D₃×C₁₃×C₁₇₃ #27212573

order	13494
isoclinism	isoclinic class of D₃
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₃×C₁₇₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	D₃×C₁₃×C₁₇₃
unambiguous semidirect product	C₃⋊(C₂×C₁₃×C₁₇₃)

Splittings

Fitting, socle, Hall	C₃×C₁₃×C₁₇₃ ⋊ C₂
derived, Sylow	C₃ ⋊ C₂×C₁₃×C₁₇₃
Hall	C₃×C₁₇₃ ⋊ C₂×C₁₃
Hall	C₃×C₁₃ ⋊ C₂×C₁₇₃

Subgroups

	subgroup	G/H
center, hypercenter	C₁₃×C₁₇₃	D₃
commutator, cosocle, hypocenter	C₃	C₂×C₁₃×C₁₇₃
Frattini	C₁	G
Fitting, socle	C₃×C₁₃×C₁₇₃	C₂
LCS	G ▷ C₃
UCS	C₁ ◁ C₁₃×C₁₇₃

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 189
transitive degree	μ_t(G) = 6747

A minimal faithful permutation representation — 2 generators on 189 points:

(2,3)(4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176)(177,178,179,180,181,182,183,184,185,186,187,188,189);(1,2,3)

Conjugacy classes & character table

6747 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₃×C₁₃×D₁₇₃, D₃×D₁₃×C₁₇₃, D₃×D₂₂₄₉, …
maximal normal subgroup of	D₃×C₁₃×D₁₇₃, D₃×D₁₃×C₁₇₃, D₃×D₂₂₄₉, …
maximal quotient of	Dic₃×C₁₃×C₁₇₃, C₂×D₃×C₁₃×C₁₇₃

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13494, 6);

D3×C13×C173 #27212573