#27279305

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1283 ⋊ C₂
Sylow	C₅₃ ⋊ #9052

Subgroups

	subgroup	G/H
center	C₂×C₄	#13133187
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1283	C₂
socle	C₂²×C₅₃	#386
prime radical	#16810759	C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1283	D₅₃
hypocenter	C₅₃	#9052
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1283

488 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 93 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) = 3392

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

Polycyclic presentation:

2^8,53|a^2=f,b^2=dfg,d^2=g,e^2=g,f^2=gh,b^a=bd,c^a=ce,e^a=egh,d^b=dg,e^b=eh,f^b=fg,i^b=i^52,d^c=dh,e^c=eg,f^c=fh,i^c=i^52

Conjugacy classes & character table

1190 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 64781);