direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C8.D6, C24.8C23, M4(2)⋊18D6, C12.59C24, C23.58D12, Dic12⋊8C22, D12.22C23, Dic6.22C23, C4.49(C2×D12), (C2×C8).101D6, (C2×C4).58D12, C8.8(C22×S3), C24⋊C2⋊9C22, (C2×C12).204D4, C12.293(C2×D4), C6⋊1(C8.C22), (C6×M4(2))⋊4C2, (C2×M4(2))⋊4S3, C4.56(S3×C23), C6.26(C22×D4), (C2×Dic12)⋊14C2, (C2×C24).69C22, (C22×C4).282D6, C22.74(C2×D12), C2.28(C22×D12), (C22×C6).119D4, (C2×C12).512C23, (C22×Dic6)⋊18C2, (C2×Dic6)⋊63C22, C4○D12.50C22, (C2×D12).230C22, (C3×M4(2))⋊20C22, (C22×C12).267C22, (C2×C24⋊C2)⋊5C2, C3⋊1(C2×C8.C22), (C2×C6).63(C2×D4), (C2×C4○D12).23C2, (C2×C4).224(C22×S3), SmallGroup(192,1306)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C8.D6
G = < a,b,c,d | a2=b8=1, c6=d2=b4, ab=ba, ac=ca, ad=da, cbc-1=b5, dbd-1=b-1, dcd-1=c5 >
Subgroups: 664 in 258 conjugacy classes, 111 normal (25 characteristic)
C1, C2, C2, C2, C3, C4, C4, C4, C22, C22, C22, S3, C6, C6, C6, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, C12, C12, D6, C2×C6, C2×C6, C2×C6, C2×C8, M4(2), SD16, Q16, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, C24, Dic6, Dic6, C4×S3, D12, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×C6, C2×M4(2), C2×SD16, C2×Q16, C8.C22, C22×Q8, C2×C4○D4, C24⋊C2, Dic12, C2×C24, C3×M4(2), C2×Dic6, C2×Dic6, C2×Dic6, S3×C2×C4, C2×D12, C4○D12, C4○D12, C22×Dic3, C2×C3⋊D4, C22×C12, C2×C8.C22, C2×C24⋊C2, C2×Dic12, C8.D6, C6×M4(2), C22×Dic6, C2×C4○D12, C2×C8.D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C24, D12, C22×S3, C8.C22, C22×D4, C2×D12, S3×C23, C2×C8.C22, C8.D6, C22×D12, C2×C8.D6
►On 96 points
(1 47)(2 48)(3 37)(4 38)(5 39)(6 40)(7 41)(8 42)(9 43)(10 44)(11 45)(12 46)(13 26)(14 27)(15 28)(16 29)(17 30)(18 31)(19 32)(20 33)(21 34)(22 35)(23 36)(24 25)(49 87)(50 88)(51 89)(52 90)(53 91)(54 92)(55 93)(56 94)(57 95)(58 96)(59 85)(60 86)(61 80)(62 81)(63 82)(64 83)(65 84)(66 73)(67 74)(68 75)(69 76)(70 77)(71 78)(72 79)
(1 50 30 61 7 56 36 67)(2 57 31 68 8 51 25 62)(3 52 32 63 9 58 26 69)(4 59 33 70 10 53 27 64)(5 54 34 65 11 60 28 71)(6 49 35 72 12 55 29 66)(13 76 37 90 19 82 43 96)(14 83 38 85 20 77 44 91)(15 78 39 92 21 84 45 86)(16 73 40 87 22 79 46 93)(17 80 41 94 23 74 47 88)(18 75 42 89 24 81 48 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 6 7 12)(2 11 8 5)(3 4 9 10)(13 20 19 14)(15 18 21 24)(16 23 22 17)(25 28 31 34)(26 33 32 27)(29 36 35 30)(37 38 43 44)(39 48 45 42)(40 41 46 47)(49 61 55 67)(50 66 56 72)(51 71 57 65)(52 64 58 70)(53 69 59 63)(54 62 60 68)(73 94 79 88)(74 87 80 93)(75 92 81 86)(76 85 82 91)(77 90 83 96)(78 95 84 89)
G:=sub<Sym(96)| (1,47)(2,48)(3,37)(4,38)(5,39)(6,40)(7,41)(8,42)(9,43)(10,44)(11,45)(12,46)(13,26)(14,27)(15,28)(16,29)(17,30)(18,31)(19,32)(20,33)(21,34)(22,35)(23,36)(24,25)(49,87)(50,88)(51,89)(52,90)(53,91)(54,92)(55,93)(56,94)(57,95)(58,96)(59,85)(60,86)(61,80)(62,81)(63,82)(64,83)(65,84)(66,73)(67,74)(68,75)(69,76)(70,77)(71,78)(72,79), (1,50,30,61,7,56,36,67)(2,57,31,68,8,51,25,62)(3,52,32,63,9,58,26,69)(4,59,33,70,10,53,27,64)(5,54,34,65,11,60,28,71)(6,49,35,72,12,55,29,66)(13,76,37,90,19,82,43,96)(14,83,38,85,20,77,44,91)(15,78,39,92,21,84,45,86)(16,73,40,87,22,79,46,93)(17,80,41,94,23,74,47,88)(18,75,42,89,24,81,48,95), (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), (1,6,7,12)(2,11,8,5)(3,4,9,10)(13,20,19,14)(15,18,21,24)(16,23,22,17)(25,28,31,34)(26,33,32,27)(29,36,35,30)(37,38,43,44)(39,48,45,42)(40,41,46,47)(49,61,55,67)(50,66,56,72)(51,71,57,65)(52,64,58,70)(53,69,59,63)(54,62,60,68)(73,94,79,88)(74,87,80,93)(75,92,81,86)(76,85,82,91)(77,90,83,96)(78,95,84,89)>;
G:=Group( (1,47)(2,48)(3,37)(4,38)(5,39)(6,40)(7,41)(8,42)(9,43)(10,44)(11,45)(12,46)(13,26)(14,27)(15,28)(16,29)(17,30)(18,31)(19,32)(20,33)(21,34)(22,35)(23,36)(24,25)(49,87)(50,88)(51,89)(52,90)(53,91)(54,92)(55,93)(56,94)(57,95)(58,96)(59,85)(60,86)(61,80)(62,81)(63,82)(64,83)(65,84)(66,73)(67,74)(68,75)(69,76)(70,77)(71,78)(72,79), (1,50,30,61,7,56,36,67)(2,57,31,68,8,51,25,62)(3,52,32,63,9,58,26,69)(4,59,33,70,10,53,27,64)(5,54,34,65,11,60,28,71)(6,49,35,72,12,55,29,66)(13,76,37,90,19,82,43,96)(14,83,38,85,20,77,44,91)(15,78,39,92,21,84,45,86)(16,73,40,87,22,79,46,93)(17,80,41,94,23,74,47,88)(18,75,42,89,24,81,48,95), (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), (1,6,7,12)(2,11,8,5)(3,4,9,10)(13,20,19,14)(15,18,21,24)(16,23,22,17)(25,28,31,34)(26,33,32,27)(29,36,35,30)(37,38,43,44)(39,48,45,42)(40,41,46,47)(49,61,55,67)(50,66,56,72)(51,71,57,65)(52,64,58,70)(53,69,59,63)(54,62,60,68)(73,94,79,88)(74,87,80,93)(75,92,81,86)(76,85,82,91)(77,90,83,96)(78,95,84,89) );
G=PermutationGroup([[(1,47),(2,48),(3,37),(4,38),(5,39),(6,40),(7,41),(8,42),(9,43),(10,44),(11,45),(12,46),(13,26),(14,27),(15,28),(16,29),(17,30),(18,31),(19,32),(20,33),(21,34),(22,35),(23,36),(24,25),(49,87),(50,88),(51,89),(52,90),(53,91),(54,92),(55,93),(56,94),(57,95),(58,96),(59,85),(60,86),(61,80),(62,81),(63,82),(64,83),(65,84),(66,73),(67,74),(68,75),(69,76),(70,77),(71,78),(72,79)], [(1,50,30,61,7,56,36,67),(2,57,31,68,8,51,25,62),(3,52,32,63,9,58,26,69),(4,59,33,70,10,53,27,64),(5,54,34,65,11,60,28,71),(6,49,35,72,12,55,29,66),(13,76,37,90,19,82,43,96),(14,83,38,85,20,77,44,91),(15,78,39,92,21,84,45,86),(16,73,40,87,22,79,46,93),(17,80,41,94,23,74,47,88),(18,75,42,89,24,81,48,95)], [(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)], [(1,6,7,12),(2,11,8,5),(3,4,9,10),(13,20,19,14),(15,18,21,24),(16,23,22,17),(25,28,31,34),(26,33,32,27),(29,36,35,30),(37,38,43,44),(39,48,45,42),(40,41,46,47),(49,61,55,67),(50,66,56,72),(51,71,57,65),(52,64,58,70),(53,69,59,63),(54,62,60,68),(73,94,79,88),(74,87,80,93),(75,92,81,86),(76,85,82,91),(77,90,83,96),(78,95,84,89)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 12 | ··· | 12 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 |
42 irreducible representations
| dim | 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 | S3 | D4 | D4 | D6 | D6 | D6 | D12 | D12 | C8.C22 | C8.D6 |
| kernel | C2×C8.D6 | C2×C24⋊C2 | C2×Dic12 | C8.D6 | C6×M4(2) | C22×Dic6 | C2×C4○D12 | C2×M4(2) | C2×C12 | C22×C6 | C2×C8 | M4(2) | C22×C4 | C2×C4 | C23 | C6 | C2 |
| # reps | 1 | 2 | 2 | 8 | 1 | 1 | 1 | 1 | 3 | 1 | 2 | 4 | 1 | 6 | 2 | 2 | 4 |
Matrix representation of C2×C8.D6 ►in GL8(𝔽73)
| 72 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 72 |
| 72 | 71 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 59 | 0 | 0 | 0 | 0 |
| 0 | 0 | 14 | 66 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 62 | 1 |
| 0 | 0 | 0 | 0 | 17 | 71 | 72 | 0 |
| 0 | 0 | 0 | 0 | 66 | 72 | 62 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 | 8 | 65 |
| 0 | 0 | 0 | 0 | 9 | 6 | 45 | 5 |
| 0 | 0 | 0 | 0 | 39 | 27 | 40 | 27 |
| 0 | 0 | 0 | 0 | 55 | 27 | 68 | 72 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 72 | 72 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 | 8 | 65 |
| 0 | 0 | 0 | 0 | 14 | 67 | 45 | 51 |
| 0 | 0 | 0 | 0 | 40 | 46 | 40 | 39 |
| 0 | 0 | 0 | 0 | 56 | 46 | 68 | 11 |
G:=sub<GL(8,GF(73))| [72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72],[72,1,0,0,0,0,0,0,71,1,0,0,0,0,0,0,0,0,7,14,0,0,0,0,0,0,59,66,0,0,0,0,0,0,0,0,1,0,17,66,0,0,0,0,0,0,71,72,0,0,0,0,3,62,72,62,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,28,9,39,55,0,0,0,0,0,6,27,27,0,0,0,0,8,45,40,68,0,0,0,0,65,5,27,72],[1,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,28,14,40,56,0,0,0,0,0,67,46,46,0,0,0,0,8,45,40,68,0,0,0,0,65,51,39,11] >;
C2×C8.D6 in GAP, Magma, Sage, TeX
C_2\times C_8.D_6
% in TeX
G:=Group("C2xC8.D6"); // GroupNames label
G:=SmallGroup(192,1306);
// by ID
G=gap.SmallGroup(192,1306);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,184,675,297,80,1684,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^8=1,c^6=d^2=b^4,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^5,d*b*d^-1=b^-1,d*c*d^-1=c^5>;
// generators/relations