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## G = Dic12.C4order 192 = 26·3

### 3rd non-split extension by Dic12 of C4 acting via C4/C2=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C24 — Dic12.C4
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×C24 — C4○D24 — Dic12.C4
 Lower central C3 — C6 — C12 — C24 — Dic12.C4
 Upper central C1 — C4 — C2×C4 — C2×C8 — C8.C4

Generators and relations for Dic12.C4
G = < a,b,c | a24=1, b2=c4=a12, bab-1=a-1, cac-1=a7, cbc-1=a9b >

Smallest permutation representation of Dic12.C4
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 33 13 45)(2 32 14 44)(3 31 15 43)(4 30 16 42)(5 29 17 41)(6 28 18 40)(7 27 19 39)(8 26 20 38)(9 25 21 37)(10 48 22 36)(11 47 23 35)(12 46 24 34)(49 73 61 85)(50 96 62 84)(51 95 63 83)(52 94 64 82)(53 93 65 81)(54 92 66 80)(55 91 67 79)(56 90 68 78)(57 89 69 77)(58 88 70 76)(59 87 71 75)(60 86 72 74)
(1 95 33 54 13 83 45 66)(2 78 34 61 14 90 46 49)(3 85 35 68 15 73 47 56)(4 92 36 51 16 80 48 63)(5 75 37 58 17 87 25 70)(6 82 38 65 18 94 26 53)(7 89 39 72 19 77 27 60)(8 96 40 55 20 84 28 67)(9 79 41 62 21 91 29 50)(10 86 42 69 22 74 30 57)(11 93 43 52 23 81 31 64)(12 76 44 59 24 88 32 71)

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

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

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

36 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 6A 6B 8A 8B 8C 8D 8E 8F 12A 12B 12C 16A ··· 16H 24A 24B 24C 24D 24E 24F 24G 24H order 1 2 2 2 3 4 4 4 4 6 6 8 8 8 8 8 8 12 12 12 16 ··· 16 24 24 24 24 24 24 24 24 size 1 1 2 24 2 1 1 2 24 2 4 2 2 2 2 8 8 2 2 4 6 ··· 6 4 4 4 4 8 8 8 8

36 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + image C1 C2 C2 C2 C4 C4 S3 D4 D4 D6 D8 SD16 C4×S3 D12 C3⋊D4 D8.C4 D4⋊S3 Q8⋊2S3 Dic12.C4 kernel Dic12.C4 C2×C3⋊C16 C3×C8.C4 C4○D24 D24 Dic12 C8.C4 C24 C2×C12 C2×C8 C12 C2×C6 C8 C8 C2×C4 C3 C4 C22 C1 # reps 1 1 1 1 2 2 1 1 1 1 2 2 2 2 2 8 1 1 4

Matrix representation of Dic12.C4 in GL4(𝔽97) generated by

 0 1 0 0 96 96 0 0 0 0 0 14 0 0 90 14
,
 96 0 0 0 1 1 0 0 0 0 75 0 0 0 75 22
,
 22 0 0 0 0 22 0 0 0 0 61 78 0 0 3 36
G:=sub<GL(4,GF(97))| [0,96,0,0,1,96,0,0,0,0,0,90,0,0,14,14],[96,1,0,0,0,1,0,0,0,0,75,75,0,0,0,22],[22,0,0,0,0,22,0,0,0,0,61,3,0,0,78,36] >;

Dic12.C4 in GAP, Magma, Sage, TeX

{\rm Dic}_{12}.C_4
% in TeX

G:=Group("Dic12.C4");
// GroupNames label

G:=SmallGroup(192,56);
// by ID

G=gap.SmallGroup(192,56);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,141,36,184,675,346,192,1684,851,102,6278]);
// Polycyclic

G:=Group<a,b,c|a^24=1,b^2=c^4=a^12,b*a*b^-1=a^-1,c*a*c^-1=a^7,c*b*c^-1=a^9*b>;
// generators/relations

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