#69885

order	384
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Sylow

C₃ ⋊ #2153

Subgroups

	subgroup	G/H
center	C₈	C₂×D₁₂
commutator	C₃×C₄	C₂²×C₈
Frattini	C₂×C₈	C₂²×D₃
Fitting	C₃×#405	C₂
socle	C₂×C₃	C₂×C₄⋊C₈
prime radical	C₂³×C₃	M₁₆
cosocle	C₂×C₃×C₈	C₂³
hypercenter	#405	D₃
hypocenter	C₃	#2153
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #405

152 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^7,3|a^2=e,b^2=d,d^2=g,e^2=f,f^2=g,b^a=bd,d^a=dg,h^a=h^2,c^b=cg

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10706704, #10706705, #10706709, …
maximal normal subgroup of	#10706704, #10706705, #10706709, …
maximal quotient of	#10706735, #10706761, #10706795, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 2461);