#27271640

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1295 ⋊ C₂
Sylow	C₅₃ ⋊ #7416

Subgroups

	subgroup	G/H
center	C₂²	#16810411
commutator	C₂×C₄⋊C₄×C₅₃	C₂×C₄
Frattini	#382	C₂×D₅₃
Fitting	C₅₃×#1295	C₂
socle	C₂²×C₅₃	AΣL(1,𝔽₁₆)
prime radical	#16811204	C₄
cosocle	C₅₃×#382	C₂²
hypercenter	#1295	D₅₃
hypocenter	C₅₃	#7416
LCS	G ▷ C₂×C₄⋊C₄×C₅₃ ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄ ◁ #382 ◁ #1295

224 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 41 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) = 1696

A minimal faithful permutation representation — 2 generators on 77 points:

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

Polycyclic presentation:

2^8,53|a^2=d,c^2=f,d^2=g,e^2=h,b^a=bc,c^a=ce,e^a=eg,c^b=cf,d^b=defh,e^b=eh,g^b=gh,i^b=i^52,d^c=dgh,e^c=eh

Conjugacy classes & character table

1019 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 57116);