#27400052

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #22461

Subgroups

	subgroup	G/H
center	C₄	D₄×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting, prime radical	C₅₃×#3584	C₂
socle	C₂×C₅₃	D₄×C₂²⋊C₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#3584	D₅₃
hypocenter	C₅₃	#22461
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #3584

1316 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 1696

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

Polycyclic presentation:

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

Conjugacy classes & character table

1196 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 185528);