#67591

order	384
isoclinism	isoclinic class of D₉₆
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Sylow

C₃ ⋊ #1423

Subgroups

	subgroup	G/H
center	C₂²	D₄₈
commutator	C₃×C₁₆	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₃
Fitting	C₂×C₃×C₃₂	C₂
socle, prime radical	C₂²×C₃	D₁₆
cosocle	C₂×C₃×C₁₆	C₂²
hypercenter	C₂×C₃₂	D₃
hypocenter	C₃	#1423
LCS	G ▷ C₃×C₁₆ ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂

56 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 39
transitive degree	μ_t(G) = 384

A minimal faithful permutation representation — 2 generators on 39 points:

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

Polycyclic presentation:

2^7,3|a^2=d,b^2=ce,c^2=ef,e^2=fg,f^2=g,b^a=bc,c^a=ce,e^a=ef,f^a=fg,h^a=h^2

Conjugacy classes & character table

102 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10683065, #10683066, #10683068, …
maximal normal subgroup of	#10683065, #10683066, #10683068, …
maximal quotient of	#10682864, #10682888, #10683087, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 167);