#27412537

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2005 ⋊ C₂
Sylow	C₅₃ ⋊ #23590

Subgroups

	subgroup	G/H
center	C₂²	#120×D₅₃
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#2005	C₂
socle	C₂²×C₅₃	C₂×#120
prime radical	#16811201	C₂²
hypercenter	#2005	D₅₃
hypocenter	C₅₃	#23590
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #2005

1074 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:

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

Polycyclic presentation:

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

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, 198013);