#27466307

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Splittings

Fitting	C₈×C₂²⋊C₄×C₅₃ ⋊ C₂
Sylow	C₅₃ ⋊ #27696

Subgroups

	subgroup	G/H
center	C₂³	C₂²×D₂₁₂
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₈×C₂²⋊C₄×C₅₃	C₂
socle	C₂³×C₅₃	C₂²×D₄
prime radical	C₂×#13133191	C₂²
hypercenter	C₈×C₂²⋊C₄	D₅₃
hypocenter	C₅₃	#27696
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄

752 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 6784

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

Polycyclic presentation:

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

Conjugacy classes & character table

2138 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 251783);