direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C14×D8, C28.41D4, C56⋊12C22, C28.44C23, (C2×C8)⋊3C14, C8⋊2(C2×C14), C4.6(C7×D4), (C2×C56)⋊11C2, D4⋊1(C2×C14), (C2×D4)⋊4C14, (D4×C14)⋊13C2, C14.74(C2×D4), C2.11(D4×C14), (C2×C14).52D4, (C7×D4)⋊10C22, C4.1(C22×C14), C22.14(C7×D4), (C2×C28).129C22, (C2×C4).25(C2×C14), SmallGroup(224,167)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C14×D8
G = < a,b,c | a14=b8=c2=1, ab=ba, ac=ca, cbc=b-1 >
Subgroups: 140 in 76 conjugacy classes, 44 normal (16 characteristic)
C1, C2, C2, C2, C4, C22, C22, C7, C8, C2×C4, D4, D4, C23, C14, C14, C14, C2×C8, D8, C2×D4, C28, C2×C14, C2×C14, C2×D8, C56, C2×C28, C7×D4, C7×D4, C22×C14, C2×C56, C7×D8, D4×C14, C14×D8
Quotients: C1, C2, C22, C7, D4, C23, C14, D8, C2×D4, C2×C14, C2×D8, C7×D4, C22×C14, C7×D8, D4×C14, C14×D8
►On 112 points
Generators in S112
(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 97 98)(99 100 101 102 103 104 105 106 107 108 109 110 111 112)
(1 82 66 102 47 18 33 93)(2 83 67 103 48 19 34 94)(3 84 68 104 49 20 35 95)(4 71 69 105 50 21 36 96)(5 72 70 106 51 22 37 97)(6 73 57 107 52 23 38 98)(7 74 58 108 53 24 39 85)(8 75 59 109 54 25 40 86)(9 76 60 110 55 26 41 87)(10 77 61 111 56 27 42 88)(11 78 62 112 43 28 29 89)(12 79 63 99 44 15 30 90)(13 80 64 100 45 16 31 91)(14 81 65 101 46 17 32 92)
(1 93)(2 94)(3 95)(4 96)(5 97)(6 98)(7 85)(8 86)(9 87)(10 88)(11 89)(12 90)(13 91)(14 92)(15 63)(16 64)(17 65)(18 66)(19 67)(20 68)(21 69)(22 70)(23 57)(24 58)(25 59)(26 60)(27 61)(28 62)(29 78)(30 79)(31 80)(32 81)(33 82)(34 83)(35 84)(36 71)(37 72)(38 73)(39 74)(40 75)(41 76)(42 77)(43 112)(44 99)(45 100)(46 101)(47 102)(48 103)(49 104)(50 105)(51 106)(52 107)(53 108)(54 109)(55 110)(56 111)
G:=sub<Sym(112)| (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)(85,86,87,88,89,90,91,92,93,94,95,96,97,98)(99,100,101,102,103,104,105,106,107,108,109,110,111,112), (1,82,66,102,47,18,33,93)(2,83,67,103,48,19,34,94)(3,84,68,104,49,20,35,95)(4,71,69,105,50,21,36,96)(5,72,70,106,51,22,37,97)(6,73,57,107,52,23,38,98)(7,74,58,108,53,24,39,85)(8,75,59,109,54,25,40,86)(9,76,60,110,55,26,41,87)(10,77,61,111,56,27,42,88)(11,78,62,112,43,28,29,89)(12,79,63,99,44,15,30,90)(13,80,64,100,45,16,31,91)(14,81,65,101,46,17,32,92), (1,93)(2,94)(3,95)(4,96)(5,97)(6,98)(7,85)(8,86)(9,87)(10,88)(11,89)(12,90)(13,91)(14,92)(15,63)(16,64)(17,65)(18,66)(19,67)(20,68)(21,69)(22,70)(23,57)(24,58)(25,59)(26,60)(27,61)(28,62)(29,78)(30,79)(31,80)(32,81)(33,82)(34,83)(35,84)(36,71)(37,72)(38,73)(39,74)(40,75)(41,76)(42,77)(43,112)(44,99)(45,100)(46,101)(47,102)(48,103)(49,104)(50,105)(51,106)(52,107)(53,108)(54,109)(55,110)(56,111)>;
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)(85,86,87,88,89,90,91,92,93,94,95,96,97,98)(99,100,101,102,103,104,105,106,107,108,109,110,111,112), (1,82,66,102,47,18,33,93)(2,83,67,103,48,19,34,94)(3,84,68,104,49,20,35,95)(4,71,69,105,50,21,36,96)(5,72,70,106,51,22,37,97)(6,73,57,107,52,23,38,98)(7,74,58,108,53,24,39,85)(8,75,59,109,54,25,40,86)(9,76,60,110,55,26,41,87)(10,77,61,111,56,27,42,88)(11,78,62,112,43,28,29,89)(12,79,63,99,44,15,30,90)(13,80,64,100,45,16,31,91)(14,81,65,101,46,17,32,92), (1,93)(2,94)(3,95)(4,96)(5,97)(6,98)(7,85)(8,86)(9,87)(10,88)(11,89)(12,90)(13,91)(14,92)(15,63)(16,64)(17,65)(18,66)(19,67)(20,68)(21,69)(22,70)(23,57)(24,58)(25,59)(26,60)(27,61)(28,62)(29,78)(30,79)(31,80)(32,81)(33,82)(34,83)(35,84)(36,71)(37,72)(38,73)(39,74)(40,75)(41,76)(42,77)(43,112)(44,99)(45,100)(46,101)(47,102)(48,103)(49,104)(50,105)(51,106)(52,107)(53,108)(54,109)(55,110)(56,111) );
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),(85,86,87,88,89,90,91,92,93,94,95,96,97,98),(99,100,101,102,103,104,105,106,107,108,109,110,111,112)], [(1,82,66,102,47,18,33,93),(2,83,67,103,48,19,34,94),(3,84,68,104,49,20,35,95),(4,71,69,105,50,21,36,96),(5,72,70,106,51,22,37,97),(6,73,57,107,52,23,38,98),(7,74,58,108,53,24,39,85),(8,75,59,109,54,25,40,86),(9,76,60,110,55,26,41,87),(10,77,61,111,56,27,42,88),(11,78,62,112,43,28,29,89),(12,79,63,99,44,15,30,90),(13,80,64,100,45,16,31,91),(14,81,65,101,46,17,32,92)], [(1,93),(2,94),(3,95),(4,96),(5,97),(6,98),(7,85),(8,86),(9,87),(10,88),(11,89),(12,90),(13,91),(14,92),(15,63),(16,64),(17,65),(18,66),(19,67),(20,68),(21,69),(22,70),(23,57),(24,58),(25,59),(26,60),(27,61),(28,62),(29,78),(30,79),(31,80),(32,81),(33,82),(34,83),(35,84),(36,71),(37,72),(38,73),(39,74),(40,75),(41,76),(42,77),(43,112),(44,99),(45,100),(46,101),(47,102),(48,103),(49,104),(50,105),(51,106),(52,107),(53,108),(54,109),(55,110),(56,111)]])
C14×D8 is a maximal subgroup of
C14.SD32 D8.Dic7 D8.D14 Dic7⋊D8 C56⋊5D4 D8⋊Dic7 (C2×D8).D7 C56⋊11D4 C56.22D4 D28⋊D4 C56⋊6D4 Dic14⋊D4 C56⋊12D4 C56.23D4 D8⋊13D14
98 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 7A | ··· | 7F | 8A | 8B | 8C | 8D | 14A | ··· | 14R | 14S | ··· | 14AP | 28A | ··· | 28L | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 7 | ··· | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | ··· | 2 | 2 | ··· | 2 |
98 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C7 | C14 | C14 | C14 | D4 | D4 | D8 | C7×D4 | C7×D4 | C7×D8 |
| kernel | C14×D8 | C2×C56 | C7×D8 | D4×C14 | C2×D8 | C2×C8 | D8 | C2×D4 | C28 | C2×C14 | C14 | C4 | C22 | C2 |
| # reps | 1 | 1 | 4 | 2 | 6 | 6 | 24 | 12 | 1 | 1 | 4 | 6 | 6 | 24 |
Matrix representation of C14×D8 ►in GL4(𝔽113) generated by
| 7 | 0 | 0 | 0 |
| 0 | 7 | 0 | 0 |
| 0 | 0 | 83 | 0 |
| 0 | 0 | 0 | 83 |
| 0 | 112 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 31 |
| 0 | 0 | 51 | 62 |
| 0 | 112 | 0 | 0 |
| 112 | 0 | 0 | 0 |
| 0 | 0 | 62 | 82 |
| 0 | 0 | 62 | 51 |
G:=sub<GL(4,GF(113))| [7,0,0,0,0,7,0,0,0,0,83,0,0,0,0,83],[0,1,0,0,112,0,0,0,0,0,0,51,0,0,31,62],[0,112,0,0,112,0,0,0,0,0,62,62,0,0,82,51] >;
C14×D8 in GAP, Magma, Sage, TeX
C_{14}\times D_8 % in TeX
G:=Group("C14xD8"); // GroupNames label
G:=SmallGroup(224,167);
// by ID
G=gap.SmallGroup(224,167);
# by ID
G:=PCGroup([6,-2,-2,-2,-7,-2,-2,697,5044,2530,88]);
// Polycyclic
G:=Group<a,b,c|a^14=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations