C₇×D₇×C₁₂₇ #27159232

order	12446
isoclinism	isoclinic class of D₇
automorphism	?
homology	Abelianization (H₁) = C₂×C₇×C₁₂₇ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₇×D₇×C₁₂₇
unambiguous semidirect product	C₇⋊(C₂×C₇×C₁₂₇)

Splittings

Fitting, socle, Hall	C₇²×C₁₂₇ ⋊ C₂
derived	C₇ ⋊ C₂×C₇×C₁₂₇
Sylow	C₇² ⋊ C₂×C₁₂₇

Subgroups

	subgroup	G/H
center, hypercenter	C₇×C₁₂₇	D₇
commutator, cosocle, hypocenter	C₇	C₂×C₇×C₁₂₇
Frattini	C₁	G
Fitting, socle	C₇²×C₁₂₇	C₂
LCS	G ▷ C₇
UCS	C₁ ◁ C₇×C₁₂₇

22 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 141
transitive degree	μ_t(G) = 1778

A minimal faithful permutation representation — 2 generators on 141 points:

(1,2,3,4,5,6,7)(9,10)(11,12)(13,14)(15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141);(8,9,11,13,14,12,10)

Conjugacy classes & character table

4445 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₇×D₇×D₁₂₇, D₇²×C₁₂₇, D₇×D₈₈₉, …
maximal normal subgroup of	C₇×D₇×D₁₂₇, D₇²×C₁₂₇, D₇×D₈₈₉, …
maximal quotient of	C₇×Dic₇×C₁₂₇, C₂×C₇×D₇×C₁₂₇, C₃×C₇×D₇×C₁₂₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12446, 15);

C7×D7×C127 #27159232