#27400033

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

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #22459

Subgroups

	subgroup	G/H
center	C₂	D₄×C₂²⋊Dic₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting, prime radical	C₂×C₅₃×#577	C₂
socle	C₂×C₅₃	D₄×C₂²⋊C₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂×#577	D₅₃
hypocenter	C₅₃	#22459
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#577

1312 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)(54,57)(55,63)(56,68)(58,75)(59,71)(60,72)(61,77)(62,65)(64,80)(66,76)(67,74)(69,85)(70,84)(73,81)(78,83)(79,82);(54,57)(55,62)(56,67)(58,71)(59,75)(60,72)(61,76)(63,65)(64,80)(66,77)(68,74)(69,84)(70,85)(73,81)(78,82)(79,83);(54,56,60,70)(55,61,65,78)(57,67,72,85)(58,74,59,69)(62,76,63,82)(64,79,73,66)(68,75,84,71)(77,80,83,81);(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)(54,55,59,64,60,65,58,73)(56,66,69,61,70,79,74,78)(57,63,71,80,72,62,75,81)(67,83,68,76,85,77,84,82)

Polycyclic presentation:

2^8,53|a^2=g,b^2=h,e^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,i^a=i^52,d^b=de,e^b=eh,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, 185509);