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## G = C12.77D12order 288 = 25·32

### 8th non-split extension by C12 of D12 acting via D12/D6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C12.77D12
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C6×C12 — S3×C2×C12 — C12.77D12
 Lower central C32 — C3×C6 — C12.77D12
 Upper central C1 — C2×C4

Generators and relations for C12.77D12
G = < a,b,c | a12=1, b12=a6, c2=a9, bab-1=cac-1=a5, cbc-1=a3b11 >

Subgroups: 306 in 111 conjugacy classes, 46 normal (42 characteristic)
C1, C2 [×3], C2 [×2], C3 [×2], C3, C4 [×2], C4, C22, C22 [×4], S3 [×2], C6 [×6], C6 [×5], C8 [×2], C2×C4, C2×C4 [×3], C23, C32, Dic3, C12 [×4], C12 [×3], D6 [×2], D6 [×2], C2×C6 [×2], C2×C6 [×5], C2×C8 [×2], C22×C4, C3×S3 [×2], C3×C6 [×3], C3⋊C8 [×5], C24, C4×S3 [×2], C2×Dic3, C2×C12 [×2], C2×C12 [×4], C22×S3, C22×C6, C22⋊C8, C3×Dic3, C3×C12 [×2], S3×C6 [×2], S3×C6 [×2], C62, C2×C3⋊C8, C2×C3⋊C8 [×3], C2×C24, S3×C2×C4, C22×C12, C3×C3⋊C8, C324C8, S3×C12 [×2], C6×Dic3, C6×C12, S3×C2×C6, D6⋊C8, C12.55D4, C6×C3⋊C8, C2×C324C8, S3×C2×C12, C12.77D12
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×2], C8 [×2], C2×C4, D4 [×2], Dic3 [×2], D6 [×2], C22⋊C4, C2×C8, M4(2), C3⋊C8 [×2], C4×S3, D12, C2×Dic3, C3⋊D4 [×3], C22⋊C8, S32, S3×C8, C8⋊S3, C2×C3⋊C8, C4.Dic3, D6⋊C4, C6.D4, S3×Dic3, D6⋊S3, C3⋊D12, D6⋊C8, C12.55D4, S3×C3⋊C8, D6.Dic3, D6⋊Dic3, C12.77D12

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

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

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

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

60 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 4A 4B 4C 4D 4E 4F 6A ··· 6F 6G 6H 6I 6J 6K 6L 6M 8A 8B 8C 8D 8E 8F 8G 8H 12A ··· 12H 12I 12J 12K 12L 12M 12N 12O 12P 24A ··· 24H order 1 2 2 2 2 2 3 3 3 4 4 4 4 4 4 6 ··· 6 6 6 6 6 6 6 6 8 8 8 8 8 8 8 8 12 ··· 12 12 12 12 12 12 12 12 12 24 ··· 24 size 1 1 1 1 6 6 2 2 4 1 1 1 1 6 6 2 ··· 2 4 4 4 6 6 6 6 6 6 6 6 18 18 18 18 2 ··· 2 4 4 4 4 6 6 6 6 6 ··· 6

60 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + - + - + + - + - image C1 C2 C2 C2 C4 C4 C8 S3 S3 D4 Dic3 D6 Dic3 M4(2) D12 C3⋊D4 C3⋊C8 C4×S3 S3×C8 C8⋊S3 C4.Dic3 S32 D6⋊S3 C3⋊D12 S3×Dic3 S3×C3⋊C8 D6.Dic3 kernel C12.77D12 C6×C3⋊C8 C2×C32⋊4C8 S3×C2×C12 C6×Dic3 S3×C2×C6 S3×C6 C2×C3⋊C8 S3×C2×C4 C3×C12 C2×Dic3 C2×C12 C22×S3 C3×C6 C12 C12 D6 C2×C6 C6 C6 C6 C2×C4 C4 C4 C22 C2 C2 # reps 1 1 1 1 2 2 8 1 1 2 1 2 1 2 2 6 4 2 4 4 4 1 1 1 1 2 2

Matrix representation of C12.77D12 in GL6(𝔽73)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 46 0 0 0 0 0 0 46 0 0 0 0 0 0 0 72 0 0 0 0 1 72
,
 0 1 0 0 0 0 72 0 0 0 0 0 0 0 51 22 0 0 0 0 51 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 30 14 0 0 0 0 14 43 0 0 0 0 0 0 0 51 0 0 0 0 51 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,46,0,0,0,0,0,0,46,0,0,0,0,0,0,0,1,0,0,0,0,72,72],[0,72,0,0,0,0,1,0,0,0,0,0,0,0,51,51,0,0,0,0,22,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[30,14,0,0,0,0,14,43,0,0,0,0,0,0,0,51,0,0,0,0,51,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C12.77D12 in GAP, Magma, Sage, TeX

C_{12}._{77}D_{12}
% in TeX

G:=Group("C12.77D12");
// GroupNames label

G:=SmallGroup(288,204);
// by ID

G=gap.SmallGroup(288,204);
# by ID

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

G:=Group<a,b,c|a^12=1,b^12=a^6,c^2=a^9,b*a*b^-1=c*a*c^-1=a^5,c*b*c^-1=a^3*b^11>;
// generators/relations

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