#79650

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

Aliases

No aliases for this group.

Splittings

Fitting	C₃×C₄×SD₁₆ ⋊ C₂
Sylow	C₃ ⋊ #2952

Subgroups

	subgroup	G/H
center	C₂²	C₂×D₃×D₄
commutator	C₃×C₄	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₃×C₄×SD₁₆	C₂
socle	C₂²×C₃	C₂²×D₄
prime radical	C₂×#916	C₂
cosocle	C₂×C₃×C₄	C₂⁴
hypercenter	C₄×SD₁₆	D₃
hypocenter	C₃	#2952
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×SD₁₆

484 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 4 generators on 23 points:

(2,3)(8,12,14,10)(9,17,18,20)(11,19,13,21)(15,16,23,22);(1,2,3)(4,6,7,5)(8,11,14,13)(9,16,18,22)(10,19,12,21)(15,20,23,17);(8,10,14,12)(9,15,18,23)(11,19,13,21)(16,20,22,17);(4,5,7,6)(8,9,14,18)(10,17,12,20)(11,16,13,22)(15,21,23,19)

Polycyclic presentation:

2^7,3|a^2=f,b^2=g,c^2=f,d^2=g,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	D₃×#2952, C₂×#79650, #11659354, …
maximal normal subgroup of	D₃×#2952, C₂×#79650, #11659354, …
maximal quotient of	#10764671, #10764900, #10764906, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 12226);