#27302282

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₄×M₃₂×C₅₃ ⋊ C₂
Sylow	C₅₃ ⋊ #13197

Subgroups

	subgroup	G/H
center	C₈	#13133191
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₄×M₃₂×C₅₃	C₂
socle	C₂×C₅₃	#1954
prime radical	C₂×#13133239	C₄
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₄×M₃₂	D₅₃
hypocenter	C₅₃	#13197
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₄×M₃₂

424 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

Polycyclic presentation:

2^8,53|a^2=f,b^2=df,c^2=e,d^2=h,e^2=h,f^2=g,g^2=h,b^a=bd,c^a=ce,e^a=eh,i^a=i^52,d^b=dh,f^b=fh

Conjugacy classes & character table

2144 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 87758);