#27277772

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #8790

Subgroups

	subgroup	G/H
center	C₂²	#16810488
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1846	C₂
socle	C₂²×C₅₃	#382
prime radical	C₂³×C₅₃	#124
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1846	D₅₃
hypocenter	C₅₃	#8790
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1846

294 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 93
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 3 generators on 93 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)(54,56,57,55,59,60,58,61)(62,65,66,77,67,78,79,92)(63,69,70,82,71,76,83,91)(64,73,74,87,75,88,89,93)(68,80,72,84,81,90,85,86);(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)(54,55,58,56,59,61,57,60)(62,64)(63,68,71,81)(65,73)(66,74)(67,75)(69,80,76,90)(70,72,83,85)(77,87)(78,88)(79,89)(82,84,91,86)(92,93);(55,60)(56,61)(57,58)(62,63)(64,72,75,85)(65,76,78,69)(66,70)(67,71)(68,74,81,89)(73,86)(77,91,92,82)(79,83)(80,87)(84,88)(90,93)

Polycyclic presentation:

2^8,53|a^2=f,b^2=e,c^2=d,d^2=h,e^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^b=fgh,i^b=i^52,f^c=fgh,i^c=i^52

Conjugacy classes & character table

1196 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 63248);