direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C8⋊D14, C56⋊2C23, D56⋊9C22, D28⋊5C23, C28.58C24, C23.52D28, M4(2)⋊17D14, Dic14⋊5C23, (C2×C8)⋊4D14, C8⋊2(C22×D7), (C2×D56)⋊14C2, (C2×C56)⋊7C22, (C2×C4).57D28, C4.48(C2×D28), C14⋊1(C8⋊C22), (C2×C28).203D4, C28.238(C2×D4), C56⋊C2⋊8C22, (C2×M4(2))⋊3D7, C4.55(C23×D7), C4○D28⋊18C22, (C2×D28)⋊49C22, (C22×D28)⋊17C2, (C14×M4(2))⋊3C2, C14.25(C22×D4), C22.73(C2×D28), C2.27(C22×D28), (C2×C28).511C23, (C22×C4).265D14, (C22×C14).118D4, (C2×Dic14)⋊57C22, (C7×M4(2))⋊19C22, (C22×C28).266C22, C7⋊1(C2×C8⋊C22), (C2×C56⋊C2)⋊4C2, (C2×C4○D28)⋊26C2, (C2×C14).62(C2×D4), (C2×C4).223(C22×D7), SmallGroup(448,1199)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C8⋊D14
G = < a,b,c,d | a2=b8=c14=d2=1, ab=ba, ac=ca, ad=da, cbc-1=b5, dbd=b-1, dcd=c-1 >
Subgroups: 1892 in 298 conjugacy classes, 111 normal (25 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, D7, C14, C14, C14, C2×C8, M4(2), D8, SD16, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, C24, Dic7, C28, C28, D14, C2×C14, C2×C14, C2×C14, C2×M4(2), C2×D8, C2×SD16, C8⋊C22, C22×D4, C2×C4○D4, C56, Dic14, Dic14, C4×D7, D28, D28, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C22×D7, C22×C14, C2×C8⋊C22, C56⋊C2, D56, C2×C56, C7×M4(2), C2×Dic14, C2×C4×D7, C2×D28, C2×D28, C2×D28, C4○D28, C4○D28, C2×C7⋊D4, C22×C28, C23×D7, C2×C56⋊C2, C2×D56, C8⋊D14, C14×M4(2), C22×D28, C2×C4○D28, C2×C8⋊D14
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C24, D14, C8⋊C22, C22×D4, D28, C22×D7, C2×C8⋊C22, C2×D28, C23×D7, C8⋊D14, C22×D28, C2×C8⋊D14
►On 112 points
Generators in S112
(1 8)(2 9)(3 10)(4 11)(5 12)(6 13)(7 14)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 49)(22 50)(23 51)(24 52)(25 53)(26 54)(27 55)(28 56)(29 36)(30 37)(31 38)(32 39)(33 40)(34 41)(35 42)(57 64)(58 65)(59 66)(60 67)(61 68)(62 69)(63 70)(71 78)(72 79)(73 80)(74 81)(75 82)(76 83)(77 84)(85 108)(86 109)(87 110)(88 111)(89 112)(90 99)(91 100)(92 101)(93 102)(94 103)(95 104)(96 105)(97 106)(98 107)
(1 93 64 53 39 109 82 18)(2 110 65 19 40 94 83 54)(3 95 66 55 41 111 84 20)(4 112 67 21 42 96 71 56)(5 97 68 43 29 99 72 22)(6 100 69 23 30 98 73 44)(7 85 70 45 31 101 74 24)(8 102 57 25 32 86 75 46)(9 87 58 47 33 103 76 26)(10 104 59 27 34 88 77 48)(11 89 60 49 35 105 78 28)(12 106 61 15 36 90 79 50)(13 91 62 51 37 107 80 16)(14 108 63 17 38 92 81 52)
(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 14)(2 13)(3 12)(4 11)(5 10)(6 9)(7 8)(15 111)(16 110)(17 109)(18 108)(19 107)(20 106)(21 105)(22 104)(23 103)(24 102)(25 101)(26 100)(27 99)(28 112)(29 34)(30 33)(31 32)(35 42)(36 41)(37 40)(38 39)(43 88)(44 87)(45 86)(46 85)(47 98)(48 97)(49 96)(50 95)(51 94)(52 93)(53 92)(54 91)(55 90)(56 89)(57 74)(58 73)(59 72)(60 71)(61 84)(62 83)(63 82)(64 81)(65 80)(66 79)(67 78)(68 77)(69 76)(70 75)
G:=sub<Sym(112)| (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(57,64)(58,65)(59,66)(60,67)(61,68)(62,69)(63,70)(71,78)(72,79)(73,80)(74,81)(75,82)(76,83)(77,84)(85,108)(86,109)(87,110)(88,111)(89,112)(90,99)(91,100)(92,101)(93,102)(94,103)(95,104)(96,105)(97,106)(98,107), (1,93,64,53,39,109,82,18)(2,110,65,19,40,94,83,54)(3,95,66,55,41,111,84,20)(4,112,67,21,42,96,71,56)(5,97,68,43,29,99,72,22)(6,100,69,23,30,98,73,44)(7,85,70,45,31,101,74,24)(8,102,57,25,32,86,75,46)(9,87,58,47,33,103,76,26)(10,104,59,27,34,88,77,48)(11,89,60,49,35,105,78,28)(12,106,61,15,36,90,79,50)(13,91,62,51,37,107,80,16)(14,108,63,17,38,92,81,52), (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), (1,14)(2,13)(3,12)(4,11)(5,10)(6,9)(7,8)(15,111)(16,110)(17,109)(18,108)(19,107)(20,106)(21,105)(22,104)(23,103)(24,102)(25,101)(26,100)(27,99)(28,112)(29,34)(30,33)(31,32)(35,42)(36,41)(37,40)(38,39)(43,88)(44,87)(45,86)(46,85)(47,98)(48,97)(49,96)(50,95)(51,94)(52,93)(53,92)(54,91)(55,90)(56,89)(57,74)(58,73)(59,72)(60,71)(61,84)(62,83)(63,82)(64,81)(65,80)(66,79)(67,78)(68,77)(69,76)(70,75)>;
G:=Group( (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(57,64)(58,65)(59,66)(60,67)(61,68)(62,69)(63,70)(71,78)(72,79)(73,80)(74,81)(75,82)(76,83)(77,84)(85,108)(86,109)(87,110)(88,111)(89,112)(90,99)(91,100)(92,101)(93,102)(94,103)(95,104)(96,105)(97,106)(98,107), (1,93,64,53,39,109,82,18)(2,110,65,19,40,94,83,54)(3,95,66,55,41,111,84,20)(4,112,67,21,42,96,71,56)(5,97,68,43,29,99,72,22)(6,100,69,23,30,98,73,44)(7,85,70,45,31,101,74,24)(8,102,57,25,32,86,75,46)(9,87,58,47,33,103,76,26)(10,104,59,27,34,88,77,48)(11,89,60,49,35,105,78,28)(12,106,61,15,36,90,79,50)(13,91,62,51,37,107,80,16)(14,108,63,17,38,92,81,52), (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), (1,14)(2,13)(3,12)(4,11)(5,10)(6,9)(7,8)(15,111)(16,110)(17,109)(18,108)(19,107)(20,106)(21,105)(22,104)(23,103)(24,102)(25,101)(26,100)(27,99)(28,112)(29,34)(30,33)(31,32)(35,42)(36,41)(37,40)(38,39)(43,88)(44,87)(45,86)(46,85)(47,98)(48,97)(49,96)(50,95)(51,94)(52,93)(53,92)(54,91)(55,90)(56,89)(57,74)(58,73)(59,72)(60,71)(61,84)(62,83)(63,82)(64,81)(65,80)(66,79)(67,78)(68,77)(69,76)(70,75) );
G=PermutationGroup([[(1,8),(2,9),(3,10),(4,11),(5,12),(6,13),(7,14),(15,43),(16,44),(17,45),(18,46),(19,47),(20,48),(21,49),(22,50),(23,51),(24,52),(25,53),(26,54),(27,55),(28,56),(29,36),(30,37),(31,38),(32,39),(33,40),(34,41),(35,42),(57,64),(58,65),(59,66),(60,67),(61,68),(62,69),(63,70),(71,78),(72,79),(73,80),(74,81),(75,82),(76,83),(77,84),(85,108),(86,109),(87,110),(88,111),(89,112),(90,99),(91,100),(92,101),(93,102),(94,103),(95,104),(96,105),(97,106),(98,107)], [(1,93,64,53,39,109,82,18),(2,110,65,19,40,94,83,54),(3,95,66,55,41,111,84,20),(4,112,67,21,42,96,71,56),(5,97,68,43,29,99,72,22),(6,100,69,23,30,98,73,44),(7,85,70,45,31,101,74,24),(8,102,57,25,32,86,75,46),(9,87,58,47,33,103,76,26),(10,104,59,27,34,88,77,48),(11,89,60,49,35,105,78,28),(12,106,61,15,36,90,79,50),(13,91,62,51,37,107,80,16),(14,108,63,17,38,92,81,52)], [(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)], [(1,14),(2,13),(3,12),(4,11),(5,10),(6,9),(7,8),(15,111),(16,110),(17,109),(18,108),(19,107),(20,106),(21,105),(22,104),(23,103),(24,102),(25,101),(26,100),(27,99),(28,112),(29,34),(30,33),(31,32),(35,42),(36,41),(37,40),(38,39),(43,88),(44,87),(45,86),(46,85),(47,98),(48,97),(49,96),(50,95),(51,94),(52,93),(53,92),(54,91),(55,90),(56,89),(57,74),(58,73),(59,72),(60,71),(61,84),(62,83),(63,82),(64,81),(65,80),(66,79),(67,78),(68,77),(69,76),(70,75)]])
82 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | ··· | 2K | 4A | 4B | 4C | 4D | 4E | 4F | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28L | 28M | ··· | 28R | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 28 | ··· | 28 | 2 | 2 | 2 | 2 | 28 | 28 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
82 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 | D4 | D4 | D7 | D14 | D14 | D14 | D28 | D28 | C8⋊C22 | C8⋊D14 |
| kernel | C2×C8⋊D14 | C2×C56⋊C2 | C2×D56 | C8⋊D14 | C14×M4(2) | C22×D28 | C2×C4○D28 | C2×C28 | C22×C14 | C2×M4(2) | C2×C8 | M4(2) | C22×C4 | C2×C4 | C23 | C14 | C2 |
| # reps | 1 | 2 | 2 | 8 | 1 | 1 | 1 | 3 | 1 | 3 | 6 | 12 | 3 | 18 | 6 | 2 | 12 |
Matrix representation of C2×C8⋊D14 ►in GL6(𝔽113)
| 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 112 | 2 | 0 | 0 | 0 | 0 |
| 112 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 67 | 27 | 44 |
| 0 | 0 | 46 | 104 | 25 | 44 |
| 0 | 0 | 95 | 94 | 58 | 46 |
| 0 | 0 | 1 | 44 | 3 | 55 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 25 | 0 | 0 |
| 0 | 0 | 88 | 88 | 0 | 0 |
| 0 | 0 | 107 | 0 | 104 | 88 |
| 0 | 0 | 107 | 107 | 104 | 0 |
| 112 | 0 | 0 | 0 | 0 | 0 |
| 112 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 25 | 0 | 0 |
| 0 | 0 | 112 | 79 | 0 | 0 |
| 0 | 0 | 63 | 89 | 109 | 13 |
| 0 | 0 | 44 | 26 | 51 | 4 |
G:=sub<GL(6,GF(113))| [112,0,0,0,0,0,0,112,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[112,112,0,0,0,0,2,1,0,0,0,0,0,0,9,46,95,1,0,0,67,104,94,44,0,0,27,25,58,3,0,0,44,44,46,55],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,34,88,107,107,0,0,25,88,0,107,0,0,0,0,104,104,0,0,0,0,88,0],[112,112,0,0,0,0,0,1,0,0,0,0,0,0,34,112,63,44,0,0,25,79,89,26,0,0,0,0,109,51,0,0,0,0,13,4] >;
C2×C8⋊D14 in GAP, Magma, Sage, TeX
C_2\times C_8\rtimes D_{14} % in TeX
G:=Group("C2xC8:D14"); // GroupNames label
G:=SmallGroup(448,1199);
// by ID
G=gap.SmallGroup(448,1199);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,675,297,80,1684,102,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^8=c^14=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^5,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations