D₃₇×C₁₆₉ #27159772

order	12506
isoclinism	isoclinic class of D₃₇
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₆₉ · Schur multiplier (H₂) = C₁

Aliases

direct product	D₃₇×C₁₆₉
unambiguous semidirect product	C₃₇⋊(C₂×C₁₆₉)

Splittings

Fitting, Hall	C₃₇×C₁₆₉ ⋊ C₂
derived, Sylow	C₃₇ ⋊ C₂×C₁₆₉

Subgroups

	subgroup	G/H
center, hypercenter	C₁₆₉	D₃₇
commutator, hypocenter	C₃₇	C₂×C₁₆₉
Frattini	C₁₃	C₁₃×D₃₇
Fitting	C₃₇×C₁₆₉	C₂
socle, cosocle	C₁₃×C₃₇	C₂×C₁₃
prime radical	C₁₃×D₃₇	C₁₃
LCS	G ▷ C₃₇
UCS	C₁ ◁ C₁₆₉

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 206
transitive degree	μ_t(G) = 6253

A minimal faithful permutation representation — 2 generators on 206 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)(38,39,41,44,48,53,59,66,74,83,93,104,116,40,42,45,49,54,60,67,75,84,94,105,117,129,43,46,50,55,61,68,76,85,95,106,118,130,141,47,51,56,62,69,77,86,96,107,119,131,142,152,52,57,63,70,78,87,97,108,120,132,143,153,162,58,64,71,79,88,98,109,121,133,144,154,163,171,65,72,80,89,99,110,122,134,145,155,164,172,179,73,81,90,100,111,123,135,146,156,165,173,180,186,82,91,101,112,124,136,147,157,166,174,181,187,192,92,102,113,125,137,148,158,167,175,182,188,193,197,103,114,126,138,149,159,168,176,183,189,194,198,201,115,127,139,150,160,169,177,184,190,195,199,202,204,128,140,151,161,170,178,185,191,196,200,203,205,206);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)

Conjugacy classes & character table

3380 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	#3866×C₁₆₉, #38736118, D₃₇×D₁₆₉, …
maximal normal subgroup of	#3866×C₁₆₉, #38736118, D₃₇×D₁₆₉, …
maximal quotient of	Dic₃₇×C₁₆₉, C₂×D₃₇×C₁₆₉, C₃×D₃₇×C₁₆₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12506, 3);

D37×C169 #27159772