#27302886

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Fitting	C₂²×SD₃₂×C₅₃ ⋊ C₂
Sylow	C₅₃ ⋊ #13320

Subgroups

	subgroup	G/H
center	C₂²	#16810755
commutator, cosocle	C₂²×C₈×C₅₃	C₂³
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂²×SD₃₂×C₅₃	C₂
socle	C₂²×C₅₃	#447
prime radical	C₂×#16810890	C₂
hypercenter	C₂²×SD₃₂	D₅₃
hypocenter	C₅₃	#13320
LCS	G ▷ C₂²×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×SD₃₂

540 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 99 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) = 89
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 3 generators on 89 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)(55,57)(58,60)(59,66)(61,75)(62,78)(63,70)(64,72)(65,79)(67,86)(68,84)(69,85)(71,77)(73,89)(74,82)(76,87)(80,88)(81,83);(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)(54,55)(56,57)(58,59,64,69)(60,66,72,85)(61,74,77,80)(62,68,63,67)(65,76,81,73)(70,86,78,84)(71,88,75,82)(79,87,83,89);(59,65)(60,70)(61,73)(62,63)(66,82)(67,74)(68,80)(69,81)(71,75)(72,78)(76,77)(79,84)(83,86)(85,88)

Polycyclic presentation:

2^8,53|b^2=h,d^2=gh,e^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gh,c^b=cf,d^b=dg,e^b=eh,g^b=gh,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, 88362);