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G = C12.50D8order 192 = 26·3

4th non-split extension by C12 of D8 acting via D8/D4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D4:3Dic6, C12.50D8, C42.44D6, (C3xD4):3Q8, (C4xD4).2S3, C6.50(C2xD8), C4:C4.238D6, C12:C8:17C2, (D4xC12).2C2, C3:4(D4:Q8), (C2xC12).57D4, C12.25(C2xQ8), C4.9(C2xDic6), C4.23(D4:S3), C12:2Q8:14C2, (C2xD4).185D6, C6.Q16:30C2, C4.59(C4oD12), C12.45(C4oD4), (C4xC12).78C22, D4:Dic3.7C2, C6.61(C22:Q8), (C2xC12).332C23, C2.7(Q8.14D6), (C6xD4).227C22, C6.107(C8.C22), C4:Dic3.136C22, C2.12(C12.48D4), C2.6(C2xD4:S3), (C2xC6).463(C2xD4), (C2xC3:C8).89C22, (C2xC4).243(C3:D4), (C3xC4:C4).269C22, (C2xC4).432(C22xS3), C22.146(C2xC3:D4), SmallGroup(192,566)

Series: Derived Chief Lower central Upper central

C1C2xC12 — C12.50D8
C1C3C6C12C2xC12C4:Dic3C12:2Q8 — C12.50D8
C3C6C2xC12 — C12.50D8
C1C22C42C4xD4

Generators and relations for C12.50D8
 G = < a,b,c | a12=b8=1, c2=a6, bab-1=cac-1=a-1, cbc-1=a6b-1 >

Subgroups: 280 in 108 conjugacy classes, 47 normal (31 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, C6, C6, C8, C2xC4, C2xC4, D4, D4, Q8, C23, Dic3, C12, C12, C12, C2xC6, C2xC6, C42, C22:C4, C4:C4, C4:C4, C2xC8, C22xC4, C2xD4, C2xQ8, C3:C8, Dic6, C2xDic3, C2xC12, C2xC12, C3xD4, C3xD4, C22xC6, D4:C4, C4:C8, C2.D8, C4xD4, C4:Q8, C2xC3:C8, C4:Dic3, C4:Dic3, C4xC12, C3xC22:C4, C3xC4:C4, C2xDic6, C22xC12, C6xD4, D4:Q8, C12:C8, C6.Q16, D4:Dic3, C12:2Q8, D4xC12, C12.50D8
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, D8, C2xD4, C2xQ8, C4oD4, Dic6, C3:D4, C22xS3, C22:Q8, C2xD8, C8.C22, D4:S3, C2xDic6, C4oD12, C2xC3:D4, D4:Q8, C12.48D4, C2xD4:S3, Q8.14D6, C12.50D8

Smallest permutation representation of C12.50D8
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 16 32 42 66 51 88 79)(2 15 33 41 67 50 89 78)(3 14 34 40 68 49 90 77)(4 13 35 39 69 60 91 76)(5 24 36 38 70 59 92 75)(6 23 25 37 71 58 93 74)(7 22 26 48 72 57 94 73)(8 21 27 47 61 56 95 84)(9 20 28 46 62 55 96 83)(10 19 29 45 63 54 85 82)(11 18 30 44 64 53 86 81)(12 17 31 43 65 52 87 80)
(1 73 7 79)(2 84 8 78)(3 83 9 77)(4 82 10 76)(5 81 11 75)(6 80 12 74)(13 91 19 85)(14 90 20 96)(15 89 21 95)(16 88 22 94)(17 87 23 93)(18 86 24 92)(25 52 31 58)(26 51 32 57)(27 50 33 56)(28 49 34 55)(29 60 35 54)(30 59 36 53)(37 71 43 65)(38 70 44 64)(39 69 45 63)(40 68 46 62)(41 67 47 61)(42 66 48 72)

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

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

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

39 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F4G4H4I6A6B6C6D6E6F6G8A8B8C8D12A12B12C12D12E···12L
order1222223444444444666666688881212121212···12
size11114422222444242422244441212121222224···4

39 irreducible representations

dim11111122222222222444
type++++++++-++++--+-
imageC1C2C2C2C2C2S3D4Q8D6D6D6D8C4oD4C3:D4Dic6C4oD12C8.C22D4:S3Q8.14D6
kernelC12.50D8C12:C8C6.Q16D4:Dic3C12:2Q8D4xC12C4xD4C2xC12C3xD4C42C4:C4C2xD4C12C12C2xC4D4C4C6C4C2
# reps11221112211142444122

Matrix representation of C12.50D8 in GL4(F73) generated by

72000
07200
0030
003349
,
165700
161600
00327
005241
,
165700
575700
00327
001041
G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,3,33,0,0,0,49],[16,16,0,0,57,16,0,0,0,0,32,52,0,0,7,41],[16,57,0,0,57,57,0,0,0,0,32,10,0,0,7,41] >;

C12.50D8 in GAP, Magma, Sage, TeX

C_{12}._{50}D_8
% in TeX

G:=Group("C12.50D8");
// GroupNames label

G:=SmallGroup(192,566);
// by ID

G=gap.SmallGroup(192,566);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,253,120,254,1123,297,136,6278]);
// Polycyclic

G:=Group<a,b,c|a^12=b^8=1,c^2=a^6,b*a*b^-1=c*a*c^-1=a^-1,c*b*c^-1=a^6*b^-1>;
// generators/relations

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