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## G = C4.Dic6order 96 = 25·3

### 3rd non-split extension by C4 of Dic6 acting via Dic6/Dic3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C4.Dic6
 Chief series C1 — C3 — C6 — C2×C6 — C2×Dic3 — C4×Dic3 — C4.Dic6
 Lower central C3 — C2×C6 — C4.Dic6
 Upper central C1 — C22 — C4⋊C4

Generators and relations for C4.Dic6
G = < a,b,c | a12=b4=1, c2=b2, bab-1=a7, cac-1=a5, cbc-1=a6b-1 >

Subgroups: 106 in 56 conjugacy classes, 33 normal (19 characteristic)
C1, C2, C3, C4, C4, C22, C6, C2×C4, C2×C4, C2×C4, Dic3, C12, C12, C2×C6, C42, C4⋊C4, C4⋊C4, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C42.C2, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C4⋊Dic3, C3×C4⋊C4, C4.Dic6
Quotients: C1, C2, C22, S3, Q8, C23, D6, C2×Q8, C4○D4, Dic6, C22×S3, C42.C2, C2×Dic6, D42S3, Q83S3, C4.Dic6

Character table of C4.Dic6

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 6A 6B 6C 12A 12B 12C 12D 12E 12F size 1 1 1 1 2 2 2 4 4 6 6 6 6 12 12 2 2 2 4 4 4 4 4 4 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 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 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 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 linear of order 2 ρ5 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 ρ6 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 ρ7 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 ρ8 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 ρ9 2 2 2 2 -1 -2 -2 -2 2 0 0 0 0 0 0 -1 -1 -1 1 1 -1 1 -1 1 orthogonal lifted from D6 ρ10 2 2 2 2 -1 2 2 2 2 0 0 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ11 2 2 2 2 -1 2 2 -2 -2 0 0 0 0 0 0 -1 -1 -1 -1 1 1 -1 1 1 orthogonal lifted from D6 ρ12 2 2 2 2 -1 -2 -2 2 -2 0 0 0 0 0 0 -1 -1 -1 1 -1 1 1 1 -1 orthogonal lifted from D6 ρ13 2 2 -2 -2 2 -2 2 0 0 0 0 0 0 0 0 -2 2 -2 2 0 0 -2 0 0 symplectic lifted from Q8, Schur index 2 ρ14 2 2 -2 -2 2 2 -2 0 0 0 0 0 0 0 0 -2 2 -2 -2 0 0 2 0 0 symplectic lifted from Q8, Schur index 2 ρ15 2 2 -2 -2 -1 2 -2 0 0 0 0 0 0 0 0 1 -1 1 1 √3 -√3 -1 √3 -√3 symplectic lifted from Dic6, Schur index 2 ρ16 2 2 -2 -2 -1 2 -2 0 0 0 0 0 0 0 0 1 -1 1 1 -√3 √3 -1 -√3 √3 symplectic lifted from Dic6, Schur index 2 ρ17 2 2 -2 -2 -1 -2 2 0 0 0 0 0 0 0 0 1 -1 1 -1 √3 √3 1 -√3 -√3 symplectic lifted from Dic6, Schur index 2 ρ18 2 2 -2 -2 -1 -2 2 0 0 0 0 0 0 0 0 1 -1 1 -1 -√3 -√3 1 √3 √3 symplectic lifted from Dic6, Schur index 2 ρ19 2 -2 -2 2 2 0 0 0 0 0 -2i 0 2i 0 0 -2 -2 2 0 0 0 0 0 0 complex lifted from C4○D4 ρ20 2 -2 2 -2 2 0 0 0 0 -2i 0 2i 0 0 0 2 -2 -2 0 0 0 0 0 0 complex lifted from C4○D4 ρ21 2 -2 2 -2 2 0 0 0 0 2i 0 -2i 0 0 0 2 -2 -2 0 0 0 0 0 0 complex lifted from C4○D4 ρ22 2 -2 -2 2 2 0 0 0 0 0 2i 0 -2i 0 0 -2 -2 2 0 0 0 0 0 0 complex lifted from C4○D4 ρ23 4 -4 -4 4 -2 0 0 0 0 0 0 0 0 0 0 2 2 -2 0 0 0 0 0 0 orthogonal lifted from Q8⋊3S3, Schur index 2 ρ24 4 -4 4 -4 -2 0 0 0 0 0 0 0 0 0 0 -2 2 2 0 0 0 0 0 0 symplectic lifted from D4⋊2S3, Schur index 2

Smallest permutation representation of C4.Dic6
Regular action 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 90 78 70)(2 85 79 65)(3 92 80 72)(4 87 81 67)(5 94 82 62)(6 89 83 69)(7 96 84 64)(8 91 73 71)(9 86 74 66)(10 93 75 61)(11 88 76 68)(12 95 77 63)(13 33 37 54)(14 28 38 49)(15 35 39 56)(16 30 40 51)(17 25 41 58)(18 32 42 53)(19 27 43 60)(20 34 44 55)(21 29 45 50)(22 36 46 57)(23 31 47 52)(24 26 48 59)
(1 22 78 46)(2 15 79 39)(3 20 80 44)(4 13 81 37)(5 18 82 42)(6 23 83 47)(7 16 84 40)(8 21 73 45)(9 14 74 38)(10 19 75 43)(11 24 76 48)(12 17 77 41)(25 89 58 69)(26 94 59 62)(27 87 60 67)(28 92 49 72)(29 85 50 65)(30 90 51 70)(31 95 52 63)(32 88 53 68)(33 93 54 61)(34 86 55 66)(35 91 56 71)(36 96 57 64)

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

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

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

Matrix representation of C4.Dic6 in GL4(𝔽13) generated by

 4 0 0 0 3 10 0 0 0 0 8 3 0 0 0 5
,
 8 0 0 0 5 5 0 0 0 0 12 11 0 0 1 1
,
 12 11 0 0 1 1 0 0 0 0 5 0 0 0 0 5
G:=sub<GL(4,GF(13))| [4,3,0,0,0,10,0,0,0,0,8,0,0,0,3,5],[8,5,0,0,0,5,0,0,0,0,12,1,0,0,11,1],[12,1,0,0,11,1,0,0,0,0,5,0,0,0,0,5] >;

C4.Dic6 in GAP, Magma, Sage, TeX

C_4.{\rm Dic}_6
% in TeX

G:=Group("C4.Dic6");
// GroupNames label

G:=SmallGroup(96,97);
// by ID

G=gap.SmallGroup(96,97);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,48,103,506,188,50,2309]);
// Polycyclic

G:=Group<a,b,c|a^12=b^4=1,c^2=b^2,b*a*b^-1=a^7,c*a*c^-1=a^5,c*b*c^-1=a^6*b^-1>;
// generators/relations

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