#27477117

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2090 ⋊ C₂
Sylow	C₅₃ ⋊ #28594

Subgroups

	subgroup	G/H
center	C₂²	#121×D₅₃
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#2090	C₂
socle	C₂²×C₅₃	C₂×#121
prime radical	#16811135	C₂²
hypercenter	#2090	D₅₃
hypocenter	C₅₃	#28594
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #2090

780 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 93
transitive degree	μ_t(G) = 3392

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

Polycyclic presentation:

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

Conjugacy classes & character table

716 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 262593);