#27293341

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1326 ⋊ C₂
Sylow	C₅₃ ⋊ #11471

Subgroups

	subgroup	G/H
center	C₂²	#99×D₅₃
commutator	C₂³×C₅₃	C₂²×C₈
Frattini	C₂×C₂²⋊C₄	C₂²×D₅₃
Fitting	C₅₃×#1326	C₂
socle	C₂²×C₅₃	C₂×#99
prime radical	C₂×#13133187	C₄
cosocle	C₂×C₂²⋊C₄×C₅₃	C₂³
hypercenter	#1326	D₅₃
hypocenter	C₅₃	#11471
LCS	G ▷ C₂³×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂×C₂²⋊C₄ ◁ #1326

756 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 756 conjugacy classes of subgroups — too many to draw. Only its 89 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) = 69
transitive degree	μ_t(G) = 1696

A minimal faithful permutation representation — 3 generators on 69 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)(54,56,58,57)(55,59,61,60);(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)(63,66)(67,68);(54,55,57,60,58,61,56,59)(62,63)(64,68,69,66)(65,67)

Polycyclic presentation:

2^8,53|a^2=e,c^2=g,e^2=g,b^a=bd,d^a=df,f^a=fh,e^b=ef,i^b=i^52,i^c=i^52,e^d=eh

Conjugacy classes & character table

728 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 78817);