#27303121

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅₃×#355 ⋊ C₂
Sylow	C₅₃ ⋊ #13357

Subgroups

	subgroup	G/H
center	C₂²	#16810811
commutator, cosocle	C₂²×Q₈×C₅₃	C₂³
Frattini	C₂²×Q₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#355	C₂
socle	C₂²×C₅₃	B(4,𝔽₂)
hypercenter	C₂×#355	D₅₃
hypocenter	C₅₃	#13357
LCS	G ▷ C₂²×Q₈×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×Q₈ ◁ C₂×#355

772 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 772 conjugacy classes of subgroups — too many to draw. Only its 79 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) = 1696

A minimal faithful permutation representation — 3 generators on 73 points:

(2,4)(6,11)(8,17)(10,19)(12,13)(15,20)(16,18)(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)(61,62)(63,64)(65,66)(67,68)(69,70)(71,72);(5,6)(7,15)(8,11)(9,13)(10,14)(12,17)(16,20)(18,19)(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)(60,61)(62,63)(64,65)(66,67)(68,69)(70,71)(72,73);(1,2)(3,4)(6,10)(7,14)(8,17)(11,19)(12,20)(13,15)

Polycyclic presentation:

2^8,53|d^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fgh,c^b=cf,d^b=dh,f^b=fh,g^b=gh,i^b=i^52,d^c=dg,f^c=fh,i^c=i^52,f^d=fh

Conjugacy classes & character table

707 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 88597);