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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: D43Dic6, C12.50D8, C42.44D6, (C3×D4)⋊3Q8, (C4×D4).2S3, C6.50(C2×D8), C4⋊C4.238D6, C12⋊C817C2, (D4×C12).2C2, C34(D4⋊Q8), (C2×C12).57D4, C12.25(C2×Q8), C4.9(C2×Dic6), C4.23(D4⋊S3), C122Q814C2, (C2×D4).185D6, C6.Q1630C2, C4.59(C4○D12), C12.45(C4○D4), (C4×C12).78C22, D4⋊Dic3.7C2, C6.61(C22⋊Q8), (C2×C12).332C23, C2.7(Q8.14D6), (C6×D4).227C22, C6.107(C8.C22), C4⋊Dic3.136C22, C2.12(C12.48D4), C2.6(C2×D4⋊S3), (C2×C6).463(C2×D4), (C2×C3⋊C8).89C22, (C2×C4).243(C3⋊D4), (C3×C4⋊C4).269C22, (C2×C4).432(C22×S3), C22.146(C2×C3⋊D4), SmallGroup(192,566)

Series: Derived Chief Lower central Upper central

C1C2×C12 — C12.50D8
C1C3C6C12C2×C12C4⋊Dic3C122Q8 — C12.50D8
C3C6C2×C12 — C12.50D8
C1C22C42C4×D4

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 [×3], C2 [×2], C3, C4 [×2], C4 [×2], C4 [×4], C22, C22 [×4], C6 [×3], C6 [×2], C8 [×2], C2×C4 [×3], C2×C4 [×5], D4 [×2], D4, Q8 [×2], C23, Dic3 [×2], C12 [×2], C12 [×2], C12 [×2], C2×C6, C2×C6 [×4], C42, C22⋊C4, C4⋊C4, C4⋊C4 [×3], C2×C8 [×2], C22×C4, C2×D4, C2×Q8, C3⋊C8 [×2], Dic6 [×2], C2×Dic3 [×2], C2×C12 [×3], C2×C12 [×3], C3×D4 [×2], C3×D4, C22×C6, D4⋊C4 [×2], C4⋊C8, C2.D8 [×2], C4×D4, C4⋊Q8, C2×C3⋊C8 [×2], C4⋊Dic3 [×2], C4⋊Dic3, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C2×Dic6, C22×C12, C6×D4, D4⋊Q8, C12⋊C8, C6.Q16 [×2], D4⋊Dic3 [×2], C122Q8, D4×C12, C12.50D8
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], Q8 [×2], C23, D6 [×3], D8 [×2], C2×D4, C2×Q8, C4○D4, Dic6 [×2], C3⋊D4 [×2], C22×S3, C22⋊Q8, C2×D8, C8.C22, D4⋊S3 [×2], C2×Dic6, C4○D12, C2×C3⋊D4, D4⋊Q8, C12.48D4, C2×D4⋊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 18 48 35 62 75 96 54)(2 17 37 34 63 74 85 53)(3 16 38 33 64 73 86 52)(4 15 39 32 65 84 87 51)(5 14 40 31 66 83 88 50)(6 13 41 30 67 82 89 49)(7 24 42 29 68 81 90 60)(8 23 43 28 69 80 91 59)(9 22 44 27 70 79 92 58)(10 21 45 26 71 78 93 57)(11 20 46 25 72 77 94 56)(12 19 47 36 61 76 95 55)
(1 60 7 54)(2 59 8 53)(3 58 9 52)(4 57 10 51)(5 56 11 50)(6 55 12 49)(13 89 19 95)(14 88 20 94)(15 87 21 93)(16 86 22 92)(17 85 23 91)(18 96 24 90)(25 72 31 66)(26 71 32 65)(27 70 33 64)(28 69 34 63)(29 68 35 62)(30 67 36 61)(37 80 43 74)(38 79 44 73)(39 78 45 84)(40 77 46 83)(41 76 47 82)(42 75 48 81)

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

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

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

39 conjugacy classes

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

39 irreducible representations

dim11111122222222222444
type++++++++-++++--+-
imageC1C2C2C2C2C2S3D4Q8D6D6D6D8C4○D4C3⋊D4Dic6C4○D12C8.C22D4⋊S3Q8.14D6
kernelC12.50D8C12⋊C8C6.Q16D4⋊Dic3C122Q8D4×C12C4×D4C2×C12C3×D4C42C4⋊C4C2×D4C12C12C2×C4D4C4C6C4C2
# 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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