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## G = Dic6.C8order 192 = 26·3

### 2nd non-split extension by Dic6 of C8 acting via C8/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — Dic6.C8
 Chief series C1 — C3 — C6 — C12 — C24 — C2×C24 — C8○D12 — Dic6.C8
 Lower central C3 — C6 — C12 — Dic6.C8
 Upper central C1 — C8 — C2×C8 — M5(2)

Generators and relations for Dic6.C8
G = < a,b,c | a12=1, b2=c8=a6, bab-1=a-1, cac-1=a7, cbc-1=a9b >

Smallest permutation representation of Dic6.C8
On 96 points
Generators in S96
(1 33 56 96 22 73 9 41 64 88 30 65)(2 42 57 89 23 66 10 34 49 81 31 74)(3 35 58 82 24 75 11 43 50 90 32 67)(4 44 59 91 25 68 12 36 51 83 17 76)(5 37 60 84 26 77 13 45 52 92 18 69)(6 46 61 93 27 70 14 38 53 85 19 78)(7 39 62 86 28 79 15 47 54 94 20 71)(8 48 63 95 29 72 16 40 55 87 21 80)
(1 5 9 13)(2 85 10 93)(3 7 11 15)(4 87 12 95)(6 89 14 81)(8 91 16 83)(17 80 25 72)(18 64 26 56)(19 66 27 74)(20 50 28 58)(21 68 29 76)(22 52 30 60)(23 70 31 78)(24 54 32 62)(33 69 41 77)(34 61 42 53)(35 71 43 79)(36 63 44 55)(37 73 45 65)(38 49 46 57)(39 75 47 67)(40 51 48 59)(82 94 90 86)(84 96 92 88)
(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)

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

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

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

48 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 6A 6B 8A 8B 8C 8D 8E 8F 8G 8H 12A 12B 12C 16A 16B 16C 16D 16E ··· 16L 24A 24B 24C 24D 24E 24F 48A ··· 48H order 1 2 2 2 3 4 4 4 4 6 6 8 8 8 8 8 8 8 8 12 12 12 16 16 16 16 16 ··· 16 24 24 24 24 24 24 48 ··· 48 size 1 1 2 12 2 1 1 2 12 2 4 1 1 1 1 2 2 12 12 2 2 4 4 4 4 4 6 ··· 6 2 2 2 2 4 4 4 ··· 4

48 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 type + + + + + + + + image C1 C2 C2 C2 C4 C4 C8 C8 S3 D4 D6 M4(2) D12 C3⋊D4 C4×S3 S3×C8 C8⋊S3 D4.C8 Dic6.C8 kernel Dic6.C8 C2×C3⋊C16 C3×M5(2) C8○D12 C4.Dic3 C4○D12 Dic6 D12 M5(2) C24 C2×C8 C2×C6 C8 C8 C2×C4 C4 C22 C3 C1 # reps 1 1 1 1 2 2 4 4 1 2 1 2 2 2 2 4 4 8 4

Matrix representation of Dic6.C8 in GL4(𝔽97) generated by

 96 1 0 0 96 0 0 0 0 0 96 3 0 0 64 1
,
 96 1 0 0 0 1 0 0 0 0 22 0 0 0 47 75
,
 64 0 0 0 0 64 0 0 0 0 0 26 0 0 92 0
G:=sub<GL(4,GF(97))| [96,96,0,0,1,0,0,0,0,0,96,64,0,0,3,1],[96,0,0,0,1,1,0,0,0,0,22,47,0,0,0,75],[64,0,0,0,0,64,0,0,0,0,0,92,0,0,26,0] >;

Dic6.C8 in GAP, Magma, Sage, TeX

{\rm Dic}_6.C_8
% in TeX

G:=Group("Dic6.C8");
// GroupNames label

G:=SmallGroup(192,74);
// by ID

G=gap.SmallGroup(192,74);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,141,36,100,1123,570,136,102,6278]);
// Polycyclic

G:=Group<a,b,c|a^12=1,b^2=c^8=a^6,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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