#27270929

order	13568
automorphism	?
homology	Abelianization (H₁) = C₈² · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #7140

Subgroups

	subgroup	G/H
center	C₂×C₈	#11719921
commutator	C₂²×C₅₃	C₈²
Frattini	C₂×C₄×C₈	C₂×D₅₃
Fitting	C₄×M₃₂×C₅₃	C₂
socle	C₂²×C₅₃	#336
prime radical	C₂³×C₅₃	C₄×C₈
cosocle	C₂×C₄×C₈×C₅₃	C₂²
hypercenter	C₄×M₃₂	D₅₃
hypocenter	C₅₃	#7140
LCS	G ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

246 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 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)(54,55,56,58,57,59,60,61)(62,64)(63,68,70,72)(65,74)(66,75)(67,73)(69,80,82,85)(71,81,84,86)(76,88)(77,79)(78,87)(83,91,92,93)(89,90);(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)(62,63,65,69,66,71,76,83,67,70,77,82,78,84,89,92)(64,72,74,85,75,86,88,93,73,68,79,80,87,81,90,91)

Polycyclic presentation:

2^8,53|a^2=d,b^2=e,d^2=f,e^2=g,f^2=h,b^a=bc,e^a=eh,i^a=i^52,c^b=ch,i^b=i^52

Conjugacy classes & character table

2168 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 56405);