#82749

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

Aliases

No aliases for this group.

Splittings

Sylow

C₃ ⋊ #2368

Subgroups

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₃
commutator	C₂²×C₃	C₂³×C₄
Fitting	C₂×C₃×C₂²⋊Q₈	C₂
socle, cosocle	C₂³×C₃	C₂⁴
prime radical	C₂⁴×C₃	C₂³
hypercenter	C₂×C₂²⋊Q₈	D₃
hypocenter	C₃	#2368
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₂²⋊Q₈

548 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:

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

Polycyclic presentation:

2^7,3|a^2=g,b^2=f,c^2=efg,b^a=be,c^a=cfg,h^a=h^2,d^b=de,h^b=h^2

Conjugacy classes & character table

78 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10691751, #10691755, #10691765, …
maximal normal subgroup of	#10691751, #10691755, #10691765, …
maximal quotient of	#10716622, #10716698, #10716747, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 15325);