C₁₂₇⋊C₃ #67420

order	381
isoclinism	isoclinic class of C₁₂₇⋊C₃
automorphism	?
homology	Abelianization (H₁) = C₃ · Schur multiplier (H₂) = C₁

Aliases

unambiguous semidirect product

C₁₂₇⋊C₃

Splittings

Fitting, socle, derived, Sylow

C₁₂₇ ⋊ C₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₁₂₇	C₃
LCS	G ▷ C₁₂₇
UCS	C₁

4 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 127
transitive degree	μ_t(G) = 127

A minimal faithful permutation representation — 2 generators on 127 points:

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

Polycyclic presentation:

3,127|b^a=b^107

Conjugacy classes & character table

45 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₁₂₇⋊C₃, C₂×C₁₂₇⋊C₃, #11765919, …
maximal normal subgroup of	D₁₂₇⋊C₃, C₂×C₁₂₇⋊C₃, #11765919, …
maximal quotient of	C₂×C₁₂₇⋊C₃, #11765917, C₃×C₁₂₇⋊C₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(381, 1);

C127⋊C3 #67420