#67514

order	384
isoclinism	isoclinic class of #4513
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

Sylow

C₃ ⋊ #1369

Subgroups

	subgroup	G/H
center	C₄	#817
commutator	C₃×C₈	C₂×C₈
Frattini	C₄×C₈	C₂×D₃
Fitting	C₃×#346	C₂
socle	C₂×C₃	#334
prime radical	C₂²×C₃	#108
cosocle	C₃×C₄×C₈	C₂²
hypercenter	#346	D₃
hypocenter	C₃	#1369
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #346

60 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 35
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 2 generators on 35 points:

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

Polycyclic presentation:

2^7,3|a^2=d,b^2=c,c^2=e,d^2=f,e^2=g,f^2=g,b^a=bc,c^a=ce,e^a=eg,h^a=h^2,d^b=dg

Conjugacy classes & character table

72 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10682658, #10682659, #10682661, …
maximal normal subgroup of	#10682658, #10682659, #10682661, …
maximal quotient of	#10681925, #10682073, C₂×#67514, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 90);