#27468201

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1953 ⋊ C₂
Sylow	C₅₃ ⋊ #27834

Subgroups

	subgroup	G/H
center	C₄	#16811337
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1953	C₂
socle	C₂×C₅₃	#2495
prime radical	C₂×#11719945	C₂³
hypercenter	#1953	D₅₃
hypocenter	C₅₃	#27834
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #1953

438 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)(54,58,59,70,60,77,61,56)(55,64,65,62,67,68,66,79)(57,73,74,84,76,72,75,69)(63,81,82,78,83,80,71,85);(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)(54,57)(55,63)(56,69)(58,73)(59,74)(60,76)(61,75)(62,78)(64,81)(65,82)(66,71)(67,83)(68,80)(70,84)(72,77)(79,85);(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)(54,56,59,58,60,70,61,77)(55,62,65,68,67,79,66,64)(57,72,75,84,76,73,74,69)(63,80,71,78,83,81,82,85);(54,55,61,65,60,67,59,66)(56,68,77,79,70,64,58,62)(57,71,75,63,76,82,74,83)(69,78,72,80,84,85,73,81)

Polycyclic presentation:

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

Conjugacy classes & character table

872 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 253677);