#71736

order	384
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Sylow

C₃ ⋊ #1534

Subgroups

	subgroup	G/H
center	C₂²	C₂×#217
commutator	C₂×C₃×C₄	C₂²×C₄
Frattini	C₄²	C₂²×D₃
Fitting	C₂×C₃×#125	C₂
socle	C₂²×C₃	C₂×C₂²⋊C₄
prime radical	C₂³×C₃	C₂²⋊C₄
cosocle	C₃×C₄²	C₂³
hypercenter	C₂×#125	D₃
hypocenter	C₃	#1534
LCS	G ▷ C₂×C₃×C₄ ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#125

196 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 35
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 3 generators on 35 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

54 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10687717, #10687720, #10687722, …
maximal normal subgroup of	#10687717, #10687720, #10687722, …
maximal quotient of	#10683267, #10683305, #10683467, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 4312);