#27303168

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

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #13366

Subgroups

	subgroup	G/H
center	C₂³	C₂×#11719922
commutator, cosocle	C₂²×C₈×C₅₃	C₂³
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#499	C₂
socle, prime radical	C₂³×C₅₃	C₂×D₈
hypercenter	C₂×#499	D₅₃
hypocenter	C₅₃	#13366
LCS	G ▷ C₂²×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#499

382 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 117 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) = 97
transitive degree	μ_t(G) = 13568

A minimal faithful permutation representation — 3 generators on 97 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)(66,68,69,77)(67,73,72,87)(70,83,80,97)(71,86,81,79)(74,92,88,96)(75,93,85,90)(76,95,94,82)(78,91,89,84);(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)(58,60,61,65)(59,62,63,64)(67,72)(73,87)(74,88)(75,85)(76,94)(82,95)(90,93)(92,96);(58,59,61,63)(60,64,65,62)(66,67,69,72)(68,76,77,94)(70,82,80,95)(71,85,81,75)(73,89,87,78)(74,91,88,84)(79,96,86,92)(83,93,97,90)

Polycyclic presentation:

2^8,53|a^2=d,b^2=dh,c^2=h,e^2=gh,g^2=h,b^a=bd,c^a=ce,e^a=eg,g^a=gh,c^b=cf,e^c=eg,g^c=gh,i^c=i^52

Conjugacy classes & character table

1196 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 88644);