metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D6⋊C8⋊23C2, C22⋊C8⋊12S3, (C2×D12).8C4, (C4×S3).47D4, C6.8(C8○D4), C4.196(S3×D4), (C2×C8).163D6, C12.355(C2×D4), C23.17(C4×S3), (C2×Dic6).8C4, (C22×C4).94D6, C2.10(C8○D12), C12.55D4⋊2C2, D6.1(C22⋊C4), (C2×C12).822C23, C2.10(D12.C4), (C2×C24).214C22, Dic3.2(C22⋊C4), (C22×C12).93C22, (S3×C2×C8)⋊14C2, (C2×C4).31(C4×S3), (C2×C3⋊D4).4C4, (C2×C8⋊S3)⋊12C2, C6.9(C2×C22⋊C4), (C3×C22⋊C8)⋊21C2, (C2×C12).39(C2×C4), (C2×C4○D12).1C2, C2.10(S3×C22⋊C4), C22.104(S3×C2×C4), C3⋊1((C22×C8)⋊C2), (C2×C3⋊C8).301C22, (S3×C2×C4).274C22, (C2×C6).77(C22×C4), (C22×C6).40(C2×C4), (C22×S3).12(C2×C4), (C2×C4).764(C22×S3), (C2×Dic3).50(C2×C4), SmallGroup(192,286)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D6⋊C8⋊C2
G = < a,b,c,d | a6=b2=c8=d2=1, bab=a-1, ac=ca, ad=da, cbc-1=a3b, dbd=bc4, dcd=a3c5 >
Subgroups: 416 in 158 conjugacy classes, 55 normal (47 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, C3⋊C8, C24, Dic6, C4×S3, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×C6, C22⋊C8, C22⋊C8, C22×C8, C2×M4(2), C2×C4○D4, S3×C8, C8⋊S3, C2×C3⋊C8, C2×C24, C2×Dic6, S3×C2×C4, C2×D12, C4○D12, C2×C3⋊D4, C22×C12, (C22×C8)⋊C2, D6⋊C8, C12.55D4, C3×C22⋊C8, S3×C2×C8, C2×C8⋊S3, C2×C4○D12, D6⋊C8⋊C2
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22⋊C4, C22×C4, C2×D4, C4×S3, C22×S3, C2×C22⋊C4, C8○D4, S3×C2×C4, S3×D4, (C22×C8)⋊C2, S3×C22⋊C4, C8○D12, D12.C4, D6⋊C8⋊C2
►On 96 points
(1 63 30 79 93 19)(2 64 31 80 94 20)(3 57 32 73 95 21)(4 58 25 74 96 22)(5 59 26 75 89 23)(6 60 27 76 90 24)(7 61 28 77 91 17)(8 62 29 78 92 18)(9 85 67 52 38 42)(10 86 68 53 39 43)(11 87 69 54 40 44)(12 88 70 55 33 45)(13 81 71 56 34 46)(14 82 72 49 35 47)(15 83 65 50 36 48)(16 84 66 51 37 41)
(1 26)(2 24)(3 28)(4 18)(5 30)(6 20)(7 32)(8 22)(9 42)(10 68)(11 44)(12 70)(13 46)(14 72)(15 48)(16 66)(17 73)(19 75)(21 77)(23 79)(25 78)(27 80)(29 74)(31 76)(34 81)(36 83)(38 85)(40 87)(41 51)(43 53)(45 55)(47 49)(50 65)(52 67)(54 69)(56 71)(57 61)(58 92)(59 63)(60 94)(62 96)(64 90)(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 53)(2 15)(3 55)(4 9)(5 49)(6 11)(7 51)(8 13)(10 79)(12 73)(14 75)(16 77)(17 66)(18 46)(19 68)(20 48)(21 70)(22 42)(23 72)(24 44)(25 67)(26 47)(27 69)(28 41)(29 71)(30 43)(31 65)(32 45)(33 57)(34 92)(35 59)(36 94)(37 61)(38 96)(39 63)(40 90)(50 80)(52 74)(54 76)(56 78)(58 85)(60 87)(62 81)(64 83)(82 89)(84 91)(86 93)(88 95)
G:=sub<Sym(96)| (1,63,30,79,93,19)(2,64,31,80,94,20)(3,57,32,73,95,21)(4,58,25,74,96,22)(5,59,26,75,89,23)(6,60,27,76,90,24)(7,61,28,77,91,17)(8,62,29,78,92,18)(9,85,67,52,38,42)(10,86,68,53,39,43)(11,87,69,54,40,44)(12,88,70,55,33,45)(13,81,71,56,34,46)(14,82,72,49,35,47)(15,83,65,50,36,48)(16,84,66,51,37,41), (1,26)(2,24)(3,28)(4,18)(5,30)(6,20)(7,32)(8,22)(9,42)(10,68)(11,44)(12,70)(13,46)(14,72)(15,48)(16,66)(17,73)(19,75)(21,77)(23,79)(25,78)(27,80)(29,74)(31,76)(34,81)(36,83)(38,85)(40,87)(41,51)(43,53)(45,55)(47,49)(50,65)(52,67)(54,69)(56,71)(57,61)(58,92)(59,63)(60,94)(62,96)(64,90)(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), (1,53)(2,15)(3,55)(4,9)(5,49)(6,11)(7,51)(8,13)(10,79)(12,73)(14,75)(16,77)(17,66)(18,46)(19,68)(20,48)(21,70)(22,42)(23,72)(24,44)(25,67)(26,47)(27,69)(28,41)(29,71)(30,43)(31,65)(32,45)(33,57)(34,92)(35,59)(36,94)(37,61)(38,96)(39,63)(40,90)(50,80)(52,74)(54,76)(56,78)(58,85)(60,87)(62,81)(64,83)(82,89)(84,91)(86,93)(88,95)>;
G:=Group( (1,63,30,79,93,19)(2,64,31,80,94,20)(3,57,32,73,95,21)(4,58,25,74,96,22)(5,59,26,75,89,23)(6,60,27,76,90,24)(7,61,28,77,91,17)(8,62,29,78,92,18)(9,85,67,52,38,42)(10,86,68,53,39,43)(11,87,69,54,40,44)(12,88,70,55,33,45)(13,81,71,56,34,46)(14,82,72,49,35,47)(15,83,65,50,36,48)(16,84,66,51,37,41), (1,26)(2,24)(3,28)(4,18)(5,30)(6,20)(7,32)(8,22)(9,42)(10,68)(11,44)(12,70)(13,46)(14,72)(15,48)(16,66)(17,73)(19,75)(21,77)(23,79)(25,78)(27,80)(29,74)(31,76)(34,81)(36,83)(38,85)(40,87)(41,51)(43,53)(45,55)(47,49)(50,65)(52,67)(54,69)(56,71)(57,61)(58,92)(59,63)(60,94)(62,96)(64,90)(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), (1,53)(2,15)(3,55)(4,9)(5,49)(6,11)(7,51)(8,13)(10,79)(12,73)(14,75)(16,77)(17,66)(18,46)(19,68)(20,48)(21,70)(22,42)(23,72)(24,44)(25,67)(26,47)(27,69)(28,41)(29,71)(30,43)(31,65)(32,45)(33,57)(34,92)(35,59)(36,94)(37,61)(38,96)(39,63)(40,90)(50,80)(52,74)(54,76)(56,78)(58,85)(60,87)(62,81)(64,83)(82,89)(84,91)(86,93)(88,95) );
G=PermutationGroup([[(1,63,30,79,93,19),(2,64,31,80,94,20),(3,57,32,73,95,21),(4,58,25,74,96,22),(5,59,26,75,89,23),(6,60,27,76,90,24),(7,61,28,77,91,17),(8,62,29,78,92,18),(9,85,67,52,38,42),(10,86,68,53,39,43),(11,87,69,54,40,44),(12,88,70,55,33,45),(13,81,71,56,34,46),(14,82,72,49,35,47),(15,83,65,50,36,48),(16,84,66,51,37,41)], [(1,26),(2,24),(3,28),(4,18),(5,30),(6,20),(7,32),(8,22),(9,42),(10,68),(11,44),(12,70),(13,46),(14,72),(15,48),(16,66),(17,73),(19,75),(21,77),(23,79),(25,78),(27,80),(29,74),(31,76),(34,81),(36,83),(38,85),(40,87),(41,51),(43,53),(45,55),(47,49),(50,65),(52,67),(54,69),(56,71),(57,61),(58,92),(59,63),(60,94),(62,96),(64,90),(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)], [(1,53),(2,15),(3,55),(4,9),(5,49),(6,11),(7,51),(8,13),(10,79),(12,73),(14,75),(16,77),(17,66),(18,46),(19,68),(20,48),(21,70),(22,42),(23,72),(24,44),(25,67),(26,47),(27,69),(28,41),(29,71),(30,43),(31,65),(32,45),(33,57),(34,92),(35,59),(36,94),(37,61),(38,96),(39,63),(40,90),(50,80),(52,74),(54,76),(56,78),(58,85),(60,87),(62,81),(64,83),(82,89),(84,91),(86,93),(88,95)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 8K | 8L | 12A | 12B | 12C | 12D | 12E | 12F | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 4 | 6 | 6 | 12 | 2 | 1 | 1 | 1 | 1 | 4 | 6 | 6 | 12 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | S3 | D4 | D6 | D6 | C4×S3 | C4×S3 | C8○D4 | C8○D12 | S3×D4 | D12.C4 |
| kernel | D6⋊C8⋊C2 | D6⋊C8 | C12.55D4 | C3×C22⋊C8 | S3×C2×C8 | C2×C8⋊S3 | C2×C4○D12 | C2×Dic6 | C2×D12 | C2×C3⋊D4 | C22⋊C8 | C4×S3 | C2×C8 | C22×C4 | C2×C4 | C23 | C6 | C2 | C4 | C2 |
| # reps | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 1 | 4 | 2 | 1 | 2 | 2 | 8 | 8 | 2 | 2 |
Matrix representation of D6⋊C8⋊C2 ►in GL4(𝔽73) generated by
| 1 | 72 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 0 | 72 | 0 | 0 |
| 72 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 46 | 72 |
| 70 | 6 | 0 | 0 |
| 67 | 3 | 0 | 0 |
| 0 | 0 | 27 | 2 |
| 0 | 0 | 0 | 46 |
| 43 | 60 | 0 | 0 |
| 13 | 30 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 46 | 72 |
G:=sub<GL(4,GF(73))| [1,1,0,0,72,0,0,0,0,0,72,0,0,0,0,72],[0,72,0,0,72,0,0,0,0,0,1,46,0,0,0,72],[70,67,0,0,6,3,0,0,0,0,27,0,0,0,2,46],[43,13,0,0,60,30,0,0,0,0,1,46,0,0,0,72] >;
D6⋊C8⋊C2 in GAP, Magma, Sage, TeX
D_6\rtimes C_8\rtimes C_2
% in TeX
G:=Group("D6:C8:C2"); // GroupNames label
G:=SmallGroup(192,286);
// by ID
G=gap.SmallGroup(192,286);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,219,58,136,6278]);
// 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=b*c^4,d*c*d=a^3*c^5>;
// generators/relations