C₂×D₅₉₁₃ #72223975

order	23652
automorphism	Aut(G) order 45979488
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral	D₁₁₈₂₆
direct product	C₂×D₅₉₁₃

Splittings

Fitting	C₂×C₇₃×C₈₁ ⋊ C₂
derived, Hall	C₇₃×C₈₁ ⋊ C₂²
Sylow	C₈₁ ⋊ C₂×D₇₃
Sylow	C₇₃ ⋊ C₂×D₈₁

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₅₉₁₃
commutator, cosocle, hypocenter	C₇₃×C₈₁	C₂²
Frattini	C₂₇	C₂×D₂₁₉
Fitting	C₂×C₇₃×C₈₁	C₂
socle	C₂×C₃×C₇₃	D₂₇
LCS	G ▷ C₇₃×C₈₁
UCS	C₁ ◁ C₂

50 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 156
transitive degree	μ_t(G) = 11826

A minimal faithful permutation representation — 2 generators on 156 points:

(1,2)(3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83)(84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156);(4,83)(5,82)(6,81)(7,80)(8,79)(9,78)(10,77)(11,76)(12,75)(13,74)(14,73)(15,72)(16,71)(17,70)(18,69)(19,68)(20,67)(21,66)(22,65)(23,64)(24,63)(25,62)(26,61)(27,60)(28,59)(29,58)(30,57)(31,56)(32,55)(33,54)(34,53)(35,52)(36,51)(37,50)(38,49)(39,48)(40,47)(41,46)(42,45)(43,44)(85,156)(86,155)(87,154)(88,153)(89,152)(90,151)(91,150)(92,149)(93,148)(94,147)(95,146)(96,145)(97,144)(98,143)(99,142)(100,141)(101,140)(102,139)(103,138)(104,137)(105,136)(106,135)(107,134)(108,133)(109,132)(110,131)(111,130)(112,129)(113,128)(114,127)(115,126)(116,125)(117,124)(118,123)(119,122)(120,121)

Conjugacy classes & character table

5916 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₃₆₅₂
maximal normal subgroup of	D₂₃₆₅₂
maximal quotient of	Dic₁₁₈₂₆, D₂₃₆₅₂

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((1,2)(3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83)(84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156), (4,83)(5,82)(6,81)(7,80)(8,79)(9,78)(10,77)(11,76)(12,75)(13,74)(14,73)(15,72)(16,71)(17,70)(18,69)(19,68)(20,67)(21,66)(22,65)(23,64)(24,63)(25,62)(26,61)(27,60)(28,59)(29,58)(30,57)(31,56)(32,55)(33,54)(34,53)(35,52)(36,51)(37,50)(38,49)(39,48)(40,47)(41,46)(42,45)(43,44)(85,156)(86,155)(87,154)(88,153)(89,152)(90,151)(91,150)(92,149)(93,148)(94,147)(95,146)(96,145)(97,144)(98,143)(99,142)(100,141)(101,140)(102,139)(103,138)(104,137)(105,136)(106,135)(107,134)(108,133)(109,132)(110,131)(111,130)(112,129)(113,128)(114,127)(115,126)(116,125)(117,124)(118,123)(119,122)(120,121));

C2×D5913 #72223975