#27287697

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

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #10446

Subgroups

	subgroup	G/H
center	C₂²	#16810537
commutator, cosocle	C₂×C₄²×C₅₃	C₂³
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂×C₅₃×#462	C₂
socle	C₂²×C₅₃	#394
prime radical	C₂³×C₅₃	#123
hypercenter	C₂×#462	D₅₃
hypocenter	C₅₃	#10446
LCS	G ▷ C₂×C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#462

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

normal subgroups only

This group has 408 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) = 93
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 3 generators on 93 points:

(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)(56,61)(59,60)(62,65,66,78,67,79,82,93)(63,69,70,86,71,76,80,91)(64,73,74,90,75,77,81,92)(68,83,72,88,84,85,87,89);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(62,64,67,75)(63,68,71,84)(65,77,79,73)(66,81,82,74)(69,85,76,83)(70,87,80,72)(78,90,93,92)(86,88,91,89);(54,55)(56,58)(57,60)(59,61)(62,63,67,71)(64,72,75,87)(65,76,79,69)(66,80,82,70)(68,81,84,74)(73,89,77,88)(78,86,93,91)(83,90,85,92)

Polycyclic presentation:

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

Conjugacy classes & character table

1025 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 73173);