#578×D₁₀₁ #27185894

order	12928
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

#578×D₁₀₁

Splittings

Fitting	#578×C₁₀₁ ⋊ C₂
Sylow	C₁₀₁ ⋊ C₂×#578

Subgroups

	subgroup	G/H
center	C₂	C₂²×D₄×D₁₀₁
commutator, cosocle	C₄×C₁₀₁	C₂⁵
Frattini	C₄	C₂⁴×D₁₀₁
Fitting	#578×C₁₀₁	C₂
socle	C₂×C₁₀₁	C₂³×D₄
hypercenter	#578	D₁₀₁
hypocenter	C₁₀₁	C₂×#578
LCS	G ▷ C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂ ◁ #41 ◁ #578

1408 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 133
transitive degree	μ_t(G) = 3232

A minimal faithful permutation representation — 5 generators on 133 points:

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

Conjugacy classes & character table

1144 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12928, 19272);

#578×D101 #27185894