#27303167

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1374 ⋊ C₂
Sylow	C₅₃ ⋊ #13365

Subgroups

	subgroup	G/H
center	C₂³	C₂×D₄₂₄
commutator, cosocle	C₂²×C₈×C₅₃	C₂³
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#1374	C₂
socle	C₂³×C₅₃	C₂×D₈
prime radical	C₂²×D₄₂₄	C₂²
hypercenter	#1374	D₅₃
hypocenter	C₅₃	#13365
LCS	G ▷ C₂²×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1374

418 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 113 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) = 81
transitive degree	μ_t(G) = 6784

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

Polycyclic presentation:

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

Conjugacy classes & character table

2132 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 88643);