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C2×#15692425⋊C2 #27188686

order12996
automorphism?
homologyAbelianization (H1) = C22×C9 · Schur multiplier (H2) = C2

Aliases

direct productC2×#15692425⋊C2
unambiguous semidirect product#15692425⋊C22
unambiguous semidirect product(C2×#15692425)⋊C2

Splittings

FittingC2×C361 ⋊ C2×C9
derived, SylowC361 ⋊ C22×C9
Hall#15692425 ⋊ C22
HallC2×D361 ⋊ C9

Subgroups

subgroupG/H
center, hypercenterC2#15692425⋊C2
commutator, hypocenterC361C22×C9
FrattiniC19C2×AGL(1,𝔽19)
FittingC2×C361C2×C9
socleC2×C19AGL(1,𝔽19)
prime radicalC2×D361⋊C3C3
cosocleC361⋊C3C22×C3
LCSG ▷ C361
UCSC1 ◁ C2

45 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 363
transitive degreeμt(G) = 722

A minimal faithful permutation representation — 2 generators on 363 points:

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

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12996, 29);