C₂×C₃×D₃×C₃₆₁ #27188705

order	12996
isoclinism	isoclinic class of D₃
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₃₆₁ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₂×C₃×D₃×C₃₆₁
unambiguous semidirect product	C₃⋊(C₂²×C₃×C₃₆₁)
unambiguous semidirect product	(C₂×C₃)⋊(C₂×C₃×C₃₆₁)

Splittings

Fitting	C₂×C₃²×C₃₆₁ ⋊ C₂
derived	C₃ ⋊ C₂²×C₃×C₃₆₁
Hall	C₃²×C₃₆₁ ⋊ C₂²
Sylow	C₃² ⋊ C₂²×C₃₆₁

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₃×C₃₆₁	D₃
commutator, hypocenter	C₃	C₂²×C₃×C₃₆₁
Frattini	C₁₉	C₂×C₃×D₃×C₁₉
Fitting	C₂×C₃²×C₃₆₁	C₂
socle	C₂×C₃²×C₁₉	C₂×C₁₉
prime radical	C₂×C₃×D₃×C₁₉	C₁₉
cosocle	C₃×C₁₉	C₂²×C₃×C₁₉
LCS	G ▷ C₃
UCS	C₁ ◁ C₂×C₃×C₃₆₁

66 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 369
transitive degree	μ_t(G) = 4332

A minimal faithful permutation representation — 2 generators on 369 points:

(1,2,3)(5,6)(7,20,49,94,155,228,289,54,103,168,240,297,338,113,182,251,304,341,362,9,26,59,108,173,245,28,65,118,187,256,309,346,129,201,266,315,348,365,12,33,70,123,192,261,36,77,134,206,271,320,85,146,219,280,325,354,367,16,41,82,139,211,276,45,90,151,224,285,330,99,164,236,293,334,359,178,21,50,95,156,229,22,55,104,169,241,298,339,114,183,252,305,342,363,197,27,60,109,174,246,29,66,119,188,257,310,73,130,202,267,316,349,366,215,34,71,124,193,262,37,78,135,207,272,321,86,147,220,281,326,355,160,232,42,83,140,212,17,46,91,152,225,286,331,100,165,237,294,335,360,179,248,51,96,157,230,23,56,105,170,242,299,62,115,184,253,306,343,364,198,263,61,110,175,247,30,67,120,189,258,311,74,131,203,268,317,350,143,216,277,72,125,194,13,38,79,136,208,273,322,87,148,221,282,327,356,161,233,290,84,141,213,18,47,92,153,226,287,52,101,166,238,295,336,361,180,249,302,97,158,231,24,57,106,171,243,300,63,116,185,254,307,344,127,199,264,313,111,176,10,31,68,121,190,259,312,75,132,204,269,318,351,144,217,278,323,126,195,14,39,80,137,209,274,43,88,149,222,283,328,357,162,234,291,332,142,214,19,48,93,154,227,288,53,102,167,239,296,337,112,181,250,303,340,159,8,25,58,107,172,244,301,64,117,186,255,308,345,128,200,265,314,347,177,11,32,69,122,191,260,35,76,133,205,270,319,352,145,218,279,324,353,196,15,40,81,138,210,275,44,89,150,223,284,329,98,163,235,292,333,358);(4,5,6)(368,369)

Conjugacy classes & character table

6498 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12996, 48);

C2×C3×D3×C361 #27188705