#27208572

order	13184
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Splittings

Sylow

C₁₀₃ ⋊ #3461

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₀₃
commutator, socle, cosocle	C₂²×C₁₀₃	C₂⁵
Fitting, prime radical	#537×C₁₀₃	C₂
hypercenter	#537	D₁₀₃
hypocenter	C₁₀₃	#3461
LCS	G ▷ C₂²×C₁₀₃ ▷ C₁₀₃
UCS	C₁ ◁ C₂² ◁ #537

1012 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 127
transitive degree	μ_t(G) = 6592

A minimal faithful permutation representation — 5 generators on 127 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)(101,102)(104,106,107,110)(105,109,108,111);(113,119)(117,120)(118,127)(125,126);(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)(105,108)(109,111)(112,115,116,124)(113,118,119,127)(114,121,122,123)(117,125,120,126);(112,114,116,122)(113,117,119,120)(115,123,124,121)(118,126,127,125);(104,105)(106,109)(107,108)(110,111)(112,113)(114,120)(115,118)(116,119)(117,122)(121,126)(123,125)(124,127)

Polycyclic presentation:

2^7,103|b^2=f,c^2=f,e^2=g,b^a=bf,c^a=cg,d^a=df,c^b=cf,h^c=h^102,h^e=h^102

Conjugacy classes & character table

1166 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13184, 17938);