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), C28.238(C2×D4), (C2×C28).203D4, C56⋊C2⋊8C22, (C2×M4(2))⋊3D7, C4.55(C23×D7), C4○D28⋊18C22, (C22×D28)⋊17C2, (C2×D28)⋊49C22, (C14×M4(2))⋊3C2, C22.73(C2×D28), C14.25(C22×D4), C2.27(C22×D28), (C2×C28).511C23, (C22×C14).118D4, (C22×C4).265D14, (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
Subgroups: 1892 in 298 conjugacy classes, 111 normal (25 characteristic)
C1, C2, C2 [×2], C2 [×8], C4 [×2], C4 [×2], C4 [×2], C22, C22 [×2], C22 [×22], C7, C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×5], D4 [×17], Q8 [×3], C23, C23 [×11], D7 [×6], C14, C14 [×2], C14 [×2], C2×C8 [×2], M4(2) [×4], D8 [×8], SD16 [×8], C22×C4, C22×C4, C2×D4 [×11], C2×Q8, C4○D4 [×6], C24, Dic7 [×2], C28 [×2], C28 [×2], D14 [×20], C2×C14, C2×C14 [×2], C2×C14 [×2], C2×M4(2), C2×D8 [×2], C2×SD16 [×2], C8⋊C22 [×8], C22×D4, C2×C4○D4, C56 [×4], Dic14 [×2], Dic14, C4×D7 [×4], D28 [×6], D28 [×7], C2×Dic7, C7⋊D4 [×4], C2×C28 [×2], C2×C28 [×4], C22×D7 [×11], C22×C14, C2×C8⋊C22, C56⋊C2 [×8], D56 [×8], C2×C56 [×2], C7×M4(2) [×4], C2×Dic14, C2×C4×D7, C2×D28, C2×D28 [×6], C2×D28 [×3], C4○D28 [×4], C4○D28 [×2], C2×C7⋊D4, C22×C28, C23×D7, C2×C56⋊C2 [×2], C2×D56 [×2], C8⋊D14 [×8], C14×M4(2), C22×D28, C2×C4○D28, C2×C8⋊D14
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D7, C2×D4 [×6], C24, D14 [×7], C8⋊C22 [×2], C22×D4, D28 [×4], C22×D7 [×7], C2×C8⋊C22, C2×D28 [×6], C23×D7, C8⋊D14 [×2], C22×D28, C2×C8⋊D14
Generators and relations
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 >
►On 112 points
(1 8)(2 9)(3 10)(4 11)(5 12)(6 13)(7 14)(15 57)(16 58)(17 59)(18 60)(19 61)(20 62)(21 63)(22 64)(23 65)(24 66)(25 67)(26 68)(27 69)(28 70)(29 36)(30 37)(31 38)(32 39)(33 40)(34 41)(35 42)(43 86)(44 87)(45 88)(46 89)(47 90)(48 91)(49 92)(50 93)(51 94)(52 95)(53 96)(54 97)(55 98)(56 85)(71 78)(72 79)(73 80)(74 81)(75 82)(76 83)(77 84)(99 106)(100 107)(101 108)(102 109)(103 110)(104 111)(105 112)
(1 20 30 48 84 69 110 98)(2 70 31 85 71 21 111 49)(3 22 32 50 72 57 112 86)(4 58 33 87 73 23 99 51)(5 24 34 52 74 59 100 88)(6 60 35 89 75 25 101 53)(7 26 36 54 76 61 102 90)(8 62 37 91 77 27 103 55)(9 28 38 56 78 63 104 92)(10 64 39 93 79 15 105 43)(11 16 40 44 80 65 106 94)(12 66 41 95 81 17 107 45)(13 18 42 46 82 67 108 96)(14 68 29 97 83 19 109 47)
(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 52)(16 51)(17 50)(18 49)(19 48)(20 47)(21 46)(22 45)(23 44)(24 43)(25 56)(26 55)(27 54)(28 53)(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)(57 95)(58 94)(59 93)(60 92)(61 91)(62 90)(63 89)(64 88)(65 87)(66 86)(67 85)(68 98)(69 97)(70 96)(71 82)(72 81)(73 80)(74 79)(75 78)(76 77)(83 84)
G:=sub<Sym(112)| (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(15,57)(16,58)(17,59)(18,60)(19,61)(20,62)(21,63)(22,64)(23,65)(24,66)(25,67)(26,68)(27,69)(28,70)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(43,86)(44,87)(45,88)(46,89)(47,90)(48,91)(49,92)(50,93)(51,94)(52,95)(53,96)(54,97)(55,98)(56,85)(71,78)(72,79)(73,80)(74,81)(75,82)(76,83)(77,84)(99,106)(100,107)(101,108)(102,109)(103,110)(104,111)(105,112), (1,20,30,48,84,69,110,98)(2,70,31,85,71,21,111,49)(3,22,32,50,72,57,112,86)(4,58,33,87,73,23,99,51)(5,24,34,52,74,59,100,88)(6,60,35,89,75,25,101,53)(7,26,36,54,76,61,102,90)(8,62,37,91,77,27,103,55)(9,28,38,56,78,63,104,92)(10,64,39,93,79,15,105,43)(11,16,40,44,80,65,106,94)(12,66,41,95,81,17,107,45)(13,18,42,46,82,67,108,96)(14,68,29,97,83,19,109,47), (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)(15,52)(16,51)(17,50)(18,49)(19,48)(20,47)(21,46)(22,45)(23,44)(24,43)(25,56)(26,55)(27,54)(28,53)(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)(57,95)(58,94)(59,93)(60,92)(61,91)(62,90)(63,89)(64,88)(65,87)(66,86)(67,85)(68,98)(69,97)(70,96)(71,82)(72,81)(73,80)(74,79)(75,78)(76,77)(83,84)>;
G:=Group( (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(15,57)(16,58)(17,59)(18,60)(19,61)(20,62)(21,63)(22,64)(23,65)(24,66)(25,67)(26,68)(27,69)(28,70)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(43,86)(44,87)(45,88)(46,89)(47,90)(48,91)(49,92)(50,93)(51,94)(52,95)(53,96)(54,97)(55,98)(56,85)(71,78)(72,79)(73,80)(74,81)(75,82)(76,83)(77,84)(99,106)(100,107)(101,108)(102,109)(103,110)(104,111)(105,112), (1,20,30,48,84,69,110,98)(2,70,31,85,71,21,111,49)(3,22,32,50,72,57,112,86)(4,58,33,87,73,23,99,51)(5,24,34,52,74,59,100,88)(6,60,35,89,75,25,101,53)(7,26,36,54,76,61,102,90)(8,62,37,91,77,27,103,55)(9,28,38,56,78,63,104,92)(10,64,39,93,79,15,105,43)(11,16,40,44,80,65,106,94)(12,66,41,95,81,17,107,45)(13,18,42,46,82,67,108,96)(14,68,29,97,83,19,109,47), (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)(15,52)(16,51)(17,50)(18,49)(19,48)(20,47)(21,46)(22,45)(23,44)(24,43)(25,56)(26,55)(27,54)(28,53)(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)(57,95)(58,94)(59,93)(60,92)(61,91)(62,90)(63,89)(64,88)(65,87)(66,86)(67,85)(68,98)(69,97)(70,96)(71,82)(72,81)(73,80)(74,79)(75,78)(76,77)(83,84) );
G=PermutationGroup([(1,8),(2,9),(3,10),(4,11),(5,12),(6,13),(7,14),(15,57),(16,58),(17,59),(18,60),(19,61),(20,62),(21,63),(22,64),(23,65),(24,66),(25,67),(26,68),(27,69),(28,70),(29,36),(30,37),(31,38),(32,39),(33,40),(34,41),(35,42),(43,86),(44,87),(45,88),(46,89),(47,90),(48,91),(49,92),(50,93),(51,94),(52,95),(53,96),(54,97),(55,98),(56,85),(71,78),(72,79),(73,80),(74,81),(75,82),(76,83),(77,84),(99,106),(100,107),(101,108),(102,109),(103,110),(104,111),(105,112)], [(1,20,30,48,84,69,110,98),(2,70,31,85,71,21,111,49),(3,22,32,50,72,57,112,86),(4,58,33,87,73,23,99,51),(5,24,34,52,74,59,100,88),(6,60,35,89,75,25,101,53),(7,26,36,54,76,61,102,90),(8,62,37,91,77,27,103,55),(9,28,38,56,78,63,104,92),(10,64,39,93,79,15,105,43),(11,16,40,44,80,65,106,94),(12,66,41,95,81,17,107,45),(13,18,42,46,82,67,108,96),(14,68,29,97,83,19,109,47)], [(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),(15,52),(16,51),(17,50),(18,49),(19,48),(20,47),(21,46),(22,45),(23,44),(24,43),(25,56),(26,55),(27,54),(28,53),(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),(57,95),(58,94),(59,93),(60,92),(61,91),(62,90),(63,89),(64,88),(65,87),(66,86),(67,85),(68,98),(69,97),(70,96),(71,82),(72,81),(73,80),(74,79),(75,78),(76,77),(83,84)])
Matrix representation ►G ⊆ 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] >;
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 |
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