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## G = D4⋊5Dic6order 192 = 26·3

### 1st semidirect product of D4 and Dic6 acting through Inn(D4)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — D4⋊5Dic6
 Chief series C1 — C3 — C6 — C2×C6 — C2×Dic3 — C22×Dic3 — D4×Dic3 — D4⋊5Dic6
 Lower central C3 — C2×C6 — D4⋊5Dic6
 Upper central C1 — C22 — C4×D4

Generators and relations for D45Dic6
G = < a,b,c,d | a4=b2=c12=1, d2=c6, bab=cac-1=a-1, ad=da, cbc-1=dbd-1=a2b, dcd-1=c-1 >

Subgroups: 504 in 228 conjugacy classes, 113 normal (43 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, Dic3, Dic3, C12, C12, C2×C6, C2×C6, C2×C6, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, Dic6, C2×Dic3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C2×C12, C3×D4, C22×C6, C2×C4⋊C4, C4×D4, C4×D4, C4×Q8, C22⋊Q8, C42.C2, C4⋊Q8, C4×Dic3, Dic3⋊C4, Dic3⋊C4, C4⋊Dic3, C4⋊Dic3, C6.D4, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C2×Dic6, C2×Dic6, C22×Dic3, C22×C12, C6×D4, D43Q8, C4×Dic6, C12.6Q8, Dic3.D4, C12⋊Q8, C4.Dic6, C2×Dic3⋊C4, C12.48D4, D4×Dic3, D4×C12, D45Dic6
Quotients: C1, C2, C22, S3, Q8, C23, D6, C2×Q8, C4○D4, C24, Dic6, C22×S3, C22×Q8, C2×C4○D4, 2+ 1+4, C2×Dic6, S3×C23, D43Q8, C22×Dic6, D46D6, S3×C4○D4, D45Dic6

Smallest permutation representation of D45Dic6
On 96 points
Generators in S96
(1 57 31 44)(2 45 32 58)(3 59 33 46)(4 47 34 60)(5 49 35 48)(6 37 36 50)(7 51 25 38)(8 39 26 52)(9 53 27 40)(10 41 28 54)(11 55 29 42)(12 43 30 56)(13 80 94 64)(14 65 95 81)(15 82 96 66)(16 67 85 83)(17 84 86 68)(18 69 87 73)(19 74 88 70)(20 71 89 75)(21 76 90 72)(22 61 91 77)(23 78 92 62)(24 63 93 79)
(1 51)(2 39)(3 53)(4 41)(5 55)(6 43)(7 57)(8 45)(9 59)(10 47)(11 49)(12 37)(13 70)(14 75)(15 72)(16 77)(17 62)(18 79)(19 64)(20 81)(21 66)(22 83)(23 68)(24 73)(25 44)(26 58)(27 46)(28 60)(29 48)(30 50)(31 38)(32 52)(33 40)(34 54)(35 42)(36 56)(61 85)(63 87)(65 89)(67 91)(69 93)(71 95)(74 94)(76 96)(78 86)(80 88)(82 90)(84 92)
(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 13 7 19)(2 24 8 18)(3 23 9 17)(4 22 10 16)(5 21 11 15)(6 20 12 14)(25 88 31 94)(26 87 32 93)(27 86 33 92)(28 85 34 91)(29 96 35 90)(30 95 36 89)(37 71 43 65)(38 70 44 64)(39 69 45 63)(40 68 46 62)(41 67 47 61)(42 66 48 72)(49 76 55 82)(50 75 56 81)(51 74 57 80)(52 73 58 79)(53 84 59 78)(54 83 60 77)

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

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

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

45 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L ··· 4Q 6A 6B 6C 6D 6E 6F 6G 12A 12B 12C 12D 12E ··· 12L order 1 2 2 2 2 2 2 2 3 4 4 4 4 4 4 4 4 4 4 4 4 ··· 4 6 6 6 6 6 6 6 12 12 12 12 12 ··· 12 size 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 6 6 6 6 12 ··· 12 2 2 2 4 4 4 4 2 2 2 2 4 ··· 4

45 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + - + + + + + - + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 S3 Q8 D6 D6 D6 D6 D6 C4○D4 Dic6 2+ 1+4 D4⋊6D6 S3×C4○D4 kernel D4⋊5Dic6 C4×Dic6 C12.6Q8 Dic3.D4 C12⋊Q8 C4.Dic6 C2×Dic3⋊C4 C12.48D4 D4×Dic3 D4×C12 C4×D4 C3×D4 C42 C22⋊C4 C4⋊C4 C22×C4 C2×D4 Dic3 D4 C6 C2 C2 # reps 1 1 1 4 1 1 2 2 2 1 1 4 1 2 1 2 1 4 8 1 2 2

Matrix representation of D45Dic6 in GL6(𝔽13)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 12 0
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 1 2 0 0 0 0 12 12 0 0 0 0 0 0 1 12 0 0 0 0 1 0 0 0 0 0 0 0 5 0 0 0 0 0 0 8
,
 5 0 0 0 0 0 8 8 0 0 0 0 0 0 3 7 0 0 0 0 10 10 0 0 0 0 0 0 0 1 0 0 0 0 12 0

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,1,0],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[1,12,0,0,0,0,2,12,0,0,0,0,0,0,1,1,0,0,0,0,12,0,0,0,0,0,0,0,5,0,0,0,0,0,0,8],[5,8,0,0,0,0,0,8,0,0,0,0,0,0,3,10,0,0,0,0,7,10,0,0,0,0,0,0,0,12,0,0,0,0,1,0] >;

D45Dic6 in GAP, Magma, Sage, TeX

D_4\rtimes_5{\rm Dic}_6
% in TeX

G:=Group("D4:5Dic6");
// GroupNames label

G:=SmallGroup(192,1098);
// by ID

G=gap.SmallGroup(192,1098);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,387,675,192,6278]);
// Polycyclic

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

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