#27597146

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2734 ⋊ C₂
Sylow	C₅₃ ⋊ #36938

Subgroups

	subgroup	G/H
center, Frattini	C₂³	C₂⁴×D₅₃
commutator, socle, cosocle	C₂³×C₅₃	C₂⁵
Fitting	C₅₃×#2734	C₂
prime radical	C₂²×D₂₁₂	C₂³
hypercenter	#2734	D₅₃
hypocenter	C₅₃	#36938
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ #2734

1570 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 77
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 5 generators on 77 points:

(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)(55,59)(58,61)(62,65,66,64)(63,67,68,69)(70,73,74,72)(71,77,75,76);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(56,60)(58,61)(70,72,74,73)(71,77,75,76);(56,60)(58,61)(62,64,66,65)(63,67,68,69)(70,72,74,73)(71,76,75,77);(54,56)(55,58)(57,60)(59,61)(70,71,74,75)(72,76,73,77);(54,55)(56,58)(57,59)(60,61)(62,63,66,68)(64,69,65,67)(71,75)(72,73)

Polycyclic presentation:

2^8,53|a^2=fg,b^2=f,c^2=g,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=df,e^a=efh,c^b=ch,d^b=dfh,i^e=i^52

Conjugacy classes & character table

884 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 382622);