C₂×#20267599 #27196869

order	13184
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂×#20267599

Splittings

Fitting	C₂²×C₁₆×C₁₀₃ ⋊ C₂
Sylow	C₁₀₃ ⋊ C₂×#359

Subgroups

	subgroup	G/H
center	C₂×C₄	D₈₂₄
commutator	C₈×C₁₀₃	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₁₀₃
Fitting	C₂²×C₁₆×C₁₀₃	C₂
socle	C₂²×C₁₀₃	#102
prime radical	C₂×#15693043	C₂
cosocle	C₂×C₈×C₁₀₃	C₂³
hypercenter	C₂²×C₁₆	D₁₀₃
hypocenter	C₁₀₃	C₂×#359
LCS	G ▷ C₈×C₁₀₃ ▷ C₄×C₁₀₃ ▷ C₂×C₁₀₃ ▷ C₁₀₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

224 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 85 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 137
transitive degree	μ_t(G) = 6592

A minimal faithful permutation representation — 3 generators on 137 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)(106,108)(107,113)(109,122)(110,119)(111,127)(112,120)(114,131)(115,128)(116,130)(117,121)(118,124)(123,135)(125,137)(126,129)(132,133)(134,136);(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)(106,107,110,115,112,116,111,126)(108,117,119,133,120,134,127,131)(109,121,123,132,125,136,124,114)(113,122,128,135,130,137,129,118);(104,105)

Conjugacy classes & character table

3308 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13184, 6235);