#27302810

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

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅₃×#510 ⋊ C₂
Sylow	C₅₃ ⋊ #13308

Subgroups

	subgroup	G/H
center	C₂²	#16810755
commutator, cosocle	C₂²×C₈×C₅₃	C₂³
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#510	C₂
socle	C₂²×C₅₃	#447
prime radical	#20284381	C₂
hypercenter	C₂×#510	D₅₃
hypocenter	C₅₃	#13308
LCS	G ▷ C₂²×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#510

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

normal subgroups only

This group has 532 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) = 6784

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)(59,65)(61,70)(62,72)(63,73)(64,66)(67,84)(68,85)(69,79)(71,76)(74,82)(75,77)(78,80)(81,89)(83,87)(86,88);(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)(58,60)(59,64)(61,75)(62,71)(63,73)(65,66)(67,81)(68,83)(69,74)(70,77)(72,76)(78,88)(79,82)(80,86)(84,89)(85,87);(58,59,63,68)(60,69,73,84)(61,74,78,89)(62,79,76,67)(64,80,83,70)(65,72,85,71)(66,86,87,77)(75,81,88,82)

Polycyclic presentation:

2^8,53|a^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,g^b=gh,e^c=eh,i^c=i^52

Conjugacy classes & character table

872 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 88286);