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## G = C2.Dic12order 96 = 25·3

### 1st central extension by C2 of Dic12

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C2.Dic12
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C4⋊Dic3 — C2.Dic12
 Lower central C3 — C6 — C12 — C2.Dic12
 Upper central C1 — C22 — C2×C4 — C2×C8

Generators and relations for C2.Dic12
G = < a,b,c | a6=b8=1, c2=a3, ab=ba, cac-1=a-1, cbc-1=a3b3 >

Character table of C2.Dic12

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 4F 6A 6B 6C 8A 8B 8C 8D 12A 12B 12C 12D 24A 24B 24C 24D 24E 24F 24G 24H size 1 1 1 1 2 2 2 12 12 12 12 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 1 1 1 -1 1 -1 1 1 1 -1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ3 1 1 1 1 1 1 1 -1 1 -1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ4 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ5 1 -1 1 -1 1 1 -1 1 i -1 -i -1 1 -1 i i -i -i 1 -1 -1 1 i -i -i i -i -i i i linear of order 4 ρ6 1 -1 1 -1 1 1 -1 1 -i -1 i -1 1 -1 -i -i i i 1 -1 -1 1 -i i i -i i i -i -i linear of order 4 ρ7 1 -1 1 -1 1 1 -1 -1 -i 1 i -1 1 -1 i i -i -i 1 -1 -1 1 i -i -i i -i -i i i linear of order 4 ρ8 1 -1 1 -1 1 1 -1 -1 i 1 -i -1 1 -1 -i -i i i 1 -1 -1 1 -i i i -i i i -i -i linear of order 4 ρ9 2 2 2 2 -1 2 2 0 0 0 0 -1 -1 -1 2 2 2 2 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ10 2 2 2 2 2 -2 -2 0 0 0 0 2 2 2 0 0 0 0 -2 -2 -2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 2 2 -1 2 2 0 0 0 0 -1 -1 -1 -2 -2 -2 -2 -1 -1 -1 -1 1 1 1 1 1 1 1 1 orthogonal lifted from D6 ρ12 2 -2 2 -2 2 -2 2 0 0 0 0 -2 2 -2 0 0 0 0 -2 2 2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ13 2 2 2 2 -1 -2 -2 0 0 0 0 -1 -1 -1 0 0 0 0 1 1 1 1 -√3 -√3 -√3 √3 √3 √3 -√3 √3 orthogonal lifted from D12 ρ14 2 2 2 2 -1 -2 -2 0 0 0 0 -1 -1 -1 0 0 0 0 1 1 1 1 √3 √3 √3 -√3 -√3 -√3 √3 -√3 orthogonal lifted from D12 ρ15 2 -2 -2 2 2 0 0 0 0 0 0 -2 -2 2 -√2 √2 -√2 √2 0 0 0 0 √2 √2 -√2 √2 √2 -√2 -√2 -√2 symplectic lifted from Q16, Schur index 2 ρ16 2 -2 -2 2 2 0 0 0 0 0 0 -2 -2 2 √2 -√2 √2 -√2 0 0 0 0 -√2 -√2 √2 -√2 -√2 √2 √2 √2 symplectic lifted from Q16, Schur index 2 ρ17 2 -2 -2 2 -1 0 0 0 0 0 0 1 1 -1 -√2 √2 -√2 √2 -√3 -√3 √3 √3 ζ83ζ3+ζ83+ζ8ζ3 ζ83ζ3+ζ83+ζ8ζ3 ζ83ζ32+ζ8ζ32+ζ8 ζ87ζ3+ζ85ζ3+ζ85 ζ87ζ3+ζ85ζ3+ζ85 ζ87ζ32+ζ87+ζ85ζ32 ζ83ζ32+ζ8ζ32+ζ8 ζ87ζ32+ζ87+ζ85ζ32 symplectic lifted from Dic12, Schur index 2 ρ18 2 -2 -2 2 -1 0 0 0 0 0 0 1 1 -1 √2 -√2 √2 -√2 -√3 -√3 √3 √3 ζ83ζ32+ζ8ζ32+ζ8 ζ83ζ32+ζ8ζ32+ζ8 ζ83ζ3+ζ83+ζ8ζ3 ζ87ζ32+ζ87+ζ85ζ32 ζ87ζ32+ζ87+ζ85ζ32 ζ87ζ3+ζ85ζ3+ζ85 ζ83ζ3+ζ83+ζ8ζ3 ζ87ζ3+ζ85ζ3+ζ85 symplectic lifted from Dic12, Schur index 2 ρ19 2 -2 -2 2 -1 0 0 0 0 0 0 1 1 -1 √2 -√2 √2 -√2 √3 √3 -√3 -√3 ζ87ζ32+ζ87+ζ85ζ32 ζ87ζ32+ζ87+ζ85ζ32 ζ87ζ3+ζ85ζ3+ζ85 ζ83ζ32+ζ8ζ32+ζ8 ζ83ζ32+ζ8ζ32+ζ8 ζ83ζ3+ζ83+ζ8ζ3 ζ87ζ3+ζ85ζ3+ζ85 ζ83ζ3+ζ83+ζ8ζ3 symplectic lifted from Dic12, Schur index 2 ρ20 2 -2 -2 2 -1 0 0 0 0 0 0 1 1 -1 -√2 √2 -√2 √2 √3 √3 -√3 -√3 ζ87ζ3+ζ85ζ3+ζ85 ζ87ζ3+ζ85ζ3+ζ85 ζ87ζ32+ζ87+ζ85ζ32 ζ83ζ3+ζ83+ζ8ζ3 ζ83ζ3+ζ83+ζ8ζ3 ζ83ζ32+ζ8ζ32+ζ8 ζ87ζ32+ζ87+ζ85ζ32 ζ83ζ32+ζ8ζ32+ζ8 symplectic lifted from Dic12, Schur index 2 ρ21 2 -2 2 -2 -1 2 -2 0 0 0 0 1 -1 1 2i 2i -2i -2i -1 1 1 -1 -i i i -i i i -i -i complex lifted from C4×S3 ρ22 2 -2 2 -2 -1 -2 2 0 0 0 0 1 -1 1 0 0 0 0 1 -1 -1 1 √-3 -√-3 -√-3 -√-3 √-3 √-3 √-3 -√-3 complex lifted from C3⋊D4 ρ23 2 -2 2 -2 -1 2 -2 0 0 0 0 1 -1 1 -2i -2i 2i 2i -1 1 1 -1 i -i -i i -i -i i i complex lifted from C4×S3 ρ24 2 2 -2 -2 -1 0 0 0 0 0 0 -1 1 1 -√-2 √-2 √-2 -√-2 -√3 √3 -√3 √3 ζ87ζ32+ζ87-ζ85ζ32 ζ83ζ32+ζ83-ζ8ζ32 ζ87ζ32+ζ87-ζ85ζ32 ζ87ζ3+ζ87-ζ85ζ3 ζ83ζ3+ζ83-ζ8ζ3 ζ87ζ3+ζ87-ζ85ζ3 ζ83ζ32+ζ83-ζ8ζ32 ζ83ζ3+ζ83-ζ8ζ3 complex lifted from C24⋊C2 ρ25 2 2 -2 -2 2 0 0 0 0 0 0 2 -2 -2 √-2 -√-2 -√-2 √-2 0 0 0 0 -√-2 √-2 -√-2 -√-2 √-2 -√-2 √-2 √-2 complex lifted from SD16 ρ26 2 -2 2 -2 -1 -2 2 0 0 0 0 1 -1 1 0 0 0 0 1 -1 -1 1 -√-3 √-3 √-3 √-3 -√-3 -√-3 -√-3 √-3 complex lifted from C3⋊D4 ρ27 2 2 -2 -2 -1 0 0 0 0 0 0 -1 1 1 √-2 -√-2 -√-2 √-2 √3 -√3 √3 -√3 ζ83ζ3+ζ83-ζ8ζ3 ζ87ζ3+ζ87-ζ85ζ3 ζ83ζ3+ζ83-ζ8ζ3 ζ83ζ32+ζ83-ζ8ζ32 ζ87ζ32+ζ87-ζ85ζ32 ζ83ζ32+ζ83-ζ8ζ32 ζ87ζ3+ζ87-ζ85ζ3 ζ87ζ32+ζ87-ζ85ζ32 complex lifted from C24⋊C2 ρ28 2 2 -2 -2 -1 0 0 0 0 0 0 -1 1 1 -√-2 √-2 √-2 -√-2 √3 -√3 √3 -√3 ζ87ζ3+ζ87-ζ85ζ3 ζ83ζ3+ζ83-ζ8ζ3 ζ87ζ3+ζ87-ζ85ζ3 ζ87ζ32+ζ87-ζ85ζ32 ζ83ζ32+ζ83-ζ8ζ32 ζ87ζ32+ζ87-ζ85ζ32 ζ83ζ3+ζ83-ζ8ζ3 ζ83ζ32+ζ83-ζ8ζ32 complex lifted from C24⋊C2 ρ29 2 2 -2 -2 -1 0 0 0 0 0 0 -1 1 1 √-2 -√-2 -√-2 √-2 -√3 √3 -√3 √3 ζ83ζ32+ζ83-ζ8ζ32 ζ87ζ32+ζ87-ζ85ζ32 ζ83ζ32+ζ83-ζ8ζ32 ζ83ζ3+ζ83-ζ8ζ3 ζ87ζ3+ζ87-ζ85ζ3 ζ83ζ3+ζ83-ζ8ζ3 ζ87ζ32+ζ87-ζ85ζ32 ζ87ζ3+ζ87-ζ85ζ3 complex lifted from C24⋊C2 ρ30 2 2 -2 -2 2 0 0 0 0 0 0 2 -2 -2 -√-2 √-2 √-2 -√-2 0 0 0 0 √-2 -√-2 √-2 √-2 -√-2 √-2 -√-2 -√-2 complex lifted from SD16

Smallest permutation representation of C2.Dic12
Regular action on 96 points
Generators in S96
(1 53 74 87 10 35)(2 54 75 88 11 36)(3 55 76 81 12 37)(4 56 77 82 13 38)(5 49 78 83 14 39)(6 50 79 84 15 40)(7 51 80 85 16 33)(8 52 73 86 9 34)(17 62 90 66 25 43)(18 63 91 67 26 44)(19 64 92 68 27 45)(20 57 93 69 28 46)(21 58 94 70 29 47)(22 59 95 71 30 48)(23 60 96 72 31 41)(24 61 89 65 32 42)
(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 62 87 25)(2 28 88 57)(3 60 81 31)(4 26 82 63)(5 58 83 29)(6 32 84 61)(7 64 85 27)(8 30 86 59)(9 22 52 71)(10 66 53 17)(11 20 54 69)(12 72 55 23)(13 18 56 67)(14 70 49 21)(15 24 50 65)(16 68 51 19)(33 92 80 45)(34 48 73 95)(35 90 74 43)(36 46 75 93)(37 96 76 41)(38 44 77 91)(39 94 78 47)(40 42 79 89)

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

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

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

Matrix representation of C2.Dic12 in GL4(𝔽73) generated by

 1 1 0 0 72 0 0 0 0 0 72 72 0 0 1 0
,
 46 0 0 0 0 46 0 0 0 0 36 11 0 0 62 25
,
 34 8 0 0 47 39 0 0 0 0 3 45 0 0 42 70
G:=sub<GL(4,GF(73))| [1,72,0,0,1,0,0,0,0,0,72,1,0,0,72,0],[46,0,0,0,0,46,0,0,0,0,36,62,0,0,11,25],[34,47,0,0,8,39,0,0,0,0,3,42,0,0,45,70] >;

C2.Dic12 in GAP, Magma, Sage, TeX

C_2.{\rm Dic}_{12}
% in TeX

G:=Group("C2.Dic12");
// GroupNames label

G:=SmallGroup(96,23);
// by ID

G=gap.SmallGroup(96,23);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,48,73,79,362,86,2309]);
// Polycyclic

G:=Group<a,b,c|a^6=b^8=1,c^2=a^3,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=a^3*b^3>;
// generators/relations

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