D₇×C₁₁₃ #13096832

order	1582
isoclinism	isoclinic class of D₇
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₁₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	D₇×C₁₁₃
unambiguous semidirect product	C₇⋊(C₂×C₁₁₃)

Splittings

Fitting, socle, Hall	C₇×C₁₁₃ ⋊ C₂
derived, Sylow	C₇ ⋊ C₂×C₁₁₃

Subgroups

	subgroup	G/H
center, hypercenter	C₁₁₃	D₇
commutator, cosocle, hypocenter	C₇	C₂×C₁₁₃
Frattini	C₁	G
Fitting, socle	C₇×C₁₁₃	C₂
LCS	G ▷ C₇
UCS	C₁ ◁ C₁₁₃

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 120
transitive degree	μ_t(G) = 791

A minimal faithful permutation representation — 2 generators on 120 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);(2,7)(3,6)(4,5)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	D₇×D₁₁₃, C₂×D₇×C₁₁₃, C₃×D₇×C₁₁₃, …
maximal normal subgroup of	D₇×D₁₁₃, C₂×D₇×C₁₁₃, C₃×D₇×C₁₁₃, …
maximal quotient of	Dic₇×C₁₁₃, C₂×D₇×C₁₁₃, C₃×D₇×C₁₁₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1582, 3);

D7×C113 #13096832