#27411520

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2031 ⋊ C₂
Sylow	C₅₃ ⋊ #23472

Subgroups

	subgroup	G/H
center	C₄	#124×D₅₃
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#2031	C₂
socle	C₂×C₅₃	C₂×#393
prime radical	C₂×#41×D₅₃	C₂²
hypercenter	#2031	D₅₃
hypocenter	C₅₃	#23472
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #2031

1148 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

722 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 196996);