#68061

order	384
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₃×C₈² ⋊ C₂
Sylow	C₃ ⋊ #1711

Subgroups

	subgroup	G/H
center	C₂²	#874
commutator, cosocle	C₃×C₄²	C₂³
Frattini	C₄²	C₂²×D₃
Fitting	C₃×C₈²	C₂
socle	C₂²×C₃	#127
prime radical	#874	C₂²
hypercenter	C₈²	D₃
hypocenter	C₃	#1711
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈²

178 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 35
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 3 generators on 35 points:

(1,2,3)(5,10)(9,17)(11,16)(12,18)(20,22,24,30,23,29,31,35)(21,26,27,28,25,33,32,34);(4,6,13,19,8,15,7,14)(5,9,16,18,10,17,11,12)(21,25)(26,33)(27,32)(28,34);(2,3)(4,5)(6,12)(7,16)(8,10)(9,14)(11,13)(15,18)(17,19)(20,21)(22,28)(23,25)(24,32)(26,30)(27,31)(29,34)(33,35)

Polycyclic presentation:

2^7,3|b^2=dfg,c^2=ef,d^2=f,e^2=g,b^a=bd,c^a=ce,d^a=df,e^a=eg,h^a=h^2

Conjugacy classes & character table

102 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#1711, C₂×#68061, #10782871, …
maximal normal subgroup of	D₃×#1711, C₂×#68061, #10782871, …
maximal quotient of	#10688966, #10694932, #10694939, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 637);