#27276496

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1496 ⋊ C₂
Sylow	C₅₃ ⋊ #8500

Subgroups

	subgroup	G/H
center	C₂²	#16810478
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1496	C₂
socle	C₂²×C₅₃	#379
prime radical	#20289081	C₂
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1496	D₅₃
hypocenter	C₅₃	#8500
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1496

736 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 736 conjugacy classes of subgroups — too many to draw. Only its 103 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) = 85
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 3 generators on 85 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)(55,61)(57,58)(60,64)(62,63)(65,66)(68,69)(70,72)(71,76)(73,82)(74,81)(75,78)(77,84)(79,85)(80,83);(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)(55,60)(57,66)(58,65)(59,67)(61,64)(62,69)(63,68)(70,72)(71,75)(73,80)(74,81)(76,78)(77,79)(82,83)(84,85);(54,55,57,61,59,63,58,62)(56,64,66,68,67,69,65,60)(70,71)(72,79,81,75)(73,78)(74,77)(76,80,84,82)(83,85)

Polycyclic presentation:

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

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, 61972);