C₂×C₃⋊(D₃₆₁⋊C₃) #27188693

order	12996
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₂×C₃⋊(D₃₆₁⋊C₃)
unambiguous semidirect product	(C₂×C₃)⋊(D₃₆₁⋊C₃)
unambiguous semidirect product	(C₂×C₃₆₁⋊C₃)⋊D₃
unambiguous semidirect product	(C₂×D₁₀₈₃)⋊C₃

Splittings

Fitting	C₂×C₃×C₃₆₁ ⋊ C₂×C₃
derived	C₃×C₃₆₁ ⋊ C₂²×C₃
Hall	C₃×C₃₆₁⋊C₃ ⋊ C₂²
Sylow	C₃₆₁ ⋊ C₂×C₃×D₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂	C₃⋊(D₃₆₁⋊C₃)
commutator, cosocle, hypocenter	C₃×C₃₆₁	C₂²×C₃
Frattini	C₁₉	C₂×C₃⋊(D₁₉⋊C₃)
Fitting	C₂×C₃×C₃₆₁	C₂×C₃
socle	C₂×C₃×C₁₉	D₁₉⋊C₃
LCS	G ▷ C₃×C₃₆₁
UCS	C₁ ◁ C₂

66 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 366
transitive degree	μ_t(G) = 2166

A minimal faithful permutation representation — 2 generators on 366 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12996, 36);

C2×C3⋊(D361⋊C3) #27188693