Copied to
clipboard

## G = D4⋊6Dic6order 192 = 26·3

### 2nd semidirect product of D4 and Dic6 acting through Inn(D4)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — D4⋊6Dic6
 Chief series C1 — C3 — C6 — C2×C6 — C2×Dic3 — C22×Dic3 — D4×Dic3 — D4⋊6Dic6
 Lower central C3 — C2×C6 — D4⋊6Dic6
 Upper central C1 — C22 — C4×D4

Generators and relations for D46Dic6
G = < a,b,c,d | a4=b2=c12=1, d2=c6, bab=a-1, ac=ca, ad=da, bc=cb, dbd-1=a2b, dcd-1=c-1 >

Subgroups: 504 in 228 conjugacy classes, 115 normal (29 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, 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×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, 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, C122Q8, Dic3.D4, C4.Dic6, C12.48D4, C2×C4⋊Dic3, D4×Dic3, D4×C12, D46Dic6
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, D42S3, S3×C23, D43Q8, C22×Dic6, C2×D42S3, D4○D12, D46Dic6

Smallest permutation representation of D46Dic6
On 96 points
Generators in S96
(1 37 70 49)(2 38 71 50)(3 39 72 51)(4 40 61 52)(5 41 62 53)(6 42 63 54)(7 43 64 55)(8 44 65 56)(9 45 66 57)(10 46 67 58)(11 47 68 59)(12 48 69 60)(13 28 86 83)(14 29 87 84)(15 30 88 73)(16 31 89 74)(17 32 90 75)(18 33 91 76)(19 34 92 77)(20 35 93 78)(21 36 94 79)(22 25 95 80)(23 26 96 81)(24 27 85 82)
(1 43)(2 44)(3 45)(4 46)(5 47)(6 48)(7 37)(8 38)(9 39)(10 40)(11 41)(12 42)(13 77)(14 78)(15 79)(16 80)(17 81)(18 82)(19 83)(20 84)(21 73)(22 74)(23 75)(24 76)(25 89)(26 90)(27 91)(28 92)(29 93)(30 94)(31 95)(32 96)(33 85)(34 86)(35 87)(36 88)(49 64)(50 65)(51 66)(52 67)(53 68)(54 69)(55 70)(56 71)(57 72)(58 61)(59 62)(60 63)
(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 23 7 17)(2 22 8 16)(3 21 9 15)(4 20 10 14)(5 19 11 13)(6 18 12 24)(25 44 31 38)(26 43 32 37)(27 42 33 48)(28 41 34 47)(29 40 35 46)(30 39 36 45)(49 81 55 75)(50 80 56 74)(51 79 57 73)(52 78 58 84)(53 77 59 83)(54 76 60 82)(61 93 67 87)(62 92 68 86)(63 91 69 85)(64 90 70 96)(65 89 71 95)(66 88 72 94)

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

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

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

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 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + - + + + + + - + - + image C1 C2 C2 C2 C2 C2 C2 C2 C2 S3 Q8 D6 D6 D6 D6 D6 C4○D4 Dic6 2+ 1+4 D4⋊2S3 D4○D12 kernel D4⋊6Dic6 C4×Dic6 C12⋊2Q8 Dic3.D4 C4.Dic6 C12.48D4 C2×C4⋊Dic3 D4×Dic3 D4×C12 C4×D4 C3×D4 C42 C22⋊C4 C4⋊C4 C22×C4 C2×D4 C12 D4 C6 C4 C2 # reps 1 1 1 4 2 2 2 2 1 1 4 1 2 1 2 1 4 8 1 2 2

Matrix representation of D46Dic6 in GL4(𝔽13) generated by

 1 0 0 0 0 1 0 0 0 0 0 1 0 0 12 0
,
 12 0 0 0 0 12 0 0 0 0 0 12 0 0 12 0
,
 7 0 0 0 2 2 0 0 0 0 1 0 0 0 0 1
,
 6 11 0 0 12 7 0 0 0 0 0 5 0 0 8 0
G:=sub<GL(4,GF(13))| [1,0,0,0,0,1,0,0,0,0,0,12,0,0,1,0],[12,0,0,0,0,12,0,0,0,0,0,12,0,0,12,0],[7,2,0,0,0,2,0,0,0,0,1,0,0,0,0,1],[6,12,0,0,11,7,0,0,0,0,0,8,0,0,5,0] >;

D46Dic6 in GAP, Magma, Sage, TeX

D_4\rtimes_6{\rm Dic}_6
% in TeX

G:=Group("D4:6Dic6");
// GroupNames label

G:=SmallGroup(192,1102);
// by ID

G=gap.SmallGroup(192,1102);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,387,1571,192,6278]);
// Polycyclic

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

׿
×
𝔽