metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D6⋊C8⋊11C2, C4⋊C4.137D6, (C2×D4).27D6, C8⋊Dic3⋊13C2, C6.Q16⋊7C2, (C2×C8).116D6, D4⋊C4⋊12S3, D6⋊3D4.4C2, C6.41(C4○D8), D4⋊Dic3⋊9C2, C4.53(C4○D12), C2.15(D8⋊S3), C6.33(C8⋊C22), (C22×S3).11D4, (C6×D4).40C22, C22.177(S3×D4), C12.151(C4○D4), C4.80(D4⋊2S3), (C2×C12).219C23, (C2×C24).127C22, (C2×Dic3).141D4, C3⋊2(C23.19D4), C4⋊Dic3.73C22, C2.11(Q8.7D6), C2.14(C23.9D6), C6.22(C22.D4), C4⋊C4⋊7S3⋊3C2, (C2×C6).232(C2×D4), (C2×C3⋊C8).17C22, (S3×C2×C4).11C22, (C3×D4⋊C4)⋊12C2, (C3×C4⋊C4).20C22, (C2×C4).326(C22×S3), SmallGroup(192,338)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D6⋊C8⋊11C2
G = < a,b,c,d | a6=b2=c8=d2=1, bab=a-1, ac=ca, ad=da, cbc-1=a3b, dbd=a3bc4, dcd=a3c3 >
Subgroups: 328 in 106 conjugacy classes, 37 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, C23, Dic3, C12, C12, D6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C2×D4, C2×D4, C3⋊C8, C24, C4×S3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C22×S3, C22×C6, C22⋊C8, D4⋊C4, D4⋊C4, C4.Q8, C2.D8, C42⋊C2, C4⋊D4, C2×C3⋊C8, C4×Dic3, C4⋊Dic3, D6⋊C4, C6.D4, C3×C4⋊C4, C2×C24, S3×C2×C4, C2×C3⋊D4, C6×D4, C23.19D4, C6.Q16, C8⋊Dic3, D6⋊C8, D4⋊Dic3, C3×D4⋊C4, C4⋊C4⋊7S3, D6⋊3D4, D6⋊C8⋊11C2
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C22×S3, C22.D4, C4○D8, C8⋊C22, C4○D12, S3×D4, D4⋊2S3, C23.19D4, C23.9D6, D8⋊S3, Q8.7D6, D6⋊C8⋊11C2
►On 96 points
(1 61 33 20 94 79)(2 62 34 21 95 80)(3 63 35 22 96 73)(4 64 36 23 89 74)(5 57 37 24 90 75)(6 58 38 17 91 76)(7 59 39 18 92 77)(8 60 40 19 93 78)(9 87 71 30 45 51)(10 88 72 31 46 52)(11 81 65 32 47 53)(12 82 66 25 48 54)(13 83 67 26 41 55)(14 84 68 27 42 56)(15 85 69 28 43 49)(16 86 70 29 44 50)
(1 75)(2 38)(3 77)(4 40)(5 79)(6 34)(7 73)(8 36)(9 71)(10 52)(11 65)(12 54)(13 67)(14 56)(15 69)(16 50)(17 80)(18 35)(19 74)(20 37)(21 76)(22 39)(23 78)(24 33)(25 66)(26 55)(27 68)(28 49)(29 70)(30 51)(31 72)(32 53)(42 84)(44 86)(46 88)(48 82)(57 94)(58 62)(59 96)(60 64)(61 90)(63 92)(89 93)(91 95)
(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 30)(2 12)(3 28)(4 10)(5 26)(6 16)(7 32)(8 14)(9 20)(11 18)(13 24)(15 22)(17 29)(19 27)(21 25)(23 31)(33 51)(34 66)(35 49)(36 72)(37 55)(38 70)(39 53)(40 68)(41 57)(42 93)(43 63)(44 91)(45 61)(46 89)(47 59)(48 95)(50 76)(52 74)(54 80)(56 78)(58 86)(60 84)(62 82)(64 88)(65 77)(67 75)(69 73)(71 79)(81 92)(83 90)(85 96)(87 94)
G:=sub<Sym(96)| (1,61,33,20,94,79)(2,62,34,21,95,80)(3,63,35,22,96,73)(4,64,36,23,89,74)(5,57,37,24,90,75)(6,58,38,17,91,76)(7,59,39,18,92,77)(8,60,40,19,93,78)(9,87,71,30,45,51)(10,88,72,31,46,52)(11,81,65,32,47,53)(12,82,66,25,48,54)(13,83,67,26,41,55)(14,84,68,27,42,56)(15,85,69,28,43,49)(16,86,70,29,44,50), (1,75)(2,38)(3,77)(4,40)(5,79)(6,34)(7,73)(8,36)(9,71)(10,52)(11,65)(12,54)(13,67)(14,56)(15,69)(16,50)(17,80)(18,35)(19,74)(20,37)(21,76)(22,39)(23,78)(24,33)(25,66)(26,55)(27,68)(28,49)(29,70)(30,51)(31,72)(32,53)(42,84)(44,86)(46,88)(48,82)(57,94)(58,62)(59,96)(60,64)(61,90)(63,92)(89,93)(91,95), (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), (1,30)(2,12)(3,28)(4,10)(5,26)(6,16)(7,32)(8,14)(9,20)(11,18)(13,24)(15,22)(17,29)(19,27)(21,25)(23,31)(33,51)(34,66)(35,49)(36,72)(37,55)(38,70)(39,53)(40,68)(41,57)(42,93)(43,63)(44,91)(45,61)(46,89)(47,59)(48,95)(50,76)(52,74)(54,80)(56,78)(58,86)(60,84)(62,82)(64,88)(65,77)(67,75)(69,73)(71,79)(81,92)(83,90)(85,96)(87,94)>;
G:=Group( (1,61,33,20,94,79)(2,62,34,21,95,80)(3,63,35,22,96,73)(4,64,36,23,89,74)(5,57,37,24,90,75)(6,58,38,17,91,76)(7,59,39,18,92,77)(8,60,40,19,93,78)(9,87,71,30,45,51)(10,88,72,31,46,52)(11,81,65,32,47,53)(12,82,66,25,48,54)(13,83,67,26,41,55)(14,84,68,27,42,56)(15,85,69,28,43,49)(16,86,70,29,44,50), (1,75)(2,38)(3,77)(4,40)(5,79)(6,34)(7,73)(8,36)(9,71)(10,52)(11,65)(12,54)(13,67)(14,56)(15,69)(16,50)(17,80)(18,35)(19,74)(20,37)(21,76)(22,39)(23,78)(24,33)(25,66)(26,55)(27,68)(28,49)(29,70)(30,51)(31,72)(32,53)(42,84)(44,86)(46,88)(48,82)(57,94)(58,62)(59,96)(60,64)(61,90)(63,92)(89,93)(91,95), (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), (1,30)(2,12)(3,28)(4,10)(5,26)(6,16)(7,32)(8,14)(9,20)(11,18)(13,24)(15,22)(17,29)(19,27)(21,25)(23,31)(33,51)(34,66)(35,49)(36,72)(37,55)(38,70)(39,53)(40,68)(41,57)(42,93)(43,63)(44,91)(45,61)(46,89)(47,59)(48,95)(50,76)(52,74)(54,80)(56,78)(58,86)(60,84)(62,82)(64,88)(65,77)(67,75)(69,73)(71,79)(81,92)(83,90)(85,96)(87,94) );
G=PermutationGroup([[(1,61,33,20,94,79),(2,62,34,21,95,80),(3,63,35,22,96,73),(4,64,36,23,89,74),(5,57,37,24,90,75),(6,58,38,17,91,76),(7,59,39,18,92,77),(8,60,40,19,93,78),(9,87,71,30,45,51),(10,88,72,31,46,52),(11,81,65,32,47,53),(12,82,66,25,48,54),(13,83,67,26,41,55),(14,84,68,27,42,56),(15,85,69,28,43,49),(16,86,70,29,44,50)], [(1,75),(2,38),(3,77),(4,40),(5,79),(6,34),(7,73),(8,36),(9,71),(10,52),(11,65),(12,54),(13,67),(14,56),(15,69),(16,50),(17,80),(18,35),(19,74),(20,37),(21,76),(22,39),(23,78),(24,33),(25,66),(26,55),(27,68),(28,49),(29,70),(30,51),(31,72),(32,53),(42,84),(44,86),(46,88),(48,82),(57,94),(58,62),(59,96),(60,64),(61,90),(63,92),(89,93),(91,95)], [(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)], [(1,30),(2,12),(3,28),(4,10),(5,26),(6,16),(7,32),(8,14),(9,20),(11,18),(13,24),(15,22),(17,29),(19,27),(21,25),(23,31),(33,51),(34,66),(35,49),(36,72),(37,55),(38,70),(39,53),(40,68),(41,57),(42,93),(43,63),(44,91),(45,61),(46,89),(47,59),(48,95),(50,76),(52,74),(54,80),(56,78),(58,86),(60,84),(62,82),(64,88),(65,77),(67,75),(69,73),(71,79),(81,92),(83,90),(85,96),(87,94)]])
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 1 | 1 | 8 | 12 | 2 | 2 | 2 | 4 | 4 | 6 | 6 | 12 | 12 | 24 | 2 | 2 | 2 | 8 | 8 | 4 | 4 | 12 | 12 | 4 | 4 | 8 | 8 | 4 | 4 | 4 | 4 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | D6 | C4○D4 | C4○D8 | C4○D12 | C8⋊C22 | D4⋊2S3 | S3×D4 | D8⋊S3 | Q8.7D6 |
| kernel | D6⋊C8⋊11C2 | C6.Q16 | C8⋊Dic3 | D6⋊C8 | D4⋊Dic3 | C3×D4⋊C4 | C4⋊C4⋊7S3 | D6⋊3D4 | D4⋊C4 | C2×Dic3 | C22×S3 | C4⋊C4 | C2×C8 | C2×D4 | C12 | C6 | C4 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 1 | 1 | 1 | 2 | 2 |
Matrix representation of D6⋊C8⋊11C2 ►in GL4(𝔽73) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 72 |
| 0 | 0 | 1 | 1 |
| 72 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 72 |
| 0 | 0 | 72 | 0 |
| 51 | 0 | 0 | 0 |
| 0 | 10 | 0 | 0 |
| 0 | 0 | 66 | 59 |
| 0 | 0 | 14 | 7 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 30 | 60 |
| 0 | 0 | 13 | 43 |
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,0,1,0,0,72,1],[72,0,0,0,0,1,0,0,0,0,0,72,0,0,72,0],[51,0,0,0,0,10,0,0,0,0,66,14,0,0,59,7],[0,1,0,0,1,0,0,0,0,0,30,13,0,0,60,43] >;
D6⋊C8⋊11C2 in GAP, Magma, Sage, TeX
D_6\rtimes C_8\rtimes_{11}C_2 % in TeX
G:=Group("D6:C8:11C2"); // GroupNames label
G:=SmallGroup(192,338);
// by ID
G=gap.SmallGroup(192,338);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,64,254,219,851,438,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^2=c^8=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^3*b,d*b*d=a^3*b*c^4,d*c*d=a^3*c^3>;
// generators/relations