#27375407

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1589 ⋊ C₂
Sylow	C₅₃ ⋊ #19887

Subgroups

	subgroup	G/H
center	C₂²	D₄×D₂₁₂
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1589	C₂
socle	C₂²×C₅₃	D₄²
prime radical	#20287064	C₂
hypercenter	#1589	D₅₃
hypocenter	C₅₃	#19887
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1589

690 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 77 points:

(1,2)(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)(62,66)(63,65)(64,69)(67,71)(68,70)(72,73)(74,77)(75,76);(55,58)(56,61)(59,60)(62,65,68,67)(63,70,71,66)(64,73,75,74)(69,72,76,77);(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)(62,64,68,75)(63,69,71,76)(65,74,67,73)(66,72,70,77);(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)(54,55)(56,58)(57,60)(59,61)(62,63,68,71)(64,72,75,77)(65,70,67,66)(69,73,76,74)

Polycyclic presentation:

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

Conjugacy classes & character table

1187 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 160883);