#27357265

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #18225

Subgroups

	subgroup	G/H
center	C₂²	C₂×#13133186
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting, prime radical	C₅₃×#3534	C₂
socle	C₂²×C₅₃	C₂×#120
hypercenter	#3534	D₅₃
hypocenter	C₅₃	#18225
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #3534

1194 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)(64,71)(68,75)(72,77)(74,76);(54,56)(55,58)(57,60)(59,61)(62,64)(63,68)(65,72)(66,71)(67,74)(69,76)(70,75)(73,77);(54,55)(56,58)(57,59)(60,61)(63,67)(65,73)(68,74)(69,70)(72,77)(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)(58,61)(62,63,66,70)(64,68,71,75)(65,67,73,69)(72,74,77,76)

Polycyclic presentation:

2^8,53|a^2=g,e^2=g,b^a=be,c^a=cf,e^a=eg,i^a=i^52,e^b=eg,f^b=fh,d^c=dg,e^c=eh

Conjugacy classes & character table

1025 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 142741);