metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D28⋊3C4, C28.1D4, C4.9D28, C14.7D8, C14.7SD16, C4⋊C4⋊1D7, C4.1(C4×D7), C28.3(C2×C4), C7⋊1(D4⋊C4), C2.2(D4⋊D7), (C2×D28).5C2, (C2×C14).30D4, (C2×C4).35D14, C2.2(Q8⋊D7), C2.5(D14⋊C4), C14.3(C22⋊C4), (C2×C28).10C22, C22.14(C7⋊D4), (C2×C7⋊C8)⋊1C2, (C7×C4⋊C4)⋊1C2, SmallGroup(224,15)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C14.D8
G = < a,b,c | a14=b8=c2=1, bab-1=cac=a-1, cbc=a7b-1 >
Smallest permutation representation of C14.D8
►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 63 111 78 18 96 42 44)(2 62 112 77 19 95 29 43)(3 61 99 76 20 94 30 56)(4 60 100 75 21 93 31 55)(5 59 101 74 22 92 32 54)(6 58 102 73 23 91 33 53)(7 57 103 72 24 90 34 52)(8 70 104 71 25 89 35 51)(9 69 105 84 26 88 36 50)(10 68 106 83 27 87 37 49)(11 67 107 82 28 86 38 48)(12 66 108 81 15 85 39 47)(13 65 109 80 16 98 40 46)(14 64 110 79 17 97 41 45)
(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(15 21)(16 20)(17 19)(22 28)(23 27)(24 26)(29 110)(30 109)(31 108)(32 107)(33 106)(34 105)(35 104)(36 103)(37 102)(38 101)(39 100)(40 99)(41 112)(42 111)(43 57)(44 70)(45 69)(46 68)(47 67)(48 66)(49 65)(50 64)(51 63)(52 62)(53 61)(54 60)(55 59)(56 58)(71 96)(72 95)(73 94)(74 93)(75 92)(76 91)(77 90)(78 89)(79 88)(80 87)(81 86)(82 85)(83 98)(84 97)
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), (1,63,111,78,18,96,42,44)(2,62,112,77,19,95,29,43)(3,61,99,76,20,94,30,56)(4,60,100,75,21,93,31,55)(5,59,101,74,22,92,32,54)(6,58,102,73,23,91,33,53)(7,57,103,72,24,90,34,52)(8,70,104,71,25,89,35,51)(9,69,105,84,26,88,36,50)(10,68,106,83,27,87,37,49)(11,67,107,82,28,86,38,48)(12,66,108,81,15,85,39,47)(13,65,109,80,16,98,40,46)(14,64,110,79,17,97,41,45), (2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(15,21)(16,20)(17,19)(22,28)(23,27)(24,26)(29,110)(30,109)(31,108)(32,107)(33,106)(34,105)(35,104)(36,103)(37,102)(38,101)(39,100)(40,99)(41,112)(42,111)(43,57)(44,70)(45,69)(46,68)(47,67)(48,66)(49,65)(50,64)(51,63)(52,62)(53,61)(54,60)(55,59)(56,58)(71,96)(72,95)(73,94)(74,93)(75,92)(76,91)(77,90)(78,89)(79,88)(80,87)(81,86)(82,85)(83,98)(84,97)>;
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), (1,63,111,78,18,96,42,44)(2,62,112,77,19,95,29,43)(3,61,99,76,20,94,30,56)(4,60,100,75,21,93,31,55)(5,59,101,74,22,92,32,54)(6,58,102,73,23,91,33,53)(7,57,103,72,24,90,34,52)(8,70,104,71,25,89,35,51)(9,69,105,84,26,88,36,50)(10,68,106,83,27,87,37,49)(11,67,107,82,28,86,38,48)(12,66,108,81,15,85,39,47)(13,65,109,80,16,98,40,46)(14,64,110,79,17,97,41,45), (2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(15,21)(16,20)(17,19)(22,28)(23,27)(24,26)(29,110)(30,109)(31,108)(32,107)(33,106)(34,105)(35,104)(36,103)(37,102)(38,101)(39,100)(40,99)(41,112)(42,111)(43,57)(44,70)(45,69)(46,68)(47,67)(48,66)(49,65)(50,64)(51,63)(52,62)(53,61)(54,60)(55,59)(56,58)(71,96)(72,95)(73,94)(74,93)(75,92)(76,91)(77,90)(78,89)(79,88)(80,87)(81,86)(82,85)(83,98)(84,97) );
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)], [(1,63,111,78,18,96,42,44),(2,62,112,77,19,95,29,43),(3,61,99,76,20,94,30,56),(4,60,100,75,21,93,31,55),(5,59,101,74,22,92,32,54),(6,58,102,73,23,91,33,53),(7,57,103,72,24,90,34,52),(8,70,104,71,25,89,35,51),(9,69,105,84,26,88,36,50),(10,68,106,83,27,87,37,49),(11,67,107,82,28,86,38,48),(12,66,108,81,15,85,39,47),(13,65,109,80,16,98,40,46),(14,64,110,79,17,97,41,45)], [(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(15,21),(16,20),(17,19),(22,28),(23,27),(24,26),(29,110),(30,109),(31,108),(32,107),(33,106),(34,105),(35,104),(36,103),(37,102),(38,101),(39,100),(40,99),(41,112),(42,111),(43,57),(44,70),(45,69),(46,68),(47,67),(48,66),(49,65),(50,64),(51,63),(52,62),(53,61),(54,60),(55,59),(56,58),(71,96),(72,95),(73,94),(74,93),(75,92),(76,91),(77,90),(78,89),(79,88),(80,87),(81,86),(82,85),(83,98),(84,97)]])
C14.D8 is a maximal subgroup of
Dic7.SD16 Dic14⋊2D4 C4⋊C4.D14 D7×D4⋊C4 (D4×D7)⋊C4 D4⋊D28 D4.6D28 D28⋊3D4 Q8⋊C4⋊D7 Dic14.11D4 Q8⋊Dic7⋊C2 Q8⋊(C4×D7) Q8⋊2D7⋊C4 Q8.D28 D28⋊4D4 D28.12D4 Dic7⋊8SD16 D14.4SD16 C8⋊8D28 C56⋊7D4 C4.Q8⋊D7 D56⋊9C4 D28⋊Q8 D28.Q8 Dic7⋊5D8 D14.5D8 C8⋊7D28 C2.D8⋊D7 C8⋊3D28 C56⋊C2⋊C4 D28⋊2Q8 D28.2Q8 C4○D28⋊C4 (C2×C14).40D8 C4⋊C4.228D14 C4⋊C4⋊36D14 C4.(C2×D28) C4⋊C4.236D14 C4×D4⋊D7 C42.48D14 C28⋊7D8 D4.1D28 C4×Q8⋊D7 C42.56D14 Q8⋊D28 Q8.1D28 D28⋊16D4 D28⋊17D4 C7⋊C8⋊22D4 C4⋊D4⋊D7 D28.36D4 D28.37D4 C7⋊C8⋊24D4 C7⋊C8⋊6D4 D28.4Q8 C42.70D14 C42.216D14 D28⋊5Q8 D28⋊6Q8 C28.D8 C42.82D14
C14.D8 is a maximal quotient of
C14.C4≀C2 C28.47D8 D28⋊2C8 C4.D56 D56⋊8C4 C14.D16 C56.6D4 D56.C4 C56.8D4 Dic28.C4 C28.C42
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 28A | ··· | 28R |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 28 | 28 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 14 | 14 | 14 | 14 | 2 | ··· | 2 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C4 | D4 | D4 | D7 | D8 | SD16 | D14 | C4×D7 | D28 | C7⋊D4 | D4⋊D7 | Q8⋊D7 |
| kernel | C14.D8 | C2×C7⋊C8 | C7×C4⋊C4 | C2×D28 | D28 | C28 | C2×C14 | C4⋊C4 | C14 | C14 | C2×C4 | C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 3 | 2 | 2 | 3 | 6 | 6 | 6 | 3 | 3 |
Matrix representation of C14.D8 ►in GL4(𝔽113) generated by
| 25 | 25 | 0 | 0 |
| 88 | 79 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 112 |
| 104 | 46 | 0 | 0 |
| 13 | 9 | 0 | 0 |
| 0 | 0 | 87 | 106 |
| 0 | 0 | 16 | 0 |
| 1 | 0 | 0 | 0 |
| 79 | 112 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 77 | 112 |
G:=sub<GL(4,GF(113))| [25,88,0,0,25,79,0,0,0,0,112,0,0,0,0,112],[104,13,0,0,46,9,0,0,0,0,87,16,0,0,106,0],[1,79,0,0,0,112,0,0,0,0,1,77,0,0,0,112] >;
C14.D8 in GAP, Magma, Sage, TeX
C_{14}.D_8 % in TeX
G:=Group("C14.D8"); // GroupNames label
G:=SmallGroup(224,15);
// by ID
G=gap.SmallGroup(224,15);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,121,31,579,297,69,6917]);
// Polycyclic
G:=Group<a,b,c|a^14=b^8=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=a^7*b^-1>;
// generators/relations
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