cdot 11 \cdot 397 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 11 \cdot 397 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 3 \cdot 11 \)
• $\card{Z(G)}$: \( 2^{2} \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{4} \cdot 3^{2} \cdot 11 \cdot 397 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \cdot 3^{2} \)
• Perm deg.: $404$
• Trans deg.: $4764$
• Rank: $2$

Group information

Description:	$C_{1191}:C_{44}$
Order:	\(52404\)\(\medspace = 2^{2} \cdot 3 \cdot 11 \cdot 397 \)
Exponent:	\(52404\)\(\medspace = 2^{2} \cdot 3 \cdot 11 \cdot 397 \)
Automorphism group:	$C_5^2\wr C_2.D_4$, of order \(628848\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 11 \cdot 397 \)
Composition factors:	$C_2$ x 2, $C_3$, $C_{11}$, $C_{397}$
Derived length:	$2$

This group is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 11$.

Group statistics

Order	1	2	3	4	6	11	12	22	33	44	66	132	397	794	1191	1588	2382	4764
Elements	1	1	2	2	2	3970	4	3970	7940	7940	7940	15880	396	396	792	792	792	1584	52404
Conjugacy classes	1	1	2	2	2	10	4	10	20	20	20	40	36	36	72	72	72	144	564
Divisions	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	18
Autjugacy classes	1	1	1	1	1	10	1	10	10	10	10	10	1	1	1	1	1	1	72

Minimal presentations

Permutation degree:	$404$
Transitive degree:	$4764$
Rank:	$2$
Inequivalent generating pairs:	$2880$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	11	not computed	not computed
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	$\langle a, b \mid a^{132}=b^{397}=1, b^{a}=b^{126} \rangle$
Permutation group:	Degree $404$ $\langle(1,2,4,10,28,72,164,257,132,256,371,345,208,322,386,324,216,352,351,215,267,138,265,197,278,147,62,111,233,119,49,18,48,116,103,96,212,348,376,283,360,227,106,42,104,224,356,382,279,381,390,350,364,281,315,180,225,234,230,339,203,91,36,55,20,53,127,250,326,395,314,372,258,239,172,76,29,74,169,152,285,310,384,290,155,276,191,328,299,165,298,183,177,79,176,308,354,219,100,40,98,214,335,262,136,57,21,7,19,51,69,159,198,89,35,12,34,87,193,94,209,346,389,304,168,217,99,141,59,139,268,375,341,204,181,266,240,120,157,68,26,9,25,66,154,163,109,229,213,97,107,228,296,327,387,300,379,368,396,361,318,182,81,105,226,359,243,122,232,259,134,56,47,114,236,223,292,194,331,330,192,86,33,11,31,80,179,313,319,355,221,128,145,277,150,282,321,184,83,140,269,334,200,90,170,75,101,220,305,377,380,274,143,60,22,58,137,263,347,383,284,237,202,291,156,67,38,95,210,167,73,166,301,218,125,142,271,378,385,388,353,317,366,393,309,363,245,123,133,249,293,158,117,112,46,17,6,16,44,108,161,295,189,272,288,303,325,186,84,32,82,130,252,244,340,391,373,261,135,260,151,64,24,63,149,231,110,45,43,15,5,13,37,93,207,343,255,131,54,129,251,367,344,280,148,185,270,365,242,349,374,264,171,113,235,362,323,196,88,195,146,61,144,275,187,211,329,333,397,338,222,102,41,14,39,65,153,287,337,294,289,247,273,332,199,297,357,392,307,174,306,316,248,126,52,124,246,336,201,320,358,238,115,118,162,71,27,70,160,175,78,30,77,173,286,190,85,188,254,370,302,253,369,394,312,178,311,342,206,92,205,241,121,50,23,8,3) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 267 & 0 \\ 0 & 231 \end{array}\right), \left(\begin{array}{rr} 396 & 0 \\ 0 & 396 \end{array}\right), \left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{397})$
Direct product:	$C_4$ $\, \times\, $ $C_3$ $\, \times\, $ $(C_{397}:C_{11})$
Semidirect product:	$C_{4764}$ $\,\rtimes\,$ $C_{11}$	$C_{1588}$ $\,\rtimes\,$ $C_{33}$	$C_{1191}$ $\,\rtimes\,$ $C_{44}$	$C_{397}$ $\,\rtimes\,$ $C_{132}$	more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{794}$ . $C_{66}$	$C_{2382}$ . $C_{22}$	$(C_{397}:C_{22})$ . $C_6$	$C_6$ . $(C_{397}:C_{22})$	all 6
Aut. group:	$\Aut(C_6^2:S_4\wr C_2)$

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

Homology

Abelianization:	$C_{132} \simeq C_{4} \times C_{3} \times C_{11}$
Schur multiplier:	$C_1$
Commutator length:	$1$

Subgroups

There are 2400 subgroups in 24 conjugacy classes, 18 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_{12}$	$G/Z \simeq$ $C_{397}:C_{11}$
Commutator:	$G' \simeq$ $C_{397}$	$G/G' \simeq$ $C_{132}$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_{1191}:C_{22}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{4764}$	$G/\operatorname{Fit} \simeq$ $C_{11}$
Radical:	$R \simeq$ $C_{1191}:C_{44}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{2382}$	$G/\operatorname{soc} \simeq$ $C_{22}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
11-Sylow subgroup:	$P_{ 11 } \simeq$ $C_{11}$
397-Sylow subgroup:	$P_{ 397 } \simeq$ $C_{397}$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

Classes of subgroups up to automorphism

Normal subgroups

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 52404: $C_{1191}:C_{44}$

Order 26202: $C_{1191}:C_{22}$

Order 17468: $C_{1588}:C_{11}$

Order 13101: $C_{1191}:C_{11}$

Order 8734: $C_{397}:C_{22}$

Order 4764: $C_{4764}$

Order 4367: $C_{397}:C_{11}$

Order 2382: $C_{2382}$

Order 1588: $C_{1588}$

Order 1191: $C_{1191}$

Order 794: $C_{794}$

Order 397: $C_{397}$

Order 132: $C_{132}$

Order 66: $C_{66}$

Order 44: $C_{44}$

Order 33: $C_{33}$

Order 22: $C_{22}$

Order 12: $C_{12}$

Order 11: $C_{11}$

Order 6: $C_6$

Order 4: $C_4$

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 52404: $C_{1191}:C_{44}$

Order 26202: $C_{1191}:C_{22}$

Order 17468: $C_{1588}:C_{11}$

Order 13101: $C_{1191}:C_{11}$

Order 8734: $C_{397}:C_{22}$

Order 4764: $C_{4764}$

Order 4367: $C_{397}:C_{11}$

Order 2382: $C_{2382}$

Order 1588: $C_{1588}$

Order 1191: $C_{1191}$

Order 794: $C_{794}$

Order 397: $C_{397}$

Order 132: $C_{132}$

Order 66: $C_{66}$

Order 44: $C_{44}$

Order 33: $C_{33}$

Order 22: $C_{22}$

Order 12: $C_{12}$

Order 11: $C_{11}$

Order 6: $C_6$

Order 4: $C_4$

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 52404: $C_{1191}:C_{44}$ ($C_1$)

Order 26202: $C_{1191}:C_{22}$ ($C_2$)

Order 17468: $C_{1588}:C_{11}$ ($C_3$)

Order 13101: $C_{1191}:C_{11}$ ($C_4$)

Order 8734: $C_{397}:C_{22}$ ($C_6$)

Order 4764: $C_{4764}$ ($C_{11}$)

Order 4367: $C_{397}:C_{11}$ ($C_{12}$)

Order 2382: $C_{2382}$ ($C_{22}$)

Order 1588: $C_{1588}$ ($C_{33}$)

Order 1191: $C_{1191}$ ($C_{44}$)

Order 794: $C_{794}$ ($C_{66}$)

Order 397: $C_{397}$ ($C_{132}$)

Order 12: $C_{12}$ ($C_{397}:C_{11}$)

Order 6: $C_6$ ($C_{397}:C_{22}$)

Order 4: $C_4$ ($C_{1191}:C_{11}$)

Order 3: $C_3$ ($C_{1588}:C_{11}$)

Order 2: $C_2$ ($C_{1191}:C_{22}$)

Order 1: $C_1$ ($C_{1191}:C_{44}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 52404: $C_{1191}:C_{44}$ ($C_1$)

Order 26202: $C_{1191}:C_{22}$ ($C_2$)

Order 17468: $C_{1588}:C_{11}$ ($C_3$)

Order 13101: $C_{1191}:C_{11}$ ($C_4$)

Order 8734: $C_{397}:C_{22}$ ($C_6$)

Order 4764: $C_{4764}$ ($C_{11}$)

Order 4367: $C_{397}:C_{11}$ ($C_{12}$)

Order 2382: $C_{2382}$ ($C_{22}$)

Order 1588: $C_{1588}$ ($C_{33}$)

Order 1191: $C_{1191}$ ($C_{44}$)

Order 794: $C_{794}$ ($C_{66}$)

Order 397: $C_{397}$ ($C_{132}$)

Order 12: $C_{12}$ ($C_{397}:C_{11}$)

Order 6: $C_6$ ($C_{397}:C_{22}$)

Order 4: $C_4$ ($C_{1191}:C_{11}$)

Order 3: $C_3$ ($C_{1588}:C_{11}$)

Order 2: $C_2$ ($C_{1191}:C_{22}$)

Order 1: $C_1$ ($C_{1191}:C_{44}$)

Series

Derived series

$C_{1191}:C_{44}$

$\rhd$

$C_{397}$

$\rhd$

$C_1$

Chief series

$C_{1191}:C_{44}$

$\rhd$

$C_{1191}:C_{22}$

$\rhd$

$C_{1191}:C_{11}$

$\rhd$

$C_{397}:C_{11}$

$\rhd$

$C_{397}$

$\rhd$

$C_1$

Lower central series

$C_{1191}:C_{44}$

$\rhd$

$C_{397}$

Upper central series

$C_1$

$\lhd$

$C_{12}$

Supergroups

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

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

Character theory

Complex character table

The $564 \times 564$ character table is not available for this group.

Rational character table

The $18 \times 18$ rational character table is not available for this group.