#27303046

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅₃×#453 ⋊ C₂
Sylow	C₅₃ ⋊ #13346

Subgroups

	subgroup	G/H
center	C₂²	#16810809
commutator, cosocle	C₂²×D₄×C₅₃	C₂³
Frattini	C₂²×D₄	C₂²×D₅₃
Fitting	C₂×C₅₃×#453	C₂
socle	C₂²×C₅₃	B(4,𝔽₂)
hypercenter	C₂×#453	D₅₃
hypocenter	C₅₃	#13346
LCS	G ▷ C₂²×D₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×D₄ ◁ C₂×#453

1022 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 1022 conjugacy classes of subgroups — too many to draw. Only its 81 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) = 65
transitive degree	μ_t(G) = 1696

A minimal faithful permutation representation — 3 generators on 65 points:

(2,4)(7,11)(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)(60,61)(62,63)(64,65);(5,6)(7,12)(8,10)(9,11)(13,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15);(1,2)(3,4)(6,9)(7,11)(10,12)

Polycyclic presentation:

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

Conjugacy classes & character table

863 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 88522);