#27304617

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃₂ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×C₄⋊C₃₂ ⋊ C₂
Sylow	C₅₃ ⋊ #13639

Subgroups

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂²×D₅₃
commutator	C₂×C₅₃	C₂²×C₃₂
Fitting	C₅₃×C₄⋊C₃₂	C₂
socle	C₂²×C₅₃	C₂²×C₁₆
prime radical	C₂²×D₅₃	C₂×C₁₆
cosocle	C₂×C₁₆×C₅₃	C₂³
hypercenter	C₄⋊C₃₂	D₅₃
hypocenter	C₅₃	#13639
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₁₆ ◁ C₄⋊C₃₂

202 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 202 conjugacy classes of subgroups — too many to draw. Only its 101 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) = 6784

A minimal faithful permutation representation — 3 generators on 93 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)(54,56,57,60)(55,59,58,61);(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)(55,58)(59,61);(54,55)(56,59)(57,58)(60,61)(62,63,64,68,65,69,72,78,66,70,73,79,75,81,84,88,67,71,74,80,76,82,85,89,77,83,86,90,87,91,92,93)

Polycyclic presentation:

2^8,53|a^2=e,c^2=d,e^2=f,f^2=g,g^2=h,b^a=bd,i^b=i^52,i^c=i^52

Conjugacy classes & character table

2240 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 90093);