#27276618

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1533 ⋊ C₂
Sylow	C₅₃ ⋊ #8524

Subgroups

	subgroup	G/H
center	C₂²	#16810478
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1533	C₂
socle	C₂²×C₅₃	#379
prime radical	C₂²×#11719947	C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1533	D₅₃
hypocenter	C₅₃	#8524
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1533

910 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 910 conjugacy classes of subgroups — too many to draw. Only its 111 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

710 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 62094);