C₂²×Dic₈₀₈ #27184245

order	12928
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂²×Dic₈₀₈

Splittings

Sylow

C₁₀₁ ⋊ C₂²×Q₃₂

Subgroups

	subgroup	G/H
center	C₂³	D₈₀₈
commutator, cosocle	C₈×C₁₀₁	C₂⁴
Frattini	C₈	C₂³×D₁₀₁
Fitting	C₂²×C₁₆×C₁₀₁	C₂
socle, prime radical	C₂³×C₁₀₁	D₈
hypercenter	C₂²×C₁₆	D₁₀₁
hypocenter	C₁₀₁	C₂²×Q₃₂
LCS	G ▷ C₈×C₁₀₁ ▷ C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

360 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 137
transitive degree	μ_t(G) = 12928

A minimal faithful permutation representation — 4 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)(106,108,111,119)(107,112,115,129)(109,121,124,137)(110,126,122,118)(113,131,132,136)(114,133,125,128)(116,134,135,120)(117,130,127,123);(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)(106,107,111,115)(108,116,119,135)(109,120,124,134)(110,125,122,114)(112,127,129,117)(113,130,132,123)(118,136,126,131)(121,133,137,128);(104,105);(102,103)

Conjugacy classes & character table

3244 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12928, 17623);

C22×Dic808 #27184245