#27301034

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1380 ⋊ C₂
Sylow	C₅₃ ⋊ #12992

Subgroups

	subgroup	G/H
center	C₂³	D₈×D₅₃
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#1380	C₂
socle	C₂³×C₅₃	C₂×D₈
prime radical	C₂×#16810860	C₂
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1380	D₅₃
hypocenter	C₅₃	#12992
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1380

582 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 77
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 3 generators on 77 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)(59,61)(62,65,67,68)(63,64,70,73)(66,74,76,71)(69,72,75,77);(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)(54,56,57,55)(62,64)(63,68)(65,70)(66,75)(67,73)(69,76)(71,72)(74,77);(54,55,57,56)(58,59)(60,61)(62,63,66,69,65,73,74,77,67,70,76,75,68,64,71,72)

Polycyclic presentation:

2^8,53|a^2=f,b^2=dfg,c^2=h,d^2=gh,f^2=gh,g^2=h,b^a=bd,c^a=ce,d^b=dg,f^b=fg,g^b=gh,i^b=i^52,i^c=i^52

Conjugacy classes & character table

1202 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 86510);