#83756

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

Aliases

No aliases for this group.

Splittings

Fitting	C₂²×C₃×M₁₆ ⋊ C₂
Sylow	C₃ ⋊ #2930

Subgroups

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₃
commutator, socle	C₂²×C₃	C₂³×C₄
Fitting	C₂²×C₃×M₁₆	C₂
prime radical	#5791	C₂
cosocle	C₂×C₃×C₄	C₂⁴
hypercenter	C₂²×M₁₆	D₃
hypocenter	C₃	#2930
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆

492 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 96

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

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

Polycyclic presentation:

2^7,3|a^2=f,c^2=f,f^2=g,b^a=be,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	#10712112, #10713396, #10801048, …
maximal normal subgroup of	#10712112, #10713396, #10801048, …
maximal quotient of	#10765379, #10765388, #10765397, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 16332);