C₂²⋊C₉×C₃₆₁ #27188668

order	12996
isoclinism	isoclinic class of A₄
automorphism	?
homology	Abelianization (H₁) = C₉×C₃₆₁ · Schur multiplier (H₂) = C₂

Aliases

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

Splittings

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

Subgroups

	subgroup	G/H
center, hypercenter	C₃×C₃₆₁	A₄
commutator, hypocenter	C₂²	C₉×C₃₆₁
Frattini	C₃×C₁₉	A₄×C₁₉
Fitting	C₂²×C₃×C₃₆₁	C₃
socle, prime radical, cosocle	C₂²×C₃×C₁₉	C₃×C₁₉
LCS	G ▷ C₂²
UCS	C₁ ◁ C₃×C₃₆₁

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 374
transitive degree	μ_t(G) = 6498

A minimal faithful permutation representation — 2 generators on 374 points:

(2,3,4)(5,6,8,7,9,11,10,12,13)(14,15,17,20,24,29,35,42,50,59,69,80,92,105,119,134,150,167,185,16,18,21,25,30,36,43,51,60,70,81,93,106,120,135,151,168,186,204,19,22,26,31,37,44,52,61,71,82,94,107,121,136,152,169,187,205,222,23,27,32,38,45,53,62,72,83,95,108,122,137,153,170,188,206,223,239,28,33,39,46,54,63,73,84,96,109,123,138,154,171,189,207,224,240,255,34,40,47,55,64,74,85,97,110,124,139,155,172,190,208,225,241,256,270,41,48,56,65,75,86,98,111,125,140,156,173,191,209,226,242,257,271,284,49,57,66,76,87,99,112,126,141,157,174,192,210,227,243,258,272,285,297,58,67,77,88,100,113,127,142,158,175,193,211,228,244,259,273,286,298,309,68,78,89,101,114,128,143,159,176,194,212,229,245,260,274,287,299,310,320,79,90,102,115,129,144,160,177,195,213,230,246,261,275,288,300,311,321,330,91,103,116,130,145,161,178,196,214,231,247,262,276,289,301,312,322,331,339,104,117,131,146,162,179,197,215,232,248,263,277,290,302,313,323,332,340,347,118,132,147,163,180,198,216,233,249,264,278,291,303,314,324,333,341,348,354,133,148,164,181,199,217,234,250,265,279,292,304,315,325,334,342,349,355,360,149,165,182,200,218,235,251,266,280,293,305,316,326,335,343,350,356,361,365,166,183,201,219,236,252,267,281,294,306,317,327,336,344,351,357,362,366,369,184,202,220,237,253,268,282,295,307,318,328,337,345,352,358,363,367,370,372,203,221,238,254,269,283,296,308,319,329,338,346,353,359,364,368,371,373,374);(1,4)(2,3)

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, 11);

C22⋊C9×C361 #27188668