#27172891

order	12928
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Sylow

C₁₀₁ ⋊ #2151

Subgroups

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₁₀₁
commutator, socle	C₂²×C₁₀₁	C₂²×C₈
Fitting	#405×C₁₀₁	C₂
prime radical	C₂³×C₁₀₁	C₂×C₈
cosocle	C₂×C₈×C₁₀₁	C₂³
hypercenter	#405	D₁₀₁
hypocenter	C₁₀₁	#2151
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂×C₈ ◁ #405

160 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 137
transitive degree	μ_t(G) = 6464

A minimal faithful permutation representation — 3 generators on 137 points:

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

Polycyclic presentation:

2^7,101|a^2=e,b^2=d,e^2=f,f^2=g,b^a=bd,h^a=h^100,c^b=cg

Conjugacy classes & character table

2056 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12928, 6269);