#27393542

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1996 ⋊ C₂
Sylow	C₅₃ ⋊ #21827

Subgroups

	subgroup	G/H
center	C₂²	C₂×C₂²⋊C₄×D₅₃
commutator	C₂³×C₅₃	C₂³×C₄
Frattini	C₂⁴	C₂³×D₅₃
Fitting	C₅₃×#1996	C₂
socle	C₂²×C₅₃	C₂²×C₂²⋊C₄
prime radical	C₂⁴×D₅₃	C₂³
cosocle	C₂⁴×C₅₃	C₂⁴
hypercenter	#1996	D₅₃
hypocenter	C₅₃	#21827
LCS	G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #1996

1130 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)(55,61)(59,62)(60,69)(67,68)(72,79)(75,83)(78,85)(82,84);(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)(54,57,58,66)(55,60,61,69)(56,63,64,65)(59,67,62,68)(70,73)(71,76)(72,78)(74,81)(75,82)(77,80)(79,85)(83,84);(54,56,58,64)(55,59,61,62)(57,65,66,63)(60,68,69,67)(70,72)(71,75)(73,78)(74,79)(76,82)(77,83)(80,84)(81,85);(54,55)(56,62)(57,60)(58,61)(59,64)(63,68)(65,67)(66,69)(70,71)(72,75)(73,80,81,76)(74,77)(78,84,85,82)(79,83)

Polycyclic presentation:

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

Conjugacy classes & character table

878 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 179018);