#71429

order	384
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Sylow

C₃ ⋊ #2161

Subgroups

	subgroup	G/H
center	C₄	C₂×D₂₄
commutator	C₃×C₈	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₃
Fitting	C₂×C₃×M₃₂	C₂
socle	C₂×C₃	C₂×#107
prime radical	C₂³×C₃	Q₁₆
cosocle	C₂×C₃×C₈	C₂³
hypercenter	C₂×M₃₂	D₃
hypocenter	C₃	#2161
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂

140 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)(6,17)(11,26)(15,29)(18,33)(19,20)(25,32)(27,30)(31,34);(4,6,16,32,9,20,23,33,10,17,22,25,8,19,7,18)(5,11,21,31,13,27,28,29,14,26,35,34,24,30,12,15);(2,3)(4,5,8,13,10,14,9,24)(6,15,19,31,17,29,20,34)(7,21,22,28,23,35,16,12)(11,25,27,33,26,32,30,18)

Polycyclic presentation:

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

Conjugacy classes & character table

72 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10710614, #10710619, #10711714, …
maximal normal subgroup of	#10710614, #10710619, #10711714, …
maximal quotient of	#10706776, #10706840, #10706868, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 4005);