#27289079

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #10662

Subgroups

	subgroup	G/H
center	C₂×C₄	#13133214
commutator, cosocle	C₂×C₄²×C₅₃	C₂³
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1275	C₂
socle	C₂²×C₅₃	#399
prime radical	C₂³×C₅₃	C₂²⋊Q₈
hypercenter	#1275	D₅₃
hypocenter	C₅₃	#10662
LCS	G ▷ C₂×C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1275

320 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 101 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) = 6784

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

Polycyclic presentation:

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

Conjugacy classes & character table

1184 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 74555);