#27275138

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1746 ⋊ C₂
Sylow	C₅₃ ⋊ #8248

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#11719921
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1746	C₂
socle	C₂²×C₅₃	C₄×C₂²⋊C₄
prime radical	#13133191	C₂×C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1746	D₅₃
hypocenter	C₅₃	#8248
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #1746

392 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 3 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)(56,63)(59,67)(62,68)(64,69)(70,73,74,81,75,82,79,85)(71,76,78,84,72,80,77,83);(54,56)(55,59,61,67)(57,64)(58,63)(60,62,65,68)(66,69)(70,72)(71,75)(73,76)(74,77)(78,79)(80,82)(81,84)(83,85);(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)(54,55)(56,62)(57,65)(58,61)(59,64)(60,66)(63,68)(67,69)(70,71,74,78,75,72,79,77)(73,76,81,84,82,80,85,83)

Polycyclic presentation:

2^8,53|a^2=e,b^2=f,c^2=e,d^2=g,e^2=h,b^a=bd,d^a=dg,f^a=fgh,i^a=i^52,c^b=cg,d^b=dh

Conjugacy classes & character table

1508 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 60614);