#27466060

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2079 ⋊ C₂
Sylow	C₅₃ ⋊ #27674

Subgroups

	subgroup	G/H
center	C₂²	#121×D₅₃
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#2079	C₂
socle	C₂²×C₅₃	C₂×#121
prime radical	#16811130	C₂²
hypercenter	#2079	D₅₃
hypocenter	C₅₃	#27674
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #2079

696 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 93
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 4 generators on 93 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)(62,66,68,78,69,77,67,64)(63,72,74,70,75,86,73,83)(65,80,84,92,82,91,81,76)(71,87,89,85,88,93,79,90);(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)(56,60)(58,61)(62,65)(63,71)(64,76)(66,80)(67,81)(68,84)(69,82)(70,85)(72,87)(73,79)(74,89)(75,88)(77,91)(78,92)(83,90)(86,93);(54,56)(55,58)(57,60)(59,61)(62,64,67,77,69,78,68,66)(63,70,73,72,75,83,74,86)(65,76,81,91,82,92,84,80)(71,85,79,87,88,90,89,93);(54,55,57,59)(56,58,60,61)(62,63)(64,72)(65,79)(66,83)(67,74)(68,73)(69,75)(70,77)(71,81)(76,90)(78,86)(80,93)(82,89)(84,88)(85,92)(87,91)

Polycyclic presentation:

2^8,53|a^2=g,b^2=e,d^2=eh,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,c^b=cg,f^c=fh,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, 251536);