#27284395

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

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #9929

Subgroups

	subgroup	G/H
center	C₂³	#13133187
commutator, cosocle	C₂³×C₄×C₅₃	C₂³
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₂×C₅₃×#324	C₂
socle	C₂³×C₅₃	#120
prime radical	C₂⁴×C₅₃	C₂×D₄
hypercenter	C₂×#324	D₅₃
hypocenter	C₅₃	#9929
LCS	G ▷ C₂³×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#324

610 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 89
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 3 generators on 89 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)(58,61)(59,64)(60,66)(62,68)(63,69)(65,72)(67,73)(70,71)(74,77,79,78)(75,81,82,86)(76,84,85,87)(80,88,89,83);(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)(58,60,62,67)(59,63,65,70)(61,66,68,73)(64,71,72,69)(74,76,79,85)(75,80,82,89)(77,84,78,87)(81,88,86,83);(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)(56,57)(58,59)(60,63)(61,64)(62,65)(66,69)(67,70)(68,72)(71,73)(74,75,79,82)(76,83,85,88)(77,81,78,86)(80,87,89,84)

Polycyclic presentation:

2^8,53|a^2=g,b^2=gh,c^2=g,d^2=g,b^a=bd,c^a=ce,d^a=dg,f^a=fh,i^a=i^52,c^b=cf,d^b=dg,i^b=i^52,d^c=dh

Conjugacy classes & character table

1184 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 69871);