#27276624

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1556 ⋊ C₂
Sylow	C₅₃ ⋊ #8525

Subgroups

	subgroup	G/H
center	C₂²	#16810478
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1556	C₂
socle	C₂²×C₅₃	#379
prime radical	C₂³×D₅₃	C₂²⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1556	D₅₃
hypocenter	C₅₃	#8525
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1556

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

normal subgroups only

This group has 694 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) = 85
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 3 generators on 85 points:

(1,2)(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)(70,72)(71,76)(73,80)(74,81)(75,77)(78,79)(82,83)(84,85);(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)(54,56,58,64)(55,59,60,62)(57,65,66,63)(61,69,68,67)(70,72)(71,75)(73,82)(74,81)(76,77)(78,84)(79,85)(80,83);(54,55,57,61,58,60,66,68)(56,62,63,69,64,59,65,67)(70,71)(72,79,81,76)(73,77)(74,78)(75,80,84,82)(83,85)

Polycyclic presentation:

2^8,53|a^2=f,b^2=g,d^2=h,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,d^b=dh,f^b=fgh,i^b=i^52,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, 62100);