metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D14⋊C8, C4.19D28, C28.52D4, C14.3M4(2), (C2×C8)⋊1D7, (C2×C56)⋊1C2, C2.5(C8×D7), C7⋊1(C22⋊C8), C14.5(C2×C8), (C2×C4).93D14, C2.1(D14⋊C4), C2.3(C8⋊D7), C4.27(C7⋊D4), (C2×Dic7).4C4, (C22×D7).2C4, C22.11(C4×D7), C14.6(C22⋊C4), (C2×C28).107C22, (C2×C7⋊C8)⋊9C2, (C2×C4×D7).7C2, (C2×C14).12(C2×C4), SmallGroup(224,26)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D14⋊C8
G = < a,b,c | a14=b2=c8=1, bab=a-1, ac=ca, cbc-1=a7b >
Smallest permutation representation of D14⋊C8
►On 112 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 97 98)(99 100 101 102 103 104 105 106 107 108 109 110 111 112)
(1 110)(2 109)(3 108)(4 107)(5 106)(6 105)(7 104)(8 103)(9 102)(10 101)(11 100)(12 99)(13 112)(14 111)(15 81)(16 80)(17 79)(18 78)(19 77)(20 76)(21 75)(22 74)(23 73)(24 72)(25 71)(26 84)(27 83)(28 82)(29 57)(30 70)(31 69)(32 68)(33 67)(34 66)(35 65)(36 64)(37 63)(38 62)(39 61)(40 60)(41 59)(42 58)(43 97)(44 96)(45 95)(46 94)(47 93)(48 92)(49 91)(50 90)(51 89)(52 88)(53 87)(54 86)(55 85)(56 98)
(1 38 56 20 111 70 85 84)(2 39 43 21 112 57 86 71)(3 40 44 22 99 58 87 72)(4 41 45 23 100 59 88 73)(5 42 46 24 101 60 89 74)(6 29 47 25 102 61 90 75)(7 30 48 26 103 62 91 76)(8 31 49 27 104 63 92 77)(9 32 50 28 105 64 93 78)(10 33 51 15 106 65 94 79)(11 34 52 16 107 66 95 80)(12 35 53 17 108 67 96 81)(13 36 54 18 109 68 97 82)(14 37 55 19 110 69 98 83)
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), (1,110)(2,109)(3,108)(4,107)(5,106)(6,105)(7,104)(8,103)(9,102)(10,101)(11,100)(12,99)(13,112)(14,111)(15,81)(16,80)(17,79)(18,78)(19,77)(20,76)(21,75)(22,74)(23,73)(24,72)(25,71)(26,84)(27,83)(28,82)(29,57)(30,70)(31,69)(32,68)(33,67)(34,66)(35,65)(36,64)(37,63)(38,62)(39,61)(40,60)(41,59)(42,58)(43,97)(44,96)(45,95)(46,94)(47,93)(48,92)(49,91)(50,90)(51,89)(52,88)(53,87)(54,86)(55,85)(56,98), (1,38,56,20,111,70,85,84)(2,39,43,21,112,57,86,71)(3,40,44,22,99,58,87,72)(4,41,45,23,100,59,88,73)(5,42,46,24,101,60,89,74)(6,29,47,25,102,61,90,75)(7,30,48,26,103,62,91,76)(8,31,49,27,104,63,92,77)(9,32,50,28,105,64,93,78)(10,33,51,15,106,65,94,79)(11,34,52,16,107,66,95,80)(12,35,53,17,108,67,96,81)(13,36,54,18,109,68,97,82)(14,37,55,19,110,69,98,83)>;
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), (1,110)(2,109)(3,108)(4,107)(5,106)(6,105)(7,104)(8,103)(9,102)(10,101)(11,100)(12,99)(13,112)(14,111)(15,81)(16,80)(17,79)(18,78)(19,77)(20,76)(21,75)(22,74)(23,73)(24,72)(25,71)(26,84)(27,83)(28,82)(29,57)(30,70)(31,69)(32,68)(33,67)(34,66)(35,65)(36,64)(37,63)(38,62)(39,61)(40,60)(41,59)(42,58)(43,97)(44,96)(45,95)(46,94)(47,93)(48,92)(49,91)(50,90)(51,89)(52,88)(53,87)(54,86)(55,85)(56,98), (1,38,56,20,111,70,85,84)(2,39,43,21,112,57,86,71)(3,40,44,22,99,58,87,72)(4,41,45,23,100,59,88,73)(5,42,46,24,101,60,89,74)(6,29,47,25,102,61,90,75)(7,30,48,26,103,62,91,76)(8,31,49,27,104,63,92,77)(9,32,50,28,105,64,93,78)(10,33,51,15,106,65,94,79)(11,34,52,16,107,66,95,80)(12,35,53,17,108,67,96,81)(13,36,54,18,109,68,97,82)(14,37,55,19,110,69,98,83) );
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)], [(1,110),(2,109),(3,108),(4,107),(5,106),(6,105),(7,104),(8,103),(9,102),(10,101),(11,100),(12,99),(13,112),(14,111),(15,81),(16,80),(17,79),(18,78),(19,77),(20,76),(21,75),(22,74),(23,73),(24,72),(25,71),(26,84),(27,83),(28,82),(29,57),(30,70),(31,69),(32,68),(33,67),(34,66),(35,65),(36,64),(37,63),(38,62),(39,61),(40,60),(41,59),(42,58),(43,97),(44,96),(45,95),(46,94),(47,93),(48,92),(49,91),(50,90),(51,89),(52,88),(53,87),(54,86),(55,85),(56,98)], [(1,38,56,20,111,70,85,84),(2,39,43,21,112,57,86,71),(3,40,44,22,99,58,87,72),(4,41,45,23,100,59,88,73),(5,42,46,24,101,60,89,74),(6,29,47,25,102,61,90,75),(7,30,48,26,103,62,91,76),(8,31,49,27,104,63,92,77),(9,32,50,28,105,64,93,78),(10,33,51,15,106,65,94,79),(11,34,52,16,107,66,95,80),(12,35,53,17,108,67,96,81),(13,36,54,18,109,68,97,82),(14,37,55,19,110,69,98,83)]])
D14⋊C8 is a maximal subgroup of
C42.282D14 C8×D28 C8⋊6D28 C42.243D14 C42.182D14 C8⋊9D28 C42.185D14 D7×C22⋊C8 C7⋊D4⋊C8 D14⋊M4(2) D14⋊C8⋊C2 D14⋊2M4(2) Dic7⋊M4(2) C7⋊C8⋊26D4 D4⋊D28 D14.D8 D4.6D28 D14.SD16 C8⋊Dic7⋊C2 D4⋊3D28 D4.D28 C56⋊1C4⋊C2 D14.1SD16 Q8⋊2D28 D14⋊4Q16 D14.Q16 Q8.D28 D28⋊4D4 D14⋊C8.C2 (C2×C8).D14 C42.200D14 D28⋊C8 C42.202D14 D14⋊3M4(2) C28⋊2M4(2) C42.31D14 D14.2SD16 D14.4SD16 C4.Q8⋊D7 C28.(C4○D4) D14.5D8 D14.2Q16 C2.D8⋊D7 C2.D8⋊7D7 C8×C7⋊D4 (C22×C8)⋊D7 C56⋊32D4 D14⋊6M4(2) C56⋊D4 C56⋊18D4 (C2×D28).14C4 D28⋊D4 Dic14⋊D4 D14⋊6SD16 Dic14⋊7D4 D28⋊7D4 Dic14.16D4 D14⋊5Q16 D28.17D4
D14⋊C8 is a maximal quotient of
C4.8Dic28 C4.17D56 (C22×D7)⋊C8 (C2×Dic7)⋊C8 D28⋊2C8 Dic14⋊2C8 D14⋊C16 D28.C8 M5(2)⋊D7 Dic14.C8 (C2×C56)⋊5C4
68 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 14A | ··· | 14I | 28A | ··· | 28L | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 14 | 14 | 1 | 1 | 1 | 1 | 14 | 14 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 14 | 14 | 14 | 14 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | D4 | D7 | M4(2) | D14 | D28 | C7⋊D4 | C4×D7 | C8×D7 | C8⋊D7 |
| kernel | D14⋊C8 | C2×C7⋊C8 | C2×C56 | C2×C4×D7 | C2×Dic7 | C22×D7 | D14 | C28 | C2×C8 | C14 | C2×C4 | C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 2 | 3 | 2 | 3 | 6 | 6 | 6 | 12 | 12 |
Matrix representation of D14⋊C8 ►in GL4(𝔽113) generated by
| 34 | 88 | 0 | 0 |
| 25 | 88 | 0 | 0 |
| 0 | 0 | 10 | 10 |
| 0 | 0 | 103 | 24 |
| 0 | 112 | 0 | 0 |
| 112 | 0 | 0 | 0 |
| 0 | 0 | 10 | 10 |
| 0 | 0 | 24 | 103 |
| 44 | 0 | 0 | 0 |
| 0 | 44 | 0 | 0 |
| 0 | 0 | 84 | 106 |
| 0 | 0 | 7 | 29 |
G:=sub<GL(4,GF(113))| [34,25,0,0,88,88,0,0,0,0,10,103,0,0,10,24],[0,112,0,0,112,0,0,0,0,0,10,24,0,0,10,103],[44,0,0,0,0,44,0,0,0,0,84,7,0,0,106,29] >;
D14⋊C8 in GAP, Magma, Sage, TeX
D_{14}\rtimes C_8 % in TeX
G:=Group("D14:C8"); // GroupNames label
G:=SmallGroup(224,26);
// by ID
G=gap.SmallGroup(224,26);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,121,31,86,6917]);
// Polycyclic
G:=Group<a,b,c|a^14=b^2=c^8=1,b*a*b=a^-1,a*c=c*a,c*b*c^-1=a^7*b>;
// generators/relations
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