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G = C12.81D12order 288 = 25·32

12nd non-split extension by C12 of D12 acting via D12/D6=C2

metabelian, supersoluble, monomial

Aliases: C12.81D12, C12.14Dic6, Dic3⋊(C3⋊C8), C6.21(S3×C8), C326(C4⋊C8), (C3×Dic3)⋊1C8, C31(C12⋊C8), (C3×C12).17Q8, C32(Dic3⋊C8), (C3×C12).112D4, (C2×C12).297D6, C62.31(C2×C4), C6.1(C4⋊Dic3), (C4×Dic3).7S3, (C6×Dic3).5C4, C6.13(C8⋊S3), (C3×C6).7M4(2), C12.80(C3⋊D4), C6.1(Dic3⋊C4), (Dic3×C12).1C2, C4.7(C322Q8), C6.3(C4.Dic3), C4.30(C3⋊D12), (C6×C12).202C22, (C2×Dic3).4Dic3, C2.3(D6.Dic3), C22.11(S3×Dic3), C2.1(Dic3⋊Dic3), C6.5(C2×C3⋊C8), C2.5(S3×C3⋊C8), (C6×C3⋊C8).3C2, (C2×C3⋊C8).9S3, (C2×C4).130S32, (C2×C6).68(C4×S3), (C3×C6).20(C2×C8), (C3×C6).18(C4⋊C4), (C2×C6).15(C2×Dic3), (C2×C324C8).14C2, SmallGroup(288,219)

Series: Derived Chief Lower central Upper central

C1C3×C6 — C12.81D12
C1C3C32C3×C6C3×C12C6×C12Dic3×C12 — C12.81D12
C32C3×C6 — C12.81D12
C1C2×C4

Generators and relations for C12.81D12
 G = < a,b,c | a12=b12=1, c2=a9, bab-1=cac-1=a5, cbc-1=b-1 >

Subgroups: 210 in 87 conjugacy classes, 46 normal (42 characteristic)
C1, C2 [×3], C3 [×2], C3, C4 [×2], C4 [×3], C22, C6 [×6], C6 [×3], C8 [×2], C2×C4, C2×C4 [×2], C32, Dic3 [×2], Dic3, C12 [×4], C12 [×5], C2×C6 [×2], C2×C6, C42, C2×C8 [×2], C3×C6 [×3], C3⋊C8 [×5], C24, C2×Dic3 [×2], C2×C12 [×2], C2×C12 [×3], C4⋊C8, C3×Dic3 [×2], C3×Dic3, C3×C12 [×2], C62, C2×C3⋊C8, C2×C3⋊C8 [×3], C4×Dic3, C4×C12, C2×C24, C3×C3⋊C8, C324C8, C6×Dic3 [×2], C6×C12, C12⋊C8, Dic3⋊C8, C6×C3⋊C8, Dic3×C12, C2×C324C8, C12.81D12
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×2], C8 [×2], C2×C4, D4, Q8, Dic3 [×2], D6 [×2], C4⋊C4, C2×C8, M4(2), C3⋊C8 [×2], Dic6 [×2], C4×S3, D12, C2×Dic3, C3⋊D4, C4⋊C8, S32, S3×C8, C8⋊S3, C2×C3⋊C8, C4.Dic3, Dic3⋊C4, C4⋊Dic3, S3×Dic3, C3⋊D12, C322Q8, C12⋊C8, Dic3⋊C8, S3×C3⋊C8, D6.Dic3, Dic3⋊Dic3, C12.81D12

Smallest permutation representation of C12.81D12
On 96 points
Generators in S96
(1 2 3 4 5 6 7 8 9 10 11 12)(13 14 15 16 17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32 33 34 35 36)(37 38 39 40 41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80 81 82 83 84)(85 86 87 88 89 90 91 92 93 94 95 96)
(1 42 91 74 5 38 95 82 9 46 87 78)(2 47 92 79 6 43 96 75 10 39 88 83)(3 40 93 84 7 48 85 80 11 44 89 76)(4 45 94 77 8 41 86 73 12 37 90 81)(13 72 36 49 17 68 28 57 21 64 32 53)(14 65 25 54 18 61 29 50 22 69 33 58)(15 70 26 59 19 66 30 55 23 62 34 51)(16 63 27 52 20 71 31 60 24 67 35 56)
(1 19 10 16 7 13 4 22)(2 24 11 21 8 18 5 15)(3 17 12 14 9 23 6 20)(25 95 34 92 31 89 28 86)(26 88 35 85 32 94 29 91)(27 93 36 90 33 87 30 96)(37 58 46 55 43 52 40 49)(38 51 47 60 44 57 41 54)(39 56 48 53 45 50 42 59)(61 74 70 83 67 80 64 77)(62 79 71 76 68 73 65 82)(63 84 72 81 69 78 66 75)

G:=sub<Sym(96)| (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96), (1,42,91,74,5,38,95,82,9,46,87,78)(2,47,92,79,6,43,96,75,10,39,88,83)(3,40,93,84,7,48,85,80,11,44,89,76)(4,45,94,77,8,41,86,73,12,37,90,81)(13,72,36,49,17,68,28,57,21,64,32,53)(14,65,25,54,18,61,29,50,22,69,33,58)(15,70,26,59,19,66,30,55,23,62,34,51)(16,63,27,52,20,71,31,60,24,67,35,56), (1,19,10,16,7,13,4,22)(2,24,11,21,8,18,5,15)(3,17,12,14,9,23,6,20)(25,95,34,92,31,89,28,86)(26,88,35,85,32,94,29,91)(27,93,36,90,33,87,30,96)(37,58,46,55,43,52,40,49)(38,51,47,60,44,57,41,54)(39,56,48,53,45,50,42,59)(61,74,70,83,67,80,64,77)(62,79,71,76,68,73,65,82)(63,84,72,81,69,78,66,75)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96), (1,42,91,74,5,38,95,82,9,46,87,78)(2,47,92,79,6,43,96,75,10,39,88,83)(3,40,93,84,7,48,85,80,11,44,89,76)(4,45,94,77,8,41,86,73,12,37,90,81)(13,72,36,49,17,68,28,57,21,64,32,53)(14,65,25,54,18,61,29,50,22,69,33,58)(15,70,26,59,19,66,30,55,23,62,34,51)(16,63,27,52,20,71,31,60,24,67,35,56), (1,19,10,16,7,13,4,22)(2,24,11,21,8,18,5,15)(3,17,12,14,9,23,6,20)(25,95,34,92,31,89,28,86)(26,88,35,85,32,94,29,91)(27,93,36,90,33,87,30,96)(37,58,46,55,43,52,40,49)(38,51,47,60,44,57,41,54)(39,56,48,53,45,50,42,59)(61,74,70,83,67,80,64,77)(62,79,71,76,68,73,65,82)(63,84,72,81,69,78,66,75) );

G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32,33,34,35,36),(37,38,39,40,41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80,81,82,83,84),(85,86,87,88,89,90,91,92,93,94,95,96)], [(1,42,91,74,5,38,95,82,9,46,87,78),(2,47,92,79,6,43,96,75,10,39,88,83),(3,40,93,84,7,48,85,80,11,44,89,76),(4,45,94,77,8,41,86,73,12,37,90,81),(13,72,36,49,17,68,28,57,21,64,32,53),(14,65,25,54,18,61,29,50,22,69,33,58),(15,70,26,59,19,66,30,55,23,62,34,51),(16,63,27,52,20,71,31,60,24,67,35,56)], [(1,19,10,16,7,13,4,22),(2,24,11,21,8,18,5,15),(3,17,12,14,9,23,6,20),(25,95,34,92,31,89,28,86),(26,88,35,85,32,94,29,91),(27,93,36,90,33,87,30,96),(37,58,46,55,43,52,40,49),(38,51,47,60,44,57,41,54),(39,56,48,53,45,50,42,59),(61,74,70,83,67,80,64,77),(62,79,71,76,68,73,65,82),(63,84,72,81,69,78,66,75)])

60 conjugacy classes

class 1 2A2B2C3A3B3C4A4B4C4D4E4F4G4H6A···6F6G6H6I8A8B8C8D8E8F8G8H12A···12H12I12J12K12L12M···12T24A···24H
order1222333444444446···66668888888812···121212121212···1224···24
size1111224111166662···24446666181818182···244446···66···6

60 irreducible representations

dim111111222222222222222444444
type+++++++--+-+++--
imageC1C2C2C2C4C8S3S3D4Q8Dic3D6M4(2)C3⋊C8Dic6D12C3⋊D4C4×S3S3×C8C8⋊S3C4.Dic3S32C3⋊D12C322Q8S3×Dic3S3×C3⋊C8D6.Dic3
kernelC12.81D12C6×C3⋊C8Dic3×C12C2×C324C8C6×Dic3C3×Dic3C2×C3⋊C8C4×Dic3C3×C12C3×C12C2×Dic3C2×C12C3×C6Dic3C12C12C12C2×C6C6C6C6C2×C4C4C4C22C2C2
# reps111148111122244222444111122

Matrix representation of C12.81D12 in GL6(𝔽73)

100000
010000
00464600
0027000
000010
000001
,
22480000
34510000
0072000
001100
0000721
0000720
,
22180000
34510000
0063000
00101000
0000720
0000721

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,46,27,0,0,0,0,46,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[22,34,0,0,0,0,48,51,0,0,0,0,0,0,72,1,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[22,34,0,0,0,0,18,51,0,0,0,0,0,0,63,10,0,0,0,0,0,10,0,0,0,0,0,0,72,72,0,0,0,0,0,1] >;

C12.81D12 in GAP, Magma, Sage, TeX

C_{12}._{81}D_{12}
% in TeX

G:=Group("C12.81D12");
// GroupNames label

G:=SmallGroup(288,219);
// by ID

G=gap.SmallGroup(288,219);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,141,36,100,1356,9414]);
// Polycyclic

G:=Group<a,b,c|a^12=b^12=1,c^2=a^9,b*a*b^-1=c*a*c^-1=a^5,c*b*c^-1=b^-1>;
// generators/relations

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