#27299223

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1390 ⋊ C₂
Sylow	C₅₃ ⋊ C₄×AΣL(1,𝔽₁₆)

Subgroups

	subgroup	G/H
center	C₂²	#99×D₅₃
commutator	C₂³×C₅₃	C₂×C₄²
Frattini	C₂²×D₄	C₂²×D₅₃
Fitting	C₅₃×#1390	C₂
socle	C₂²×C₅₃	C₂×#99
prime radical	C₂×#13133185	C₂²
cosocle	C₂²×D₄×C₅₃	C₂³
hypercenter	#1390	D₅₃
hypocenter	C₅₃	C₄×AΣL(1,𝔽₁₆)
LCS	G ▷ C₂³×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×D₄ ◁ #1390

844 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 844 conjugacy classes of subgroups — too many to draw. Only its 115 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:

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

Polycyclic presentation:

2^8,53|a^2=e,b^2=f,c^2=e,b^a=bd,f^a=fg,d^b=dg,g^b=gh,i^c=i^52,f^d=fh

Conjugacy classes & character table

728 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 84699);