#27466266

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2103 ⋊ C₂
Sylow	C₅₃ ⋊ #27692

Subgroups

	subgroup	G/H
center	C₄	#16811191
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#2103	C₂
socle	C₂×C₅₃	#2487
prime radical	#16811135	C₂²
hypercenter	#2103	D₅₃
hypocenter	C₅₃	#27692
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #2103

626 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 1696

A minimal faithful permutation representation — 4 generators on 85 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)(54,57,58,71,59,70,60,74)(55,62,63,69,64,77,72,83)(56,67,68,82,66,80,73,84)(61,75,65,78,76,81,79,85);(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)(56,66)(57,70)(62,77)(68,73)(69,83)(71,74)(75,81)(78,85);(54,56,58,68,59,66,60,73)(55,61,63,65,64,76,72,79)(57,67,71,82,70,80,74,84)(62,75,69,78,77,81,83,85);(54,55,60,63,59,64,58,72)(56,65,73,76,66,79,68,61)(57,69,74,77,70,83,71,62)(67,81,84,85,80,75,82,78)

Polycyclic presentation:

2^8,53|a^2=eg,b^2=e,d^2=e,e^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,g^a=gh,c^b=cg,f^b=fh,d^c=dh,i^c=i^52,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, 251742);