D₅×C₁₂₇ #11929647

order	1270
isoclinism	isoclinic class of D₅
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₂₇ · Schur multiplier (H₂) = C₁

Aliases

direct product	D₅×C₁₂₇
unambiguous semidirect product	C₅⋊(C₂×C₁₂₇)

Splittings

Fitting, socle, Hall	C₅×C₁₂₇ ⋊ C₂
derived, Sylow	C₅ ⋊ C₂×C₁₂₇

Subgroups

	subgroup	G/H
center, hypercenter	C₁₂₇	D₅
commutator, cosocle, hypocenter	C₅	C₂×C₁₂₇
Frattini	C₁	G
Fitting, socle	C₅×C₁₂₇	C₂
LCS	G ▷ C₅
UCS	C₁ ◁ C₁₂₇

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 132
transitive degree	μ_t(G) = 635

A minimal faithful permutation representation — 2 generators on 132 points:

(1,2,3,4,5)(6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132);(2,5)(3,4)

Conjugacy classes & character table

508 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	AGL(1,𝔽₅)×C₁₂₇, C₁₂₇⋊AGL(1,𝔽₅), D₅×D₁₂₇, …
maximal normal subgroup of	AGL(1,𝔽₅)×C₁₂₇, C₁₂₇⋊AGL(1,𝔽₅), D₅×D₁₂₇, …
maximal quotient of	Dic₅×C₁₂₇, C₂×D₅×C₁₂₇, C₃×D₅×C₁₂₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1270, 1);

D5×C127 #11929647