#27439001

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2015 ⋊ C₂
Sylow	C₅₃ ⋊ #25735

Subgroups

	subgroup	G/H
center	C₂³	C₂×D₄×D₅₃
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#2015	C₂
socle	C₂³×C₅₃	C₂²×D₄
prime radical	C₂²×#11719947	C₂²
hypercenter	#2015	D₅₃
hypocenter	C₅₃	#25735
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2015

1466 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)(58,60,61,59)(64,73)(68,81)(72,87)(74,86)(80,92)(82,89)(85,93)(88,91);(55,57)(62,65,76,83,67,78,66,77)(63,69,79,90,71,84,70,75)(64,72,86,91,73,87,74,88)(68,80,89,93,81,92,82,85);(54,55)(56,57)(58,59,61,60)(62,64)(63,68)(65,72)(66,74)(67,73)(69,80)(70,82)(71,81)(75,85)(76,86)(77,88)(78,87)(79,89)(83,91)(84,92)(90,93);(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)(59,60)(62,63,67,71)(64,68,73,81)(65,75,78,90)(66,79,76,70)(69,83,84,77)(72,85,87,93)(74,89,86,82)(80,91,92,88)

Polycyclic presentation:

2^8,53|a^2=h,b^2=e,c^2=fh,d^2=e,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,i^a=i^52,c^b=cg,d^b=dh,i^d=i^52

Conjugacy classes & character table

878 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 224477);