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G = C6.Dic12order 288 = 25·32

1st non-split extension by C6 of Dic12 acting via Dic12/Dic6=C2

metabelian, supersoluble, monomial

Aliases: C6.6Dic12, C62.21D4, Dic6:1Dic3, (C3xC6).4Q16, C12.17(C4xS3), (C3xDic6):2C4, (C3xC12).36D4, (C2xC6).51D12, C4.2(S3xDic3), C6.9(C24:C2), C6.37(D6:C4), (C2xC12).274D6, (C6xDic6).3C2, (C2xDic6).1S3, (C3xC6).11SD16, C6.1(C3:Q16), C3:1(Q8:2Dic3), C12.32(C3:D4), C2.8(D6:Dic3), (C6xC12).28C22, C6.2(Q8:2S3), C3:3(C2.Dic12), C12.23(C2xDic3), C32:5(Q8:C4), C4.12(D6:S3), C12:Dic3.9C2, C6.7(C6.D4), C2.1(C32:3Q16), C2.2(C32:5SD16), C22.14(C3:D12), (C6xC3:C8).1C2, (C2xC4).54S32, (C2xC3:C8).1S3, (C3xC12).26(C2xC4), (C2xC6).32(C3:D4), (C3xC6).33(C22:C4), SmallGroup(288,214)

Series: Derived Chief Lower central Upper central

C1C3xC12 — C6.Dic12
C1C3C32C3xC6C3xC12C6xC12C6xDic6 — C6.Dic12
C32C3xC6C3xC12 — C6.Dic12
C1C22C2xC4

Generators and relations for C6.Dic12
 G = < a,b,c | a6=b24=1, c2=b12, bab-1=a-1, ac=ca, cbc-1=a3b-1 >

Subgroups: 330 in 97 conjugacy classes, 42 normal (38 characteristic)
C1, C2, C3, C3, C4, C4, C22, C6, C6, C8, C2xC4, C2xC4, Q8, C32, Dic3, C12, C12, C2xC6, C2xC6, C4:C4, C2xC8, C2xQ8, C3xC6, C3:C8, C24, Dic6, Dic6, C2xDic3, C2xC12, C2xC12, C3xQ8, Q8:C4, C3xDic3, C3:Dic3, C3xC12, C62, C2xC3:C8, C4:Dic3, C2xC24, C2xDic6, C6xQ8, C3xC3:C8, C3xDic6, C3xDic6, C6xDic3, C2xC3:Dic3, C6xC12, C2.Dic12, Q8:2Dic3, C6xC3:C8, C12:Dic3, C6xDic6, C6.Dic12
Quotients: C1, C2, C4, C22, S3, C2xC4, D4, Dic3, D6, C22:C4, SD16, Q16, C4xS3, D12, C2xDic3, C3:D4, Q8:C4, S32, C24:C2, Dic12, D6:C4, Q8:2S3, C3:Q16, C6.D4, S3xDic3, D6:S3, C3:D12, C2.Dic12, Q8:2Dic3, C32:5SD16, C32:3Q16, D6:Dic3, C6.Dic12

Smallest permutation representation of C6.Dic12
On 96 points
Generators in S96
(1 67 9 51 17 59)(2 60 18 52 10 68)(3 69 11 53 19 61)(4 62 20 54 12 70)(5 71 13 55 21 63)(6 64 22 56 14 72)(7 49 15 57 23 65)(8 66 24 58 16 50)(25 96 33 80 41 88)(26 89 42 81 34 73)(27 74 35 82 43 90)(28 91 44 83 36 75)(29 76 37 84 45 92)(30 93 46 85 38 77)(31 78 39 86 47 94)(32 95 48 87 40 79)
(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 85 13 73)(2 29 14 41)(3 83 15 95)(4 27 16 39)(5 81 17 93)(6 25 18 37)(7 79 19 91)(8 47 20 35)(9 77 21 89)(10 45 22 33)(11 75 23 87)(12 43 24 31)(26 67 38 55)(28 65 40 53)(30 63 42 51)(32 61 44 49)(34 59 46 71)(36 57 48 69)(50 86 62 74)(52 84 64 96)(54 82 66 94)(56 80 68 92)(58 78 70 90)(60 76 72 88)

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

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

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

48 conjugacy classes

class 1 2A2B2C3A3B3C4A4B4C4D4E4F6A···6F6G6H6I8A8B8C8D12A12B12C12D12E···12J12K12L12M12N24A···24H
order12223334444446···666688881212121212···121212121224···24
size111122422121236362···2444666622224···4121212126···6

48 irreducible representations

dim111112222222222222244444444
type++++++++-+-+-++---++-
imageC1C2C2C2C4S3S3D4D4Dic3D6SD16Q16C4xS3C3:D4D12C3:D4C24:C2Dic12S32Q8:2S3C3:Q16S3xDic3D6:S3C3:D12C32:5SD16C32:3Q16
kernelC6.Dic12C6xC3:C8C12:Dic3C6xDic6C3xDic6C2xC3:C8C2xDic6C3xC12C62Dic6C2xC12C3xC6C3xC6C12C12C2xC6C2xC6C6C6C2xC4C6C6C4C4C22C2C2
# reps111141111222224224411111122

Matrix representation of C6.Dic12 in GL6(F73)

7200000
0720000
0017200
001000
000010
000001
,
59550000
8260000
0002700
0027000
0000667
00006659
,
24250000
47490000
001000
000100
0000714
0000766

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,1,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[59,8,0,0,0,0,55,26,0,0,0,0,0,0,0,27,0,0,0,0,27,0,0,0,0,0,0,0,66,66,0,0,0,0,7,59],[24,47,0,0,0,0,25,49,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,7,7,0,0,0,0,14,66] >;

C6.Dic12 in GAP, Magma, Sage, TeX

C_6.{\rm Dic}_{12}
% in TeX

G:=Group("C6.Dic12");
// GroupNames label

G:=SmallGroup(288,214);
// by ID

G=gap.SmallGroup(288,214);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,85,92,422,100,1356,9414]);
// Polycyclic

G:=Group<a,b,c|a^6=b^24=1,c^2=b^12,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a^3*b^-1>;
// generators/relations

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