#27184565

order	12928
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₄×Q₈×C₁₀₁ ⋊ C₂
Sylow	C₁₀₁ ⋊ #3461

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×D₁₀₁
commutator, socle, cosocle	C₂²×C₁₀₁	C₂⁵
Frattini	C₂²	C₂⁴×D₁₀₁
Fitting	C₂×C₄×Q₈×C₁₀₁	C₂
prime radical	#20265758	C₂
hypercenter	C₂×C₄×Q₈	D₁₀₁
hypocenter	C₁₀₁	#3461
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×Q₈

1012 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 125
transitive degree	μ_t(G) = 6464

A minimal faithful permutation representation — 5 generators on 125 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)(102,104,105,108)(103,107,106,109);(111,117)(115,118)(116,125)(123,124);(103,106)(107,109)(110,113,114,122)(111,116,117,125)(112,119,120,121)(115,123,118,124);(110,112,114,120)(111,115,117,118)(113,121,122,119)(116,124,125,123);(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)(102,103)(104,107)(105,106)(108,109)(110,111)(112,118)(113,116)(114,117)(115,120)(119,124)(121,123)(122,125)

Polycyclic presentation:

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

Conjugacy classes & character table

2044 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12928, 17943);