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

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

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

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

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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