#27270649

order	13568
automorphism	?
homology	Abelianization (H₁) = C₄×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #7027

Subgroups

	subgroup	G/H
center	C₂²×C₄	#11719921
commutator	C₂×C₄×C₅₃	C₄×C₈
Frattini	C₄³	C₂×D₅₃
Fitting	C₄×C₄⋊C₈×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	#95
cosocle	C₄³×C₅₃	C₂²
hypercenter	C₄×C₄⋊C₈	D₅₃
hypocenter	C₅₃	#7027
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×C₄⋊C₈

338 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 77
transitive degree	μ_t(G) = 13568

A minimal faithful permutation representation — 2 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)(54,55,56,58,57,59,60,61)(63,66,68,67)(70,72)(71,74)(73,75)(76,77);(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)(62,63)(64,66)(65,68)(67,69)(70,71,73,76,74,72,77,75)

Polycyclic presentation:

2^8,53|a^2=d,b^2=e,c^2=f,d^2=g,e^2=h,b^a=bc,c^a=cf,e^a=efg,i^a=i^52,c^b=cg,i^b=i^52

Conjugacy classes & character table

2156 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 56125);