#27532813

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #32891

Subgroups

	subgroup	G/H
center	C₂²	C₂×#13133186
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting, prime radical	C₂²×H(5,𝔽₂)×C₅₃	C₂
socle	C₂²×C₅₃	C₂×#120
hypercenter	C₂²×H(5,𝔽₂)	D₅₃
hypocenter	C₅₃	#32891
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×H(5,𝔽₂)

3940 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 73
transitive degree	μ_t(G) = 1696

A minimal faithful permutation representation — 4 generators on 73 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)(55,57);(61,71)(64,69)(66,73)(67,70);(58,60)(59,63)(61,70)(62,68)(64,73)(65,72)(66,69)(67,71);(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)(56,57)(58,59,62,65)(60,66,68,73)(61,69,71,64)(63,70,72,67)

Polycyclic presentation:

2^8,53|a^2=h,e^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,i^a=i^52,e^b=eh,f^b=fh,e^c=eh

Conjugacy classes & character table

1817 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 318289);