#27411790

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2042 ⋊ C₂
Sylow	C₅₃ ⋊ #23506

Subgroups

	subgroup	G/H
center	C₂³	C₂×D₄×D₅₃
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#2042	C₂
socle	C₂³×C₅₃	C₂²×D₄
prime radical	C₂²×D₄×D₅₃	C₂²
hypercenter	#2042	D₅₃
hypocenter	C₅₃	#23506
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2042

1574 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)(64,74)(68,77)(72,76)(73,75);(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)(55,57)(58,60,61,59)(62,64)(63,68)(65,73)(66,75)(67,74)(69,76)(70,72)(71,77);(58,59)(60,61)(64,73)(65,66)(68,76)(69,70)(72,77)(74,75);(54,55)(56,57)(59,60)(62,63,67,71)(64,72,74,76)(65,70,66,69)(68,73,77,75)

Polycyclic presentation:

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

Conjugacy classes & character table

884 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 197266);