#27539394

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2241 ⋊ C₂
Sylow	C₅₃ ⋊ #33697

Subgroups

	subgroup	G/H
center	C₂²	C₂×D₈×D₅₃
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2241	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	#20285144	C₂
hypercenter	#2241	D₅₃
hypocenter	C₅₃	#33697
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2241

598 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 93
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 4 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)(62,66,69,83)(63,72,76,75)(64,77,79,80)(65,78,81,93)(67,68,82,84)(70,85,88,87)(71,89,90,86)(73,91,92,74);(55,59)(58,61)(62,65,67,80,83,93,84,79,69,81,82,77,66,78,68,64)(63,71,73,85,72,89,91,88,76,90,92,87,75,86,74,70);(54,56,57,60)(55,58,59,61)(62,64,68,78,66,77,82,81,69,79,84,93,83,80,67,65)(63,70,74,86,75,87,92,90,76,88,91,89,72,85,73,71);(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)(56,58)(57,59)(60,61)(62,63)(64,71)(65,70)(66,72)(67,74)(68,73)(69,76)(75,83)(77,89)(78,85)(79,90)(80,86)(81,88)(82,91)(84,92)(87,93)

Polycyclic presentation:

2^8,53|b^2=f,c^2=e,d^2=h,e^2=gh,f^2=g,g^2=h,b^a=be,c^a=cf,e^a=eg,f^a=fgh,g^a=gh,i^a=i^52,i^d=i^52

Conjugacy classes & character table

1064 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 324870);