#27287248

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #10375

Subgroups

	subgroup	G/H
center	C₂²	#16810536
commutator, cosocle	C₂³×C₄×C₅₃	C₂³
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₅₃×#1282	C₂
socle	C₂²×C₅₃	#393
prime radical	C₂⁴×C₅₃	C₂×D₄
hypercenter	#1282	D₅₃
hypocenter	C₅₃	#10375
LCS	G ▷ C₂³×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #1282

466 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 97 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)(54,55,57,59)(56,61,60,58)(62,65,66,74)(63,68,69,72)(64,70,71,73)(67,75,76,77);(62,64)(63,67)(65,73,74,70)(66,71)(68,75,72,77)(69,76);(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)(55,58)(56,60)(59,61)(62,63)(64,67)(65,72,74,68)(66,69)(70,77,73,75)(71,76)

Polycyclic presentation:

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

Conjugacy classes & character table

1181 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 72724);