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## G = Dic6⋊C4order 96 = 25·3

### 5th semidirect product of Dic6 and C4 acting via C4/C2=C2

Series: Derived Chief Lower central Upper central

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

Generators and relations for Dic6⋊C4
G = < a,b,c | a12=c4=1, b2=a6, bab-1=a-1, cac-1=a7, bc=cb >

Subgroups: 122 in 70 conjugacy classes, 43 normal (19 characteristic)
C1, C2, C3, C4, C4, C22, C6, C2×C4, C2×C4, C2×C4, Q8, Dic3, Dic3, C12, C12, C2×C6, C42, C4⋊C4, C4⋊C4, C2×Q8, Dic6, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C4×Q8, C4×Dic3, C4×Dic3, Dic3⋊C4, C3×C4⋊C4, C2×Dic6, Dic6⋊C4
Quotients: C1, C2, C4, C22, S3, C2×C4, Q8, C23, D6, C22×C4, C2×Q8, C4○D4, C4×S3, C22×S3, C4×Q8, S3×C2×C4, D42S3, S3×Q8, Dic6⋊C4

Character table of Dic6⋊C4

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 4N 4O 4P 6A 6B 6C 12A 12B 12C 12D 12E 12F size 1 1 1 1 2 2 2 2 2 2 2 3 3 3 3 6 6 6 6 6 6 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 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 -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 -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 -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 1 1 -1 -1 -1 -1 linear of order 2 ρ9 1 -1 1 -1 1 1 i -i -1 -i i i -i i -i -1 -1 i -i 1 1 -1 1 -1 i -i -i 1 i -1 linear of order 4 ρ10 1 -1 1 -1 1 1 -i i -1 i -i -i i -i i -1 -1 -i i 1 1 -1 1 -1 -i i i 1 -i -1 linear of order 4 ρ11 1 -1 1 -1 1 1 i -i -1 -i i -i i -i i 1 1 -i i -1 -1 -1 1 -1 i -i -i 1 i -1 linear of order 4 ρ12 1 -1 1 -1 1 1 -i i -1 i -i i -i i -i 1 1 i -i -1 -1 -1 1 -1 -i i i 1 -i -1 linear of order 4 ρ13 1 -1 1 -1 1 -1 i -i 1 i -i i -i i -i 1 -1 -i i -1 1 -1 1 -1 i -i i -1 -i 1 linear of order 4 ρ14 1 -1 1 -1 1 -1 -i i 1 -i i -i i -i i 1 -1 i -i -1 1 -1 1 -1 -i i -i -1 i 1 linear of order 4 ρ15 1 -1 1 -1 1 -1 i -i 1 i -i -i i -i i -1 1 i -i 1 -1 -1 1 -1 i -i i -1 -i 1 linear of order 4 ρ16 1 -1 1 -1 1 -1 -i i 1 -i i i -i i -i -1 1 -i i 1 -1 -1 1 -1 -i i -i -1 i 1 linear of order 4 ρ17 2 2 2 2 -1 2 -2 -2 2 -2 -2 0 0 0 0 0 0 0 0 0 0 -1 -1 -1 1 1 1 -1 1 -1 orthogonal lifted from D6 ρ18 2 2 2 2 -1 -2 -2 -2 -2 2 2 0 0 0 0 0 0 0 0 0 0 -1 -1 -1 1 1 -1 1 -1 1 orthogonal lifted from D6 ρ19 2 2 2 2 -1 -2 2 2 -2 -2 -2 0 0 0 0 0 0 0 0 0 0 -1 -1 -1 -1 -1 1 1 1 1 orthogonal lifted from D6 ρ20 2 2 2 2 -1 2 2 2 2 2 2 0 0 0 0 0 0 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ21 2 2 -2 -2 2 0 0 0 0 0 0 2 2 -2 -2 0 0 0 0 0 0 -2 -2 2 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ22 2 2 -2 -2 2 0 0 0 0 0 0 -2 -2 2 2 0 0 0 0 0 0 -2 -2 2 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ23 2 -2 -2 2 2 0 0 0 0 0 0 -2i 2i 2i -2i 0 0 0 0 0 0 2 -2 -2 0 0 0 0 0 0 complex lifted from C4○D4 ρ24 2 -2 2 -2 -1 -2 -2i 2i 2 -2i 2i 0 0 0 0 0 0 0 0 0 0 1 -1 1 i -i i 1 -i -1 complex lifted from C4×S3 ρ25 2 -2 2 -2 -1 2 2i -2i -2 -2i 2i 0 0 0 0 0 0 0 0 0 0 1 -1 1 -i i i -1 -i 1 complex lifted from C4×S3 ρ26 2 -2 2 -2 -1 -2 2i -2i 2 2i -2i 0 0 0 0 0 0 0 0 0 0 1 -1 1 -i i -i 1 i -1 complex lifted from C4×S3 ρ27 2 -2 2 -2 -1 2 -2i 2i -2 2i -2i 0 0 0 0 0 0 0 0 0 0 1 -1 1 i -i -i -1 i 1 complex lifted from C4×S3 ρ28 2 -2 -2 2 2 0 0 0 0 0 0 2i -2i -2i 2i 0 0 0 0 0 0 2 -2 -2 0 0 0 0 0 0 complex lifted from C4○D4 ρ29 4 4 -4 -4 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 2 -2 0 0 0 0 0 0 symplectic lifted from S3×Q8, Schur index 2 ρ30 4 -4 -4 4 -2 0 0 0 0 0 0 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 Dic6⋊C4
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 84 7 78)(2 83 8 77)(3 82 9 76)(4 81 10 75)(5 80 11 74)(6 79 12 73)(13 87 19 93)(14 86 20 92)(15 85 21 91)(16 96 22 90)(17 95 23 89)(18 94 24 88)(25 38 31 44)(26 37 32 43)(27 48 33 42)(28 47 34 41)(29 46 35 40)(30 45 36 39)(49 69 55 63)(50 68 56 62)(51 67 57 61)(52 66 58 72)(53 65 59 71)(54 64 60 70)
(1 51 41 21)(2 58 42 16)(3 53 43 23)(4 60 44 18)(5 55 45 13)(6 50 46 20)(7 57 47 15)(8 52 48 22)(9 59 37 17)(10 54 38 24)(11 49 39 19)(12 56 40 14)(25 94 81 70)(26 89 82 65)(27 96 83 72)(28 91 84 67)(29 86 73 62)(30 93 74 69)(31 88 75 64)(32 95 76 71)(33 90 77 66)(34 85 78 61)(35 92 79 68)(36 87 80 63)

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

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

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

Matrix representation of Dic6⋊C4 in GL5(𝔽13)

 1 0 0 0 0 0 12 1 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 12 0
,
 1 0 0 0 0 0 3 3 0 0 0 6 10 0 0 0 0 0 8 0 0 0 0 0 5
,
 5 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 12 0 0 0 0 0 1

G:=sub<GL(5,GF(13))| [1,0,0,0,0,0,12,12,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,1,0],[1,0,0,0,0,0,3,6,0,0,0,3,10,0,0,0,0,0,8,0,0,0,0,0,5],[5,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,1] >;

Dic6⋊C4 in GAP, Magma, Sage, TeX

{\rm Dic}_6\rtimes C_4
% in TeX

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

G:=SmallGroup(96,94);
// by ID

G=gap.SmallGroup(96,94);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,96,55,116,122,2309]);
// Polycyclic

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

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