#27288602

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #10586

Subgroups

	subgroup	G/H
center	C₂³	#13133185
commutator, cosocle	C₂×C₄²×C₅₃	C₂³
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1836	C₂
socle, prime radical	C₂³×C₅₃	#120
hypercenter	#1836	D₅₃
hypocenter	C₅₃	#10586
LCS	G ▷ C₂×C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1836

350 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 95 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) = 13568

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

Polycyclic presentation:

2^8,53|a^2=dg,b^2=d,c^2=dh,e^2=g,f^2=gh,b^a=bd,c^a=ce,e^a=eg,f^a=fh,c^b=cf,e^b=eh,f^b=fgh,i^b=i^52,e^c=eg,f^c=fgh

Conjugacy classes & character table

866 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 74078);