C₃⋊C₁₂₈ #67425

order	384
isoclinism	isoclinic class of D₃
automorphism	Aut(G) = C₂×D₃×C₃₂ (order 384) · Inn(G) = D₃ · Out(G) = C₂×C₃₂
homology	Abelianization (H₁) = C₁₂₈ · Schur multiplier (H₂) = C₁

Aliases

unambiguous semidirect product

C₃⋊C₁₂₈

Splittings

derived, Sylow

C₃ ⋊ C₁₂₈

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₆₄	D₃
commutator, hypocenter	C₃	C₁₂₈
Fitting, cosocle	C₃×C₆₄	C₂
socle, prime radical	C₂×C₃	C₆₄
LCS	G ▷ C₃
UCS	C₁ ◁ C₆₄

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 131
transitive degree	μ_t(G) = 384

A minimal faithful permutation representation — 2 generators on 131 points:

(2,3)(4,5,6,12,7,13,18,33,8,14,19,34,23,38,48,68,9,15,20,35,24,39,49,69,27,42,52,72,58,78,88,103,10,16,21,36,25,40,50,70,28,43,53,73,59,79,89,104,30,45,55,75,61,81,91,106,64,84,94,109,98,113,118,124,11,17,22,37,26,41,51,71,29,44,54,74,60,80,90,105,31,46,56,76,62,82,92,107,65,85,95,110,99,114,119,125,32,47,57,77,63,83,93,108,66,86,96,111,100,115,120,126,67,87,97,112,101,116,121,127,102,117,122,128,123,129,130,131);(1,2,3)

Polycyclic presentation:

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

Conjugacy classes & character table

192 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×C₁₂₈, C₂×C₃⋊C₁₂₈, #10683097, …
maximal normal subgroup of	D₃×C₁₂₈, C₂×C₃⋊C₁₂₈, #10683097, …
maximal quotient of	C₃⋊C₂₅₆, C₂×C₃⋊C₁₂₈, C₉⋊C₁₂₈, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 1);

C3⋊C128 #67425