#27538138

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2199 ⋊ C₂
Sylow	C₅₃ ⋊ #33550

Subgroups

	subgroup	G/H
center	C₂	B(4,𝔽₂)×D₅₃
commutator, cosocle	C₂×D₄×C₅₃	C₂⁴
Frattini	C₂×D₄	C₂³×D₅₃
Fitting	C₅₃×#2199	C₂
socle	C₂×C₅₃	C₂×B(4,𝔽₂)
hypercenter	#2199	D₅₃
hypocenter	C₅₃	#33550
LCS	G ▷ C₂×D₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×D₄ ◁ #2199

1144 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 1144 conjugacy classes of subgroups — too many to draw. Only its 137 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) = 85
transitive degree	μ_t(G) = 1696

A minimal faithful permutation representation — 4 generators on 85 points:

(1,4,6,16)(2,8,10,21)(3,13,12,25)(5,15,17,28)(7,19,18,29)(9,20,22,30)(11,24,23,31)(14,27,26,32)(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)(74,75)(76,77)(78,79)(80,81)(82,83)(84,85);(3,12)(5,17)(7,18)(11,23)(13,25)(15,28)(19,29)(24,31)(33,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,80,82,84,85,83,81,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35);(1,3)(2,7)(4,13)(5,14)(6,12)(8,19)(9,11)(10,18)(15,27)(16,25)(17,26)(20,24)(21,29)(22,23)(28,32)(30,31);(1,2)(3,11)(4,8)(5,9)(6,10)(7,14)(12,23)(13,24)(15,20)(16,21)(17,22)(18,26)(19,27)(25,31)(28,30)(29,32)

Polycyclic presentation:

2^8,53|d^2=h,b^a=be,c^a=cf,g^a=gh,c^b=ch,f^b=fgh,e^c=eg,i^d=i^52,f^e=fh

Conjugacy classes & character table

476 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 323614);