metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2×C14)⋊8D8, (C7×D4)⋊13D4, D4⋊5(C7⋊D4), C7⋊5(C22⋊D8), C14.72(C2×D8), (C22×D4)⋊1D7, C28⋊7D4⋊25C2, C22⋊3(D4⋊D7), (C2×C28).300D4, C28.205(C2×D4), C14.71C22≀C2, (C2×D4).198D14, D4⋊Dic7⋊38C2, (C2×D28)⋊14C22, C4⋊Dic7⋊21C22, C28.55D4⋊15C2, (C2×C28).472C23, (C22×C4).149D14, (C22×C14).196D4, C2.4(C24⋊D7), C23.85(C7⋊D4), C14.102(C8⋊C22), (D4×C14).240C22, C2.22(D4.D14), (C22×C28).197C22, (D4×C2×C14)⋊1C2, (C2×D4⋊D7)⋊23C2, (C2×C7⋊C8)⋊10C22, C2.26(C2×D4⋊D7), C4.58(C2×C7⋊D4), (C2×C14).553(C2×D4), (C2×C4).83(C7⋊D4), (C2×C4).558(C22×D7), C22.216(C2×C7⋊D4), SmallGroup(448,751)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C2×C14)⋊8D8
G = < a,b,c,d | a2=b14=c8=d2=1, ab=ba, cac-1=dad=ab7, cbc-1=dbd=b-1, dcd=c-1 >
Subgroups: 852 in 198 conjugacy classes, 51 normal (25 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C7, C8, C2×C4, C2×C4, D4, D4, C23, C23, D7, C14, C14, C22⋊C4, C4⋊C4, C2×C8, D8, C22×C4, C2×D4, C2×D4, C24, Dic7, C28, C28, D14, C2×C14, C2×C14, C2×C14, C22⋊C8, D4⋊C4, C4⋊D4, C2×D8, C22×D4, C7⋊C8, D28, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×D4, C7×D4, C22×D7, C22×C14, C22×C14, C22⋊D8, C2×C7⋊C8, C4⋊Dic7, D14⋊C4, D4⋊D7, C2×D28, C2×C7⋊D4, C22×C28, D4×C14, D4×C14, C23×C14, C28.55D4, D4⋊Dic7, C28⋊7D4, C2×D4⋊D7, D4×C2×C14, (C2×C14)⋊8D8
Quotients: C1, C2, C22, D4, C23, D7, D8, C2×D4, D14, C22≀C2, C2×D8, C8⋊C22, C7⋊D4, C22×D7, C22⋊D8, D4⋊D7, C2×C7⋊D4, C2×D4⋊D7, D4.D14, C24⋊D7, (C2×C14)⋊8D8
►On 112 points
Generators in S112
(1 87)(2 88)(3 89)(4 90)(5 91)(6 92)(7 93)(8 94)(9 95)(10 96)(11 97)(12 98)(13 85)(14 86)(15 112)(16 99)(17 100)(18 101)(19 102)(20 103)(21 104)(22 105)(23 106)(24 107)(25 108)(26 109)(27 110)(28 111)(29 52)(30 53)(31 54)(32 55)(33 56)(34 43)(35 44)(36 45)(37 46)(38 47)(39 48)(40 49)(41 50)(42 51)(57 82)(58 83)(59 84)(60 71)(61 72)(62 73)(63 74)(64 75)(65 76)(66 77)(67 78)(68 79)(69 80)(70 81)
(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)
(1 106 46 57 87 16 37 75)(2 105 47 70 88 15 38 74)(3 104 48 69 89 28 39 73)(4 103 49 68 90 27 40 72)(5 102 50 67 91 26 41 71)(6 101 51 66 92 25 42 84)(7 100 52 65 93 24 29 83)(8 99 53 64 94 23 30 82)(9 112 54 63 95 22 31 81)(10 111 55 62 96 21 32 80)(11 110 56 61 97 20 33 79)(12 109 43 60 98 19 34 78)(13 108 44 59 85 18 35 77)(14 107 45 58 86 17 36 76)
(1 75)(2 74)(3 73)(4 72)(5 71)(6 84)(7 83)(8 82)(9 81)(10 80)(11 79)(12 78)(13 77)(14 76)(15 47)(16 46)(17 45)(18 44)(19 43)(20 56)(21 55)(22 54)(23 53)(24 52)(25 51)(26 50)(27 49)(28 48)(29 100)(30 99)(31 112)(32 111)(33 110)(34 109)(35 108)(36 107)(37 106)(38 105)(39 104)(40 103)(41 102)(42 101)(57 87)(58 86)(59 85)(60 98)(61 97)(62 96)(63 95)(64 94)(65 93)(66 92)(67 91)(68 90)(69 89)(70 88)
G:=sub<Sym(112)| (1,87)(2,88)(3,89)(4,90)(5,91)(6,92)(7,93)(8,94)(9,95)(10,96)(11,97)(12,98)(13,85)(14,86)(15,112)(16,99)(17,100)(18,101)(19,102)(20,103)(21,104)(22,105)(23,106)(24,107)(25,108)(26,109)(27,110)(28,111)(29,52)(30,53)(31,54)(32,55)(33,56)(34,43)(35,44)(36,45)(37,46)(38,47)(39,48)(40,49)(41,50)(42,51)(57,82)(58,83)(59,84)(60,71)(61,72)(62,73)(63,74)(64,75)(65,76)(66,77)(67,78)(68,79)(69,80)(70,81), (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), (1,106,46,57,87,16,37,75)(2,105,47,70,88,15,38,74)(3,104,48,69,89,28,39,73)(4,103,49,68,90,27,40,72)(5,102,50,67,91,26,41,71)(6,101,51,66,92,25,42,84)(7,100,52,65,93,24,29,83)(8,99,53,64,94,23,30,82)(9,112,54,63,95,22,31,81)(10,111,55,62,96,21,32,80)(11,110,56,61,97,20,33,79)(12,109,43,60,98,19,34,78)(13,108,44,59,85,18,35,77)(14,107,45,58,86,17,36,76), (1,75)(2,74)(3,73)(4,72)(5,71)(6,84)(7,83)(8,82)(9,81)(10,80)(11,79)(12,78)(13,77)(14,76)(15,47)(16,46)(17,45)(18,44)(19,43)(20,56)(21,55)(22,54)(23,53)(24,52)(25,51)(26,50)(27,49)(28,48)(29,100)(30,99)(31,112)(32,111)(33,110)(34,109)(35,108)(36,107)(37,106)(38,105)(39,104)(40,103)(41,102)(42,101)(57,87)(58,86)(59,85)(60,98)(61,97)(62,96)(63,95)(64,94)(65,93)(66,92)(67,91)(68,90)(69,89)(70,88)>;
G:=Group( (1,87)(2,88)(3,89)(4,90)(5,91)(6,92)(7,93)(8,94)(9,95)(10,96)(11,97)(12,98)(13,85)(14,86)(15,112)(16,99)(17,100)(18,101)(19,102)(20,103)(21,104)(22,105)(23,106)(24,107)(25,108)(26,109)(27,110)(28,111)(29,52)(30,53)(31,54)(32,55)(33,56)(34,43)(35,44)(36,45)(37,46)(38,47)(39,48)(40,49)(41,50)(42,51)(57,82)(58,83)(59,84)(60,71)(61,72)(62,73)(63,74)(64,75)(65,76)(66,77)(67,78)(68,79)(69,80)(70,81), (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), (1,106,46,57,87,16,37,75)(2,105,47,70,88,15,38,74)(3,104,48,69,89,28,39,73)(4,103,49,68,90,27,40,72)(5,102,50,67,91,26,41,71)(6,101,51,66,92,25,42,84)(7,100,52,65,93,24,29,83)(8,99,53,64,94,23,30,82)(9,112,54,63,95,22,31,81)(10,111,55,62,96,21,32,80)(11,110,56,61,97,20,33,79)(12,109,43,60,98,19,34,78)(13,108,44,59,85,18,35,77)(14,107,45,58,86,17,36,76), (1,75)(2,74)(3,73)(4,72)(5,71)(6,84)(7,83)(8,82)(9,81)(10,80)(11,79)(12,78)(13,77)(14,76)(15,47)(16,46)(17,45)(18,44)(19,43)(20,56)(21,55)(22,54)(23,53)(24,52)(25,51)(26,50)(27,49)(28,48)(29,100)(30,99)(31,112)(32,111)(33,110)(34,109)(35,108)(36,107)(37,106)(38,105)(39,104)(40,103)(41,102)(42,101)(57,87)(58,86)(59,85)(60,98)(61,97)(62,96)(63,95)(64,94)(65,93)(66,92)(67,91)(68,90)(69,89)(70,88) );
G=PermutationGroup([[(1,87),(2,88),(3,89),(4,90),(5,91),(6,92),(7,93),(8,94),(9,95),(10,96),(11,97),(12,98),(13,85),(14,86),(15,112),(16,99),(17,100),(18,101),(19,102),(20,103),(21,104),(22,105),(23,106),(24,107),(25,108),(26,109),(27,110),(28,111),(29,52),(30,53),(31,54),(32,55),(33,56),(34,43),(35,44),(36,45),(37,46),(38,47),(39,48),(40,49),(41,50),(42,51),(57,82),(58,83),(59,84),(60,71),(61,72),(62,73),(63,74),(64,75),(65,76),(66,77),(67,78),(68,79),(69,80),(70,81)], [(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)], [(1,106,46,57,87,16,37,75),(2,105,47,70,88,15,38,74),(3,104,48,69,89,28,39,73),(4,103,49,68,90,27,40,72),(5,102,50,67,91,26,41,71),(6,101,51,66,92,25,42,84),(7,100,52,65,93,24,29,83),(8,99,53,64,94,23,30,82),(9,112,54,63,95,22,31,81),(10,111,55,62,96,21,32,80),(11,110,56,61,97,20,33,79),(12,109,43,60,98,19,34,78),(13,108,44,59,85,18,35,77),(14,107,45,58,86,17,36,76)], [(1,75),(2,74),(3,73),(4,72),(5,71),(6,84),(7,83),(8,82),(9,81),(10,80),(11,79),(12,78),(13,77),(14,76),(15,47),(16,46),(17,45),(18,44),(19,43),(20,56),(21,55),(22,54),(23,53),(24,52),(25,51),(26,50),(27,49),(28,48),(29,100),(30,99),(31,112),(32,111),(33,110),(34,109),(35,108),(36,107),(37,106),(38,105),(39,104),(40,103),(41,102),(42,101),(57,87),(58,86),(59,85),(60,98),(61,97),(62,96),(63,95),(64,94),(65,93),(66,92),(67,91),(68,90),(69,89),(70,88)]])
79 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 4A | 4B | 4C | 4D | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14U | 14V | ··· | 14AS | 28A | ··· | 28L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 56 | 2 | 2 | 4 | 56 | 2 | 2 | 2 | 28 | 28 | 28 | 28 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
79 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D7 | D8 | D14 | D14 | C7⋊D4 | C7⋊D4 | C7⋊D4 | C8⋊C22 | D4⋊D7 | D4.D14 |
| kernel | (C2×C14)⋊8D8 | C28.55D4 | D4⋊Dic7 | C28⋊7D4 | C2×D4⋊D7 | D4×C2×C14 | C2×C28 | C7×D4 | C22×C14 | C22×D4 | C2×C14 | C22×C4 | C2×D4 | C2×C4 | D4 | C23 | C14 | C22 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 1 | 4 | 1 | 3 | 4 | 3 | 6 | 6 | 24 | 6 | 1 | 6 | 6 |
Matrix representation of (C2×C14)⋊8D8 ►in GL4(𝔽113) generated by
| 1 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 112 |
| 64 | 0 | 0 | 0 |
| 0 | 83 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 112 | 0 | 0 | 0 |
| 0 | 0 | 62 | 46 |
| 0 | 0 | 27 | 0 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 62 | 46 |
| 0 | 0 | 27 | 51 |
G:=sub<GL(4,GF(113))| [1,0,0,0,0,112,0,0,0,0,112,0,0,0,0,112],[64,0,0,0,0,83,0,0,0,0,1,0,0,0,0,1],[0,112,0,0,1,0,0,0,0,0,62,27,0,0,46,0],[0,1,0,0,1,0,0,0,0,0,62,27,0,0,46,51] >;
(C2×C14)⋊8D8 in GAP, Magma, Sage, TeX
(C_2\times C_{14})\rtimes_8D_8 % in TeX
G:=Group("(C2xC14):8D8"); // GroupNames label
G:=SmallGroup(448,751);
// by ID
G=gap.SmallGroup(448,751);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,253,254,1684,851,102,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^14=c^8=d^2=1,a*b=b*a,c*a*c^-1=d*a*d=a*b^7,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations