#27487261

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #29438

Subgroups

	subgroup	G/H
center	C₂³	C₂²×C₂²⋊D₅₃
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#2320	C₂
socle, prime radical	C₂³×C₅₃	C₂²×D₄
hypercenter	#2320	D₅₃
hypocenter	C₅₃	#29438
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ #376 ◁ #2320

626 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 13568

A minimal faithful permutation representation — 4 generators on 85 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

1190 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 272737);