#27300311

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1377 ⋊ C₂
Sylow	C₅₃ ⋊ #12847

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²⋊C₄×D₅₃
commutator	C₂³×C₅₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#1377	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₈
prime radical	C₂³×D₅₃	C₂×C₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1377	D₅₃
hypocenter	C₅₃	#12847
LCS	G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #1377

434 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 123 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) = 3392

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

(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)(54,56,57,60)(55,59,58,61)(63,69)(64,72)(65,75)(71,82)(74,85)(76,87)(79,88)(90,93);(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)(59,61)(62,64)(63,68)(65,73)(66,74)(67,72)(69,78)(70,79)(71,80)(75,84)(76,86)(77,85)(81,88)(82,89)(83,90)(87,92)(91,93);(54,55)(56,59)(57,58)(60,61)(62,63,65,70,66,71,76,83,67,69,75,81,77,82,87,91)(64,68,73,79,74,80,86,90,72,78,84,88,85,89,92,93)

Polycyclic presentation:

2^8,53|a^2=f,c^2=d,f^2=g,g^2=h,b^a=bd,c^a=ce,e^a=eh,c^b=ch,i^b=i^52,f^c=fh

Conjugacy classes & character table

1208 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 85787);