#27287785

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

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅₃×Q₈⋊C₈ ⋊ C₂
Sylow	C₅₃ ⋊ #10458

Subgroups

	subgroup	G/H
center	C₂²	#16810539
commutator, cosocle	C₂×C₄²×C₅₃	C₂³
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂×C₅₃×Q₈⋊C₈	C₂
socle	C₂²×C₅₃	#394
prime radical	#16811173	C₂²
hypercenter	C₂×Q₈⋊C₈	D₅₃
hypocenter	C₅₃	#10458
LCS	G ▷ C₂×C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×Q₈⋊C₈

428 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 99 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) = 77
transitive degree	μ_t(G) = 6784

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

Polycyclic presentation:

2^8,53|b^2=g,c^2=d,d^2=g,e^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fg,i^a=i^52,c^b=cf,d^b=dg,e^b=eg,e^c=egh,f^c=fg,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, 73261);