#67599

order	384
isoclinism	isoclinic class of #213
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₆ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

Sylow

C₃ ⋊ #1422

Subgroups

	subgroup	G/H
center	C₁₆	#67
commutator	C₃×C₄	C₂×C₁₆
Frattini	C₂×C₁₆	C₂×D₃
Fitting	C₃×M₆₄	C₂
socle	C₂×C₃	C₄⋊C₁₆
prime radical	C₂²×C₃	M₃₂
cosocle	C₂×C₃×C₁₆	C₂²
hypercenter	M₆₄	D₃
hypocenter	C₃	#1422
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₁₆ ◁ C₂×C₁₆ ◁ M₆₄

58 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

96 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#1422, C₂×#67599, #10715678, …
maximal normal subgroup of	D₃×#1422, C₂×#67599, #10715678, …
maximal quotient of	#10682852, #10682859, #10682877, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 175);