#27417958

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1899 ⋊ C₂
Sylow	C₅₃ ⋊ #24113

Subgroups

	subgroup	G/H
center	C₂²	C₂×#13133187
commutator, cosocle	C₂⁴×C₅₃	C₂⁴
Frattini	C₂⁴	C₂³×D₅₃
Fitting	C₅₃×#1899	C₂
socle	C₂²×C₅₃	C₂×#120
prime radical	#16811182	C₂²
hypercenter	#1899	D₅₃
hypocenter	C₅₃	#24113
LCS	G ▷ C₂⁴×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #1899

1122 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)(55,60)(59,61)(62,65,66,72)(63,68,69,75)(64,70,71,76)(67,74,73,77);(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)(54,57,58,56)(55,59,60,61)(67,73)(74,77);(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)(54,56,58,57)(55,59,60,61)(62,64)(63,67)(65,70)(66,71)(68,74)(69,73)(72,76)(75,77);(54,55)(56,61)(57,59)(58,60)(62,63)(64,67)(65,68)(66,69)(70,74)(71,73)(72,75)(76,77)

Polycyclic presentation:

2^8,53|b^2=e,c^2=e,d^2=h,b^a=be,c^a=cf,d^a=de,g^a=gh,c^b=cg,f^b=fh,i^b=i^52,i^c=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, 203434);