#27287349

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1304 ⋊ C₂
Sylow	C₅₃ ⋊ #10392

Subgroups

	subgroup	G/H
center	C₂²	#16810536
commutator, cosocle	C₂⁵×C₅₃	C₂³
Frattini	C₂⁵	C₂²×D₅₃
Fitting	C₅₃×#1304	C₂
socle	C₂²×C₅₃	#393
prime radical	C₂³×C₂²⋊D₅₃	C₂²
hypercenter	#1304	D₅₃
hypocenter	C₅₃	#10392
LCS	G ▷ C₂⁵×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ #1304

746 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 746 conjugacy classes of subgroups — too many to draw. Only its 73 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 77
transitive degree	μ_t(G) = 1696

A minimal faithful permutation representation — 3 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)(58,61)(62,65)(63,68)(64,70)(66,74)(67,75)(69,72)(71,73)(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)(54,56)(55,58)(57,60)(59,61)(62,64)(63,67)(65,73,74,70)(66,71)(68,75,72,77)(69,76);(54,55)(56,58)(57,59)(60,61)(62,63,66,69)(64,67,71,76)(65,72)(68,74)(70,77)(73,75)

Polycyclic presentation:

2^8,53|a^2=dg,b^2=d,b^a=bd,c^a=ce,e^a=eg,f^a=fgh,c^b=cf,e^b=eh,f^b=fg,i^b=i^52,d^c=dg,i^c=i^52

Conjugacy classes & character table

704 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 72825);