#27289064

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1570 ⋊ C₂
Sylow	C₅₃ ⋊ #10659

Subgroups

	subgroup	G/H
center	C₂×C₄	#13133213
commutator, cosocle	C₂×C₄²×C₅₃	C₂³
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1570	C₂
socle	C₂²×C₅₃	#399
prime radical	C₂²×D₂₁₂	Q₈
hypercenter	#1570	D₅₃
hypocenter	C₅₃	#10659
LCS	G ▷ C₂×C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1570

356 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 89 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)(55,60)(56,57)(59,61)(63,68)(69,75);(55,59)(56,57)(60,61)(62,64,71,72,66,73,65,74)(63,67,75,70,68,76,69,77);(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)(54,55,56,60,58,61,57,59)(62,63,65,69,66,68,71,75)(64,70,72,76,73,77,74,67)

Polycyclic presentation:

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

Conjugacy classes & character table

1028 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 74540);