#27393359

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1782 ⋊ C₂
Sylow	C₅₃ ⋊ #21796

Subgroups

	subgroup	G/H
center	C₂²	C₂×C₂²⋊C₄×D₅₃
commutator	C₂³×C₅₃	C₂³×C₄
Frattini	C₂⁴	C₂³×D₅₃
Fitting	C₅₃×#1782	C₂
socle	C₂²×C₅₃	C₂²×C₂²⋊C₄
prime radical	C₂×#16811201	C₂
cosocle	C₂⁴×C₅₃	C₂⁴
hypercenter	#1782	D₅₃
hypocenter	C₅₃	#21796
LCS	G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #1782

1334 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 77
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 4 generators on 77 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)(55,59)(58,60)(62,65,66,73)(63,68,69,75)(64,71,72,77)(67,74,70,76);(56,61)(58,60)(62,64)(63,67)(65,71)(66,72)(68,74)(69,70)(73,77)(75,76);(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)(54,56,57,61)(55,58,59,60)(65,73)(68,75)(71,77)(74,76);(54,55)(56,60)(57,59)(58,61)(62,63)(64,70,72,67)(65,68)(66,69)(71,76,77,74)(73,75)

Polycyclic presentation:

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

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, 178835);