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G = C127D8order 192 = 26·3

1st semidirect product of C12 and D8 acting via D8/D4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C127D8, D41D12, C42.49D6, (C3×D4)⋊8D4, (C4×D4)⋊3S3, C43(D4⋊S3), (D4×C12)⋊3C2, C33(C4⋊D8), C6.52(C2×D8), C4⋊D128C2, C4⋊C4.243D6, C12⋊C823C2, (C2×C12).60D4, C4.13(C2×D12), C12.17(C2×D4), (C2×D4).190D6, C4.9(C4○D12), C6.D830C2, C12.50(C4○D4), C2.8(D4⋊D6), C6.64(C4⋊D4), (C4×C12).86C22, C2.12(C127D4), C6.109(C8⋊C22), (C2×C12).337C23, (C6×D4).232C22, (C2×D12).93C22, (C2×D4⋊S3)⋊7C2, C2.7(C2×D4⋊S3), (C2×C6).468(C2×D4), (C2×C3⋊C8).93C22, (C2×C4).245(C3⋊D4), (C3×C4⋊C4).274C22, (C2×C4).437(C22×S3), C22.149(C2×C3⋊D4), SmallGroup(192,574)

Series: Derived Chief Lower central Upper central

C1C2×C12 — C127D8
C1C3C6C12C2×C12C2×D12C4⋊D12 — C127D8
C3C6C2×C12 — C127D8
C1C22C42C4×D4

Generators and relations for C127D8
 G = < a,b,c | a12=b8=c2=1, bab-1=cac=a-1, cbc=b-1 >

Subgroups: 504 in 140 conjugacy classes, 47 normal (31 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×2], C4 [×2], C4 [×2], C22, C22 [×10], S3 [×2], C6 [×3], C6 [×2], C8 [×2], C2×C4 [×3], C2×C4 [×3], D4 [×2], D4 [×9], C23 [×3], C12 [×2], C12 [×2], C12 [×2], D6 [×6], C2×C6, C2×C6 [×4], C42, C22⋊C4, C4⋊C4, C2×C8 [×2], D8 [×4], C22×C4, C2×D4, C2×D4 [×4], C3⋊C8 [×2], D12 [×8], C2×C12 [×3], C2×C12 [×3], C3×D4 [×2], C3×D4, C22×S3 [×2], C22×C6, D4⋊C4 [×2], C4⋊C8, C4×D4, C41D4, C2×D8 [×2], C2×C3⋊C8 [×2], D4⋊S3 [×4], C4×C12, C3×C22⋊C4, C3×C4⋊C4, C2×D12 [×2], C2×D12 [×2], C22×C12, C6×D4, C4⋊D8, C12⋊C8, C6.D8 [×2], C4⋊D12, C2×D4⋊S3 [×2], D4×C12, C127D8
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×4], C23, D6 [×3], D8 [×2], C2×D4 [×2], C4○D4, D12 [×2], C3⋊D4 [×2], C22×S3, C4⋊D4, C2×D8, C8⋊C22, D4⋊S3 [×2], C2×D12, C4○D12, C2×C3⋊D4, C4⋊D8, C127D4, C2×D4⋊S3, D4⋊D6, C127D8

Smallest permutation representation of C127D8
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 89 35 43 54 67 16 77)(2 88 36 42 55 66 17 76)(3 87 25 41 56 65 18 75)(4 86 26 40 57 64 19 74)(5 85 27 39 58 63 20 73)(6 96 28 38 59 62 21 84)(7 95 29 37 60 61 22 83)(8 94 30 48 49 72 23 82)(9 93 31 47 50 71 24 81)(10 92 32 46 51 70 13 80)(11 91 33 45 52 69 14 79)(12 90 34 44 53 68 15 78)
(2 12)(3 11)(4 10)(5 9)(6 8)(13 26)(14 25)(15 36)(16 35)(17 34)(18 33)(19 32)(20 31)(21 30)(22 29)(23 28)(24 27)(37 61)(38 72)(39 71)(40 70)(41 69)(42 68)(43 67)(44 66)(45 65)(46 64)(47 63)(48 62)(49 59)(50 58)(51 57)(52 56)(53 55)(73 93)(74 92)(75 91)(76 90)(77 89)(78 88)(79 87)(80 86)(81 85)(82 96)(83 95)(84 94)

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

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

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

39 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A4B4C4D4E4F4G6A6B6C6D6E6F6G8A8B8C8D12A12B12C12D12E···12L
order1222222234444444666666688881212121212···12
size11114424242222244422244441212121222224···4

39 irreducible representations

dim11111122222222222444
type+++++++++++++++++
imageC1C2C2C2C2C2S3D4D4D6D6D6D8C4○D4C3⋊D4D12C4○D12C8⋊C22D4⋊S3D4⋊D6
kernelC127D8C12⋊C8C6.D8C4⋊D12C2×D4⋊S3D4×C12C4×D4C2×C12C3×D4C42C4⋊C4C2×D4C12C12C2×C4D4C4C6C4C2
# reps11212112211142444122

Matrix representation of C127D8 in GL6(𝔽73)

7200000
0720000
00271900
0004600
0000721
0000720
,
57160000
57570000
0072200
0072100
0000072
0000720
,
100000
0720000
001000
0017200
000001
000010

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,27,0,0,0,0,0,19,46,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[57,57,0,0,0,0,16,57,0,0,0,0,0,0,72,72,0,0,0,0,2,1,0,0,0,0,0,0,0,72,0,0,0,0,72,0],[1,0,0,0,0,0,0,72,0,0,0,0,0,0,1,1,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C127D8 in GAP, Magma, Sage, TeX

C_{12}\rtimes_7D_8
% in TeX

G:=Group("C12:7D8");
// GroupNames label

G:=SmallGroup(192,574);
// by ID

G=gap.SmallGroup(192,574);
# by ID

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

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

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