#79720

order	384
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₃×C₄×Q₁₆ ⋊ C₂
Sylow	C₃ ⋊ #2950

Subgroups

	subgroup	G/H
center	C₂²	C₂×D₃×D₄
commutator	C₃×C₄	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₃×C₄×Q₁₆	C₂
socle	C₂²×C₃	C₂²×D₄
prime radical	C₂²×D₃	C₂×D₄
cosocle	C₂×C₃×C₄	C₂⁴
hypercenter	C₄×Q₁₆	D₃
hypocenter	C₃	#2950
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₁₆

456 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 39
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 4 generators on 39 points:

(2,3)(9,16)(11,22)(14,28)(17,24)(18,31)(19,33)(23,35)(34,37);(1,2,3)(4,6,7,5)(8,11,13,22)(9,15,16,30)(10,19,21,33)(12,23,26,35)(14,27,28,38)(17,29,24,36)(18,32,31,39)(20,34,25,37);(8,10,13,21)(9,14,16,28)(11,19,22,33)(12,25,26,20)(15,27,30,38)(17,31,24,18)(23,37,35,34)(29,39,36,32);(4,5,7,6)(8,9)(10,18)(11,15)(12,24)(13,16)(14,20)(17,26)(19,32)(21,31)(22,30)(23,36)(25,28)(27,34)(29,35)(33,39)(37,38)

Polycyclic presentation:

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

Conjugacy classes & character table

72 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10948533, #10948535, #10948537, …
maximal normal subgroup of	#10948533, #10948535, #10948537, …
maximal quotient of	#10764656, #10764880, #10765027, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 12296);