#27271036

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

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #7181

Subgroups

	subgroup	G/H
center	C₈	#13133138
commutator	C₈×C₅₃	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₅₃
Fitting	C₂×C₅₃×#364	C₂
socle	C₂×C₅₃	#1279
prime radical	C₂³×C₅₃	#104
cosocle	C₂×C₄×C₈×C₅₃	C₂²
hypercenter	C₂×#364	D₅₃
hypocenter	C₅₃	#7181
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₂×#364

240 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 generators on 85 points:

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

Polycyclic presentation:

2^8,53|a^2=d,b^2=e,c^2=fh,d^2=g,e^2=fg,f^2=h,g^2=h,b^a=bc,c^a=cf,e^a=ef,f^a=fh,c^b=ch,i^b=i^52

Conjugacy classes & character table

1520 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 56512);