metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic6⋊20D4, C6.332+ (1+4), C4⋊D4⋊7S3, C4⋊C4.177D6, (C2×D4).90D6, C3⋊3(Q8⋊6D4), C4.109(S3×D4), Dic3⋊D4⋊17C2, C12⋊D4⋊19C2, C12⋊3D4⋊15C2, C12.225(C2×D4), C22⋊C4.46D6, C6.62(C22×D4), Dic3⋊8(C4○D4), Dic3⋊4D4⋊6C2, (C2×C12).35C23, (C2×C6).143C24, D6⋊C4.12C22, Dic3.21(C2×D4), (C22×C4).235D6, C23.14D6⋊10C2, Dic6⋊C4⋊20C2, C2.35(D4⋊6D6), (C6×D4).117C22, C23.20(C22×S3), (C22×C6).14C23, (C2×D12).142C22, Dic3⋊C4.14C22, (C22×S3).62C23, C22.164(S3×C23), (C4×Dic3).90C22, (C22×C12).237C22, (C2×Dic6).293C22, (C2×Dic3).225C23, C6.D4.110C22, (C22×Dic3).104C22, C2.35(C2×S3×D4), (C3×C4⋊D4)⋊8C2, (C4×C3⋊D4)⋊15C2, C2.34(S3×C4○D4), (C2×C4○D12)⋊19C2, C6.148(C2×C4○D4), (S3×C2×C4).82C22, (C2×D4⋊2S3)⋊11C2, (C3×C4⋊C4).139C22, (C2×C4).585(C22×S3), (C2×C3⋊D4).25C22, (C3×C22⋊C4).8C22, SmallGroup(192,1158)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 864 in 312 conjugacy classes, 105 normal (43 characteristic)
C1, C2 [×3], C2 [×6], C3, C4 [×2], C4 [×11], C22, C22 [×18], S3 [×3], C6 [×3], C6 [×3], C2×C4 [×2], C2×C4 [×2], C2×C4 [×17], D4 [×24], Q8 [×4], C23, C23 [×2], C23 [×3], Dic3 [×6], Dic3 [×2], C12 [×2], C12 [×3], D6 [×9], C2×C6, C2×C6 [×9], C42 [×3], C22⋊C4 [×2], C22⋊C4 [×4], C4⋊C4, C4⋊C4 [×3], C22×C4, C22×C4 [×5], C2×D4, C2×D4 [×2], C2×D4 [×12], C2×Q8, C4○D4 [×8], Dic6 [×4], C4×S3 [×6], D12 [×4], C2×Dic3 [×3], C2×Dic3 [×2], C2×Dic3 [×4], C3⋊D4 [×16], C2×C12 [×2], C2×C12 [×2], C2×C12 [×2], C3×D4 [×4], C22×S3, C22×S3 [×2], C22×C6, C22×C6 [×2], C4×D4 [×3], C4×Q8, C4⋊D4, C4⋊D4 [×5], C4⋊1D4 [×3], C2×C4○D4 [×2], C4×Dic3, C4×Dic3 [×2], Dic3⋊C4, Dic3⋊C4 [×2], D6⋊C4, D6⋊C4 [×2], C6.D4, C3×C22⋊C4 [×2], C3×C4⋊C4, C2×Dic6, S3×C2×C4, S3×C2×C4 [×2], C2×D12, C2×D12 [×2], C4○D12 [×4], D4⋊2S3 [×4], C22×Dic3 [×2], C2×C3⋊D4, C2×C3⋊D4 [×8], C22×C12, C6×D4, C6×D4 [×2], Q8⋊6D4, Dic3⋊4D4 [×2], Dic3⋊D4 [×2], Dic6⋊C4, C12⋊D4, C4×C3⋊D4, C23.14D6 [×2], C12⋊3D4, C12⋊3D4 [×2], C3×C4⋊D4, C2×C4○D12, C2×D4⋊2S3, Dic6⋊20D4
Quotients:
C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C4○D4 [×2], C24, C22×S3 [×7], C22×D4, C2×C4○D4, 2+ (1+4), S3×D4 [×2], S3×C23, Q8⋊6D4, C2×S3×D4, D4⋊6D6, S3×C4○D4, Dic6⋊20D4
Generators and relations
G = < a,b,c,d | a12=c4=d2=1, b2=a6, bab-1=a-1, cac-1=a7, ad=da, cbc-1=dbd=a6b, dcd=c-1 >
►On 96 points
(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 55 7 49)(2 54 8 60)(3 53 9 59)(4 52 10 58)(5 51 11 57)(6 50 12 56)(13 93 19 87)(14 92 20 86)(15 91 21 85)(16 90 22 96)(17 89 23 95)(18 88 24 94)(25 81 31 75)(26 80 32 74)(27 79 33 73)(28 78 34 84)(29 77 35 83)(30 76 36 82)(37 65 43 71)(38 64 44 70)(39 63 45 69)(40 62 46 68)(41 61 47 67)(42 72 48 66)
(1 26 70 94)(2 33 71 89)(3 28 72 96)(4 35 61 91)(5 30 62 86)(6 25 63 93)(7 32 64 88)(8 27 65 95)(9 34 66 90)(10 29 67 85)(11 36 68 92)(12 31 69 87)(13 50 75 45)(14 57 76 40)(15 52 77 47)(16 59 78 42)(17 54 79 37)(18 49 80 44)(19 56 81 39)(20 51 82 46)(21 58 83 41)(22 53 84 48)(23 60 73 43)(24 55 74 38)
(1 58)(2 59)(3 60)(4 49)(5 50)(6 51)(7 52)(8 53)(9 54)(10 55)(11 56)(12 57)(13 30)(14 31)(15 32)(16 33)(17 34)(18 35)(19 36)(20 25)(21 26)(22 27)(23 28)(24 29)(37 66)(38 67)(39 68)(40 69)(41 70)(42 71)(43 72)(44 61)(45 62)(46 63)(47 64)(48 65)(73 96)(74 85)(75 86)(76 87)(77 88)(78 89)(79 90)(80 91)(81 92)(82 93)(83 94)(84 95)
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), (1,55,7,49)(2,54,8,60)(3,53,9,59)(4,52,10,58)(5,51,11,57)(6,50,12,56)(13,93,19,87)(14,92,20,86)(15,91,21,85)(16,90,22,96)(17,89,23,95)(18,88,24,94)(25,81,31,75)(26,80,32,74)(27,79,33,73)(28,78,34,84)(29,77,35,83)(30,76,36,82)(37,65,43,71)(38,64,44,70)(39,63,45,69)(40,62,46,68)(41,61,47,67)(42,72,48,66), (1,26,70,94)(2,33,71,89)(3,28,72,96)(4,35,61,91)(5,30,62,86)(6,25,63,93)(7,32,64,88)(8,27,65,95)(9,34,66,90)(10,29,67,85)(11,36,68,92)(12,31,69,87)(13,50,75,45)(14,57,76,40)(15,52,77,47)(16,59,78,42)(17,54,79,37)(18,49,80,44)(19,56,81,39)(20,51,82,46)(21,58,83,41)(22,53,84,48)(23,60,73,43)(24,55,74,38), (1,58)(2,59)(3,60)(4,49)(5,50)(6,51)(7,52)(8,53)(9,54)(10,55)(11,56)(12,57)(13,30)(14,31)(15,32)(16,33)(17,34)(18,35)(19,36)(20,25)(21,26)(22,27)(23,28)(24,29)(37,66)(38,67)(39,68)(40,69)(41,70)(42,71)(43,72)(44,61)(45,62)(46,63)(47,64)(48,65)(73,96)(74,85)(75,86)(76,87)(77,88)(78,89)(79,90)(80,91)(81,92)(82,93)(83,94)(84,95)>;
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), (1,55,7,49)(2,54,8,60)(3,53,9,59)(4,52,10,58)(5,51,11,57)(6,50,12,56)(13,93,19,87)(14,92,20,86)(15,91,21,85)(16,90,22,96)(17,89,23,95)(18,88,24,94)(25,81,31,75)(26,80,32,74)(27,79,33,73)(28,78,34,84)(29,77,35,83)(30,76,36,82)(37,65,43,71)(38,64,44,70)(39,63,45,69)(40,62,46,68)(41,61,47,67)(42,72,48,66), (1,26,70,94)(2,33,71,89)(3,28,72,96)(4,35,61,91)(5,30,62,86)(6,25,63,93)(7,32,64,88)(8,27,65,95)(9,34,66,90)(10,29,67,85)(11,36,68,92)(12,31,69,87)(13,50,75,45)(14,57,76,40)(15,52,77,47)(16,59,78,42)(17,54,79,37)(18,49,80,44)(19,56,81,39)(20,51,82,46)(21,58,83,41)(22,53,84,48)(23,60,73,43)(24,55,74,38), (1,58)(2,59)(3,60)(4,49)(5,50)(6,51)(7,52)(8,53)(9,54)(10,55)(11,56)(12,57)(13,30)(14,31)(15,32)(16,33)(17,34)(18,35)(19,36)(20,25)(21,26)(22,27)(23,28)(24,29)(37,66)(38,67)(39,68)(40,69)(41,70)(42,71)(43,72)(44,61)(45,62)(46,63)(47,64)(48,65)(73,96)(74,85)(75,86)(76,87)(77,88)(78,89)(79,90)(80,91)(81,92)(82,93)(83,94)(84,95) );
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)], [(1,55,7,49),(2,54,8,60),(3,53,9,59),(4,52,10,58),(5,51,11,57),(6,50,12,56),(13,93,19,87),(14,92,20,86),(15,91,21,85),(16,90,22,96),(17,89,23,95),(18,88,24,94),(25,81,31,75),(26,80,32,74),(27,79,33,73),(28,78,34,84),(29,77,35,83),(30,76,36,82),(37,65,43,71),(38,64,44,70),(39,63,45,69),(40,62,46,68),(41,61,47,67),(42,72,48,66)], [(1,26,70,94),(2,33,71,89),(3,28,72,96),(4,35,61,91),(5,30,62,86),(6,25,63,93),(7,32,64,88),(8,27,65,95),(9,34,66,90),(10,29,67,85),(11,36,68,92),(12,31,69,87),(13,50,75,45),(14,57,76,40),(15,52,77,47),(16,59,78,42),(17,54,79,37),(18,49,80,44),(19,56,81,39),(20,51,82,46),(21,58,83,41),(22,53,84,48),(23,60,73,43),(24,55,74,38)], [(1,58),(2,59),(3,60),(4,49),(5,50),(6,51),(7,52),(8,53),(9,54),(10,55),(11,56),(12,57),(13,30),(14,31),(15,32),(16,33),(17,34),(18,35),(19,36),(20,25),(21,26),(22,27),(23,28),(24,29),(37,66),(38,67),(39,68),(40,69),(41,70),(42,71),(43,72),(44,61),(45,62),(46,63),(47,64),(48,65),(73,96),(74,85),(75,86),(76,87),(77,88),(78,89),(79,90),(80,91),(81,92),(82,93),(83,94),(84,95)])
Matrix representation ►G ⊆ GL6(𝔽13)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 1 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,1,0,0,0,0,0,0,0,12,12,0,0,0,0,1,0],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,8,0,0,0,0,0,0,5,0,0,0,0,0,0,0,12,0,0,0,0,12,0],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,0,5,0,0,0,0,5,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,8,0,0,0,0,5,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12] >;
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4N | 4O | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 12A | 12B | 12C | 12D | 12E | 12F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 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 | 4 | 4 | 4 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 6 | ··· | 6 | 12 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 4 | 4 | 4 | 4 | 8 | 8 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | D6 | D6 | C4○D4 | 2+ (1+4) | S3×D4 | D4⋊6D6 | S3×C4○D4 |
| kernel | Dic6⋊20D4 | Dic3⋊4D4 | Dic3⋊D4 | Dic6⋊C4 | C12⋊D4 | C4×C3⋊D4 | C23.14D6 | C12⋊3D4 | C3×C4⋊D4 | C2×C4○D12 | C2×D4⋊2S3 | C4⋊D4 | Dic6 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | Dic3 | C6 | C4 | C2 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 3 | 1 | 1 | 1 | 1 | 4 | 2 | 1 | 1 | 3 | 4 | 1 | 2 | 2 | 2 |
In GAP, Magma, Sage, TeX
Dic_6\rtimes_{20}D_4 % in TeX
G:=Group("Dic6:20D4"); // GroupNames label
G:=SmallGroup(192,1158);
// by ID
G=gap.SmallGroup(192,1158);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,477,232,184,570,185,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^12=c^4=d^2=1,b^2=a^6,b*a*b^-1=a^-1,c*a*c^-1=a^7,a*d=d*a,c*b*c^-1=d*b*d=a^6*b,d*c*d=c^-1>;
// generators/relations