#27544502

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅₃×#519 ⋊ C₂
Sylow	C₅₃ ⋊ #34211

Subgroups

	subgroup	G/H
center, Frattini	C₂³	C₂⁴×D₅₃
commutator	C₂²×C₅₃	C₂⁴×C₄
Fitting	C₂×C₅₃×#519	C₂
socle, cosocle	C₂³×C₅₃	C₂⁵
prime radical	C₂³×D₂₁₂	C₂²
hypercenter	C₂×#519	D₅₃
hypocenter	C₅₃	#34211
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×#519

3102 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 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)(62,66,67,65)(63,70,69,75)(64,72,73,74)(68,77,71,76);(62,65,67,66)(63,70,69,75)(64,72,73,74)(68,76,71,77);(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)(58,59,60,61)(63,69)(68,71)(70,75)(76,77);(54,56,57,55)(62,64)(63,68)(65,74)(66,72)(67,73)(69,71)(70,77)(75,76);(54,55,57,56)(59,61)(62,63)(64,71)(65,70)(66,75)(67,69)(68,73)(72,77)(74,76)

Polycyclic presentation:

2^8,53|a^2=h,b^2=h,c^2=fg,d^2=f,e^2=f,b^a=bf,c^a=cg,e^a=ef,d^b=df,i^c=i^52

Conjugacy classes & character table

1856 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 329978);