#27304268

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₁₆ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1397 ⋊ C₂
Sylow	C₅₃ ⋊ C₄×M₆₄

Subgroups

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂²×D₅₃
commutator	C₂×C₅₃	C₂×C₄×C₁₆
Fitting	C₅₃×#1397	C₂
socle	C₂²×C₅₃	C₂²×C₁₆
prime radical	C₂²×D₅₃	C₂×C₁₆
cosocle	C₂×C₁₆×C₅₃	C₂³
hypercenter	#1397	D₅₃
hypocenter	C₅₃	C₄×M₆₄
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₁₆ ◁ #1397

250 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 147 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) = 89
transitive degree	μ_t(G) = 6784

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

Polycyclic presentation:

2^8,53|a^2=d,b^2=e,d^2=f,f^2=g,g^2=h,c^a=ch,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, 89744);