#27184234

order	12928
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₂×M₃₂×C₁₀₁ ⋊ C₂
Sylow	C₁₀₁ ⋊ #3407

Subgroups

	subgroup	G/H
center, Frattini	C₈	C₂³×D₁₀₁
commutator, socle	C₂×C₁₀₁	C₂³×C₈
Fitting	C₂×M₃₂×C₁₀₁	C₂
prime radical	C₂×#13108074	C₄
cosocle	C₈×C₁₀₁	C₂⁴
hypercenter	C₂×M₃₂	D₁₀₁
hypocenter	C₁₀₁	#3407
LCS	G ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₈ ◁ C₂×M₃₂

360 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

2068 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12928, 17612);