#27304191

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1376 ⋊ C₂
Sylow	C₅₃ ⋊ #13533

Subgroups

	subgroup	G/H
center	C₂²	#16810914
commutator, cosocle	C₂²×C₈×C₅₃	C₂³
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#1376	C₂
socle	C₂²×C₅₃	#476
prime radical	C₂×#13133239	C₂²
hypercenter	#1376	D₅₃
hypocenter	C₅₃	#13533
LCS	G ▷ C₂²×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1376

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

normal subgroups only

This group has 346 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) = 73
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 3 generators on 73 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)(55,57)(58,59)(60,62)(61,69)(63,73)(64,67)(65,71)(66,68)(70,72);(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)(58,60,68,69,64,70,63,65)(59,62,66,61,67,72,73,71);(58,59,62,69,68,73,61,60,64,67,72,65,63,66,71,70)

Polycyclic presentation:

2^8,53|a^2=dg,b^2=deg,d^2=g,e^2=gh,g^2=h,b^a=bd,c^a=ce,e^a=eh,c^b=cf,d^b=dg,e^b=eg,g^b=gh,i^b=i^52,d^c=dg,e^c=eg,g^c=gh,i^c=i^52

Conjugacy classes & character table

1496 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 89667);