#67809

order	384
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₆ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Sylow

C₃ ⋊ #1400

Subgroups

	subgroup	G/H
center	C₈	#217
commutator	C₂²×C₃	C₂×C₁₆
Frattini	C₂×C₁₆	C₂×D₃
Fitting	C₂×C₃×M₃₂	C₂
socle	C₂×C₃	C₂²⋊C₁₆
prime radical	C₂³×C₃	C₁₆
cosocle	C₂×C₃×C₁₆	C₂²
hypercenter	C₂×M₃₂	D₃
hypocenter	C₃	#1400
LCS	G ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₈ ◁ C₂×C₁₆ ◁ C₂×M₃₂

74 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 35
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 2 generators on 35 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10682529, #10682530, #10682531, …
maximal normal subgroup of	#10682529, #10682530, #10682531, …
maximal quotient of	#10682526, #10682586, #10682628, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 385);