#27393091

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #21757

Subgroups

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊Dic₅₃
commutator	C₂³×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting, prime radical	C₂×C₅₃×#537	C₂
socle	C₂²×C₅₃	C₂²×C₂²⋊C₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂×#537	D₅₃
hypocenter	C₅₃	#21757
LCS	G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ C₂×#537

1318 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 77
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 4 generators on 77 points:

(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)(55,59)(58,60)(62,64,66,70)(63,68,67,73)(65,71,72,76)(69,75,74,77);(56,61)(58,60)(63,67)(65,72)(68,73)(71,76);(54,56)(55,58)(57,61)(59,60)(64,70)(68,73)(71,76)(75,77);(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)(54,55)(56,60)(57,59)(58,61)(62,63,65,69,66,67,72,74)(64,68,71,75,70,73,76,77)

Polycyclic presentation:

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

Conjugacy classes & character table

1190 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 178567);