C₂×D₉×C₃₆₁ #27188671

order	12996
isoclinism	isoclinic class of D₉
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃₆₁ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₂×D₉×C₃₆₁
unambiguous semidirect product	C₉⋊(C₂²×C₃₆₁)
unambiguous semidirect product	(C₂×C₉)⋊(C₂×C₃₆₁)

Splittings

Fitting	C₂×C₉×C₃₆₁ ⋊ C₂
derived, Sylow	C₉ ⋊ C₂²×C₃₆₁
Hall	C₉×C₃₆₁ ⋊ C₂²

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₃₆₁	D₉
commutator, hypocenter	C₉	C₂²×C₃₆₁
Frattini	C₃×C₁₉	C₂×D₃×C₁₉
Fitting	C₂×C₉×C₃₆₁	C₂
socle	C₂×C₃×C₁₉	D₃×C₁₉
prime radical	C₂×D₉×C₁₉	C₁₉
cosocle	C₉×C₁₉	C₂²×C₁₉
LCS	G ▷ C₉
UCS	C₁ ◁ C₂×C₃₆₁

45 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 372
transitive degree	μ_t(G) = 6498

A minimal faithful permutation representation — 2 generators on 372 points:

(2,4)(3,7)(5,8)(6,9)(10,14,22,34,50,70,94,122,154,190,226,258,286,310,330,346,358,366,370,12,18,28,42,60,82,108,138,172,209,243,273,299,321,339,353,363,369,181,15,23,35,51,71,95,123,155,191,227,259,287,311,331,347,359,367,163,200,19,29,43,61,83,109,139,173,210,244,274,300,322,340,354,364,146,182,218,24,36,52,72,96,124,156,192,228,260,288,312,332,348,360,130,164,201,235,30,44,62,84,110,140,174,211,245,275,301,323,341,355,115,147,183,219,251,37,53,73,97,125,157,193,229,261,289,313,333,349,101,131,165,202,236,266,45,63,85,111,141,175,212,246,276,302,324,342,88,116,148,184,220,252,280,54,74,98,126,158,194,230,262,290,314,334,76,102,132,166,203,237,267,293,64,86,112,142,176,213,247,277,303,325,65,89,117,149,185,221,253,281,305,75,99,127,159,195,231,263,291,315,55,77,103,133,167,204,238,268,294,316,87,113,143,177,214,248,278,304,46,66,90,118,150,186,222,254,282,306,326,100,128,160,196,232,264,292,38,56,78,104,134,168,205,239,269,295,317,335,114,144,178,215,249,279,31,47,67,91,119,151,187,223,255,283,307,327,343,129,161,197,233,265,25,39,57,79,105,135,169,206,240,270,296,318,336,350,145,179,216,250,20,32,48,68,92,120,152,188,224,256,284,308,328,344,356,162,198,234,16,26,40,58,80,106,136,170,207,241,271,297,319,337,351,361,180,217,13,21,33,49,69,93,121,153,189,225,257,285,309,329,345,357,365,199,11,17,27,41,59,81,107,137,171,208,242,272,298,320,338,352,362,368);(1,6,4,3,8,5,7,2,9)(371,372)

Conjugacy classes & character table

4332 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12996, 14);

C2×D9×C361 #27188671