#78512

order	384
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Fitting	C₃×C₄×M₁₆ ⋊ C₂
Sylow	C₃ ⋊ #2971

Subgroups

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₃
commutator, socle	C₂²×C₃	C₂³×C₄
Fitting	C₃×C₄×M₁₆	C₂
prime radical	C₂×#235	C₄
cosocle	C₂×C₃×C₄	C₂⁴
hypercenter	C₄×M₁₆	D₃
hypocenter	C₃	#2971
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆

402 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 4 generators on 19 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

90 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10690988, #10690992, #10691002, …
maximal normal subgroup of	#10690988, #10690992, #10691002, …
maximal quotient of	#10793474, #10793476, #10793625, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 11088);