#27557236

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #35018

Subgroups

	subgroup	G/H
center, Frattini	C₂³	C₂⁴×D₅₃
commutator, socle, cosocle	C₂³×C₅₃	C₂⁵
Fitting	C₅₃×#2449	C₂
prime radical	C₂²×D₄×C₅₃	C₂³
hypercenter	#2449	D₅₃
hypocenter	C₅₃	#35018
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ #2449

1732 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 77
transitive degree	μ_t(G) = 6784

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

Polycyclic presentation:

2^8,53|a^2=g,c^2=fg,e^2=f,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh,i^e=i^52

Conjugacy classes & character table

1052 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 342712);