#27209290

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

Aliases

No aliases for this group.

Splittings

Fitting	#563×C₁₀₃ ⋊ C₂
Sylow	C₁₀₃ ⋊ #3531

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₀₃
commutator, socle, cosocle	C₂²×C₁₀₃	C₂⁵
Fitting	#563×C₁₀₃	C₂
hypercenter	#563	D₁₀₃
hypocenter	C₁₀₃	#3531
LCS	G ▷ C₂²×C₁₀₃ ▷ C₁₀₃
UCS	C₁ ◁ C₂² ◁ #563

930 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 135
transitive degree	μ_t(G) = 6592

A minimal faithful permutation representation — 5 generators on 135 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)(120,123,124,131)(121,127,126,133)(122,130,129,135)(125,132,128,134);(105,111)(109,118)(110,114)(117,119)(121,126)(122,129)(127,133)(130,135);(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)(104,107,108,116)(105,110,111,114)(106,112,113,115)(109,117,118,119)(122,129)(125,128)(130,135)(132,134);(104,106,108,113)(105,109,111,118)(107,115,116,112)(110,119,114,117)(120,122)(121,125)(123,130)(124,129)(126,128)(127,132)(131,135)(133,134);(104,105)(106,109)(107,114)(108,111)(110,116)(112,119)(113,118)(115,117)(120,121)(122,128)(123,127)(124,126)(125,129)(130,134)(131,133)(132,135)

Polycyclic presentation:

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

Conjugacy classes & character table

1007 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13184, 18656);