#27299823

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #12749

Subgroups

	subgroup	G/H
center	C₂²	#16810757
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting, prime radical	C₂×C₅₃×#578	C₂
socle	C₂²×C₅₃	#447
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×#578	D₅₃
hypocenter	C₅₃	#12749
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#131 ◁ C₂×#578

882 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 89
transitive degree	μ_t(G) = 6784

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)(58,61,65,59)(60,71,75,66)(62,76,81,67)(63,68,64,69)(70,72,87,89)(73,82,74,83)(77,79,85,80)(78,86,88,84);(58,60)(59,66)(61,71)(62,78)(63,73)(64,74)(65,75)(67,84)(68,82)(69,83)(70,77)(72,79)(76,86)(80,89)(81,88)(85,87);(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)(58,59,64,69,65,61,63,68)(60,70,74,86,75,87,73,84)(62,77,80,76,81,85,79,67)(66,78,83,89,71,88,82,72)

Polycyclic presentation:

2^8,53|a^2=f,c^2=h,d^2=gh,e^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh,i^a=i^52,d^b=dg,g^b=gh,e^c=eh,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, 85299);