direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D4⋊D14, C28.33C24, D28.29C23, C7⋊C8⋊5C23, C4○D4⋊14D14, (C2×D4)⋊41D14, (C2×Q8)⋊30D14, D4⋊5(C22×D7), (C7×D4)⋊5C23, (C7×Q8)⋊5C23, Q8⋊5(C22×D7), C14⋊5(C8⋊C22), D4⋊D7⋊18C22, C28.426(C2×D4), (C2×C28).217D4, Q8⋊D7⋊17C22, C4.33(C23×D7), (D4×C14)⋊45C22, (C2×D28)⋊58C22, (C22×D28)⋊20C2, (Q8×C14)⋊37C22, (C2×C28).555C23, C14.158(C22×D4), (C22×C14).122D4, (C22×C4).281D14, C23.68(C7⋊D4), C4.Dic7⋊36C22, (C22×C28).290C22, C7⋊6(C2×C8⋊C22), (C2×C4○D4)⋊2D7, (C2×D4⋊D7)⋊31C2, (C14×C4○D4)⋊2C2, (C2×C7⋊C8)⋊22C22, (C2×Q8⋊D7)⋊31C2, C4.29(C2×C7⋊D4), (C2×C14).75(C2×D4), (C7×C4○D4)⋊16C22, (C2×C4).95(C7⋊D4), (C2×C4.Dic7)⋊30C2, C2.31(C22×C7⋊D4), (C2×C4).245(C22×D7), C22.118(C2×C7⋊D4), SmallGroup(448,1273)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1556 in 298 conjugacy classes, 111 normal (27 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 [×2], D4 [×15], Q8 [×2], Q8, C23, C23 [×11], D7 [×4], C14, C14 [×2], C14 [×4], C2×C8 [×2], M4(2) [×4], D8 [×8], SD16 [×8], C22×C4, C22×C4, C2×D4, C2×D4 [×10], C2×Q8, C4○D4 [×4], C4○D4 [×2], C24, C28 [×2], C28 [×2], C28 [×2], D14 [×16], C2×C14, C2×C14 [×2], C2×C14 [×6], C2×M4(2), C2×D8 [×2], C2×SD16 [×2], C8⋊C22 [×8], C22×D4, C2×C4○D4, C7⋊C8 [×4], D28 [×4], D28 [×6], C2×C28 [×2], C2×C28 [×4], C2×C28 [×5], C7×D4 [×2], C7×D4 [×5], C7×Q8 [×2], C7×Q8, C22×D7 [×10], C22×C14, C22×C14, C2×C8⋊C22, C2×C7⋊C8 [×2], C4.Dic7 [×4], D4⋊D7 [×8], Q8⋊D7 [×8], C2×D28 [×6], C2×D28 [×3], C22×C28, C22×C28, D4×C14, D4×C14, Q8×C14, C7×C4○D4 [×4], C7×C4○D4 [×2], C23×D7, C2×C4.Dic7, C2×D4⋊D7 [×2], C2×Q8⋊D7 [×2], D4⋊D14 [×8], C22×D28, C14×C4○D4, C2×D4⋊D14
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D7, C2×D4 [×6], C24, D14 [×7], C8⋊C22 [×2], C22×D4, C7⋊D4 [×4], C22×D7 [×7], C2×C8⋊C22, C2×C7⋊D4 [×6], C23×D7, D4⋊D14 [×2], C22×C7⋊D4, C2×D4⋊D14
Generators and relations
G = < a,b,c,d,e | a2=b4=c2=d14=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe=b-1, bd=db, dcd-1=b2c, ece=b-1c, ede=d-1 >
►On 112 points
Generators in S112
(1 8)(2 9)(3 10)(4 11)(5 12)(6 13)(7 14)(15 83)(16 84)(17 71)(18 72)(19 73)(20 74)(21 75)(22 76)(23 77)(24 78)(25 79)(26 80)(27 81)(28 82)(29 36)(30 37)(31 38)(32 39)(33 40)(34 41)(35 42)(43 93)(44 94)(45 95)(46 96)(47 97)(48 98)(49 85)(50 86)(51 87)(52 88)(53 89)(54 90)(55 91)(56 92)(57 64)(58 65)(59 66)(60 67)(61 68)(62 69)(63 70)(99 106)(100 107)(101 108)(102 109)(103 110)(104 111)(105 112)
(1 34 69 99)(2 35 70 100)(3 36 57 101)(4 37 58 102)(5 38 59 103)(6 39 60 104)(7 40 61 105)(8 41 62 106)(9 42 63 107)(10 29 64 108)(11 30 65 109)(12 31 66 110)(13 32 67 111)(14 33 68 112)(15 86 76 43)(16 87 77 44)(17 88 78 45)(18 89 79 46)(19 90 80 47)(20 91 81 48)(21 92 82 49)(22 93 83 50)(23 94 84 51)(24 95 71 52)(25 96 72 53)(26 97 73 54)(27 98 74 55)(28 85 75 56)
(1 50)(2 94)(3 52)(4 96)(5 54)(6 98)(7 56)(8 86)(9 44)(10 88)(11 46)(12 90)(13 48)(14 92)(15 41)(16 107)(17 29)(18 109)(19 31)(20 111)(21 33)(22 99)(23 35)(24 101)(25 37)(26 103)(27 39)(28 105)(30 79)(32 81)(34 83)(36 71)(38 73)(40 75)(42 77)(43 62)(45 64)(47 66)(49 68)(51 70)(53 58)(55 60)(57 95)(59 97)(61 85)(63 87)(65 89)(67 91)(69 93)(72 102)(74 104)(76 106)(78 108)(80 110)(82 112)(84 100)
(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 85)(16 98)(17 97)(18 96)(19 95)(20 94)(21 93)(22 92)(23 91)(24 90)(25 89)(26 88)(27 87)(28 86)(29 103)(30 102)(31 101)(32 100)(33 99)(34 112)(35 111)(36 110)(37 109)(38 108)(39 107)(40 106)(41 105)(42 104)(43 75)(44 74)(45 73)(46 72)(47 71)(48 84)(49 83)(50 82)(51 81)(52 80)(53 79)(54 78)(55 77)(56 76)(57 66)(58 65)(59 64)(60 63)(61 62)(67 70)(68 69)
G:=sub<Sym(112)| (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(15,83)(16,84)(17,71)(18,72)(19,73)(20,74)(21,75)(22,76)(23,77)(24,78)(25,79)(26,80)(27,81)(28,82)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(43,93)(44,94)(45,95)(46,96)(47,97)(48,98)(49,85)(50,86)(51,87)(52,88)(53,89)(54,90)(55,91)(56,92)(57,64)(58,65)(59,66)(60,67)(61,68)(62,69)(63,70)(99,106)(100,107)(101,108)(102,109)(103,110)(104,111)(105,112), (1,34,69,99)(2,35,70,100)(3,36,57,101)(4,37,58,102)(5,38,59,103)(6,39,60,104)(7,40,61,105)(8,41,62,106)(9,42,63,107)(10,29,64,108)(11,30,65,109)(12,31,66,110)(13,32,67,111)(14,33,68,112)(15,86,76,43)(16,87,77,44)(17,88,78,45)(18,89,79,46)(19,90,80,47)(20,91,81,48)(21,92,82,49)(22,93,83,50)(23,94,84,51)(24,95,71,52)(25,96,72,53)(26,97,73,54)(27,98,74,55)(28,85,75,56), (1,50)(2,94)(3,52)(4,96)(5,54)(6,98)(7,56)(8,86)(9,44)(10,88)(11,46)(12,90)(13,48)(14,92)(15,41)(16,107)(17,29)(18,109)(19,31)(20,111)(21,33)(22,99)(23,35)(24,101)(25,37)(26,103)(27,39)(28,105)(30,79)(32,81)(34,83)(36,71)(38,73)(40,75)(42,77)(43,62)(45,64)(47,66)(49,68)(51,70)(53,58)(55,60)(57,95)(59,97)(61,85)(63,87)(65,89)(67,91)(69,93)(72,102)(74,104)(76,106)(78,108)(80,110)(82,112)(84,100), (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)(15,85)(16,98)(17,97)(18,96)(19,95)(20,94)(21,93)(22,92)(23,91)(24,90)(25,89)(26,88)(27,87)(28,86)(29,103)(30,102)(31,101)(32,100)(33,99)(34,112)(35,111)(36,110)(37,109)(38,108)(39,107)(40,106)(41,105)(42,104)(43,75)(44,74)(45,73)(46,72)(47,71)(48,84)(49,83)(50,82)(51,81)(52,80)(53,79)(54,78)(55,77)(56,76)(57,66)(58,65)(59,64)(60,63)(61,62)(67,70)(68,69)>;
G:=Group( (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(15,83)(16,84)(17,71)(18,72)(19,73)(20,74)(21,75)(22,76)(23,77)(24,78)(25,79)(26,80)(27,81)(28,82)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(43,93)(44,94)(45,95)(46,96)(47,97)(48,98)(49,85)(50,86)(51,87)(52,88)(53,89)(54,90)(55,91)(56,92)(57,64)(58,65)(59,66)(60,67)(61,68)(62,69)(63,70)(99,106)(100,107)(101,108)(102,109)(103,110)(104,111)(105,112), (1,34,69,99)(2,35,70,100)(3,36,57,101)(4,37,58,102)(5,38,59,103)(6,39,60,104)(7,40,61,105)(8,41,62,106)(9,42,63,107)(10,29,64,108)(11,30,65,109)(12,31,66,110)(13,32,67,111)(14,33,68,112)(15,86,76,43)(16,87,77,44)(17,88,78,45)(18,89,79,46)(19,90,80,47)(20,91,81,48)(21,92,82,49)(22,93,83,50)(23,94,84,51)(24,95,71,52)(25,96,72,53)(26,97,73,54)(27,98,74,55)(28,85,75,56), (1,50)(2,94)(3,52)(4,96)(5,54)(6,98)(7,56)(8,86)(9,44)(10,88)(11,46)(12,90)(13,48)(14,92)(15,41)(16,107)(17,29)(18,109)(19,31)(20,111)(21,33)(22,99)(23,35)(24,101)(25,37)(26,103)(27,39)(28,105)(30,79)(32,81)(34,83)(36,71)(38,73)(40,75)(42,77)(43,62)(45,64)(47,66)(49,68)(51,70)(53,58)(55,60)(57,95)(59,97)(61,85)(63,87)(65,89)(67,91)(69,93)(72,102)(74,104)(76,106)(78,108)(80,110)(82,112)(84,100), (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)(15,85)(16,98)(17,97)(18,96)(19,95)(20,94)(21,93)(22,92)(23,91)(24,90)(25,89)(26,88)(27,87)(28,86)(29,103)(30,102)(31,101)(32,100)(33,99)(34,112)(35,111)(36,110)(37,109)(38,108)(39,107)(40,106)(41,105)(42,104)(43,75)(44,74)(45,73)(46,72)(47,71)(48,84)(49,83)(50,82)(51,81)(52,80)(53,79)(54,78)(55,77)(56,76)(57,66)(58,65)(59,64)(60,63)(61,62)(67,70)(68,69) );
G=PermutationGroup([(1,8),(2,9),(3,10),(4,11),(5,12),(6,13),(7,14),(15,83),(16,84),(17,71),(18,72),(19,73),(20,74),(21,75),(22,76),(23,77),(24,78),(25,79),(26,80),(27,81),(28,82),(29,36),(30,37),(31,38),(32,39),(33,40),(34,41),(35,42),(43,93),(44,94),(45,95),(46,96),(47,97),(48,98),(49,85),(50,86),(51,87),(52,88),(53,89),(54,90),(55,91),(56,92),(57,64),(58,65),(59,66),(60,67),(61,68),(62,69),(63,70),(99,106),(100,107),(101,108),(102,109),(103,110),(104,111),(105,112)], [(1,34,69,99),(2,35,70,100),(3,36,57,101),(4,37,58,102),(5,38,59,103),(6,39,60,104),(7,40,61,105),(8,41,62,106),(9,42,63,107),(10,29,64,108),(11,30,65,109),(12,31,66,110),(13,32,67,111),(14,33,68,112),(15,86,76,43),(16,87,77,44),(17,88,78,45),(18,89,79,46),(19,90,80,47),(20,91,81,48),(21,92,82,49),(22,93,83,50),(23,94,84,51),(24,95,71,52),(25,96,72,53),(26,97,73,54),(27,98,74,55),(28,85,75,56)], [(1,50),(2,94),(3,52),(4,96),(5,54),(6,98),(7,56),(8,86),(9,44),(10,88),(11,46),(12,90),(13,48),(14,92),(15,41),(16,107),(17,29),(18,109),(19,31),(20,111),(21,33),(22,99),(23,35),(24,101),(25,37),(26,103),(27,39),(28,105),(30,79),(32,81),(34,83),(36,71),(38,73),(40,75),(42,77),(43,62),(45,64),(47,66),(49,68),(51,70),(53,58),(55,60),(57,95),(59,97),(61,85),(63,87),(65,89),(67,91),(69,93),(72,102),(74,104),(76,106),(78,108),(80,110),(82,112),(84,100)], [(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),(15,85),(16,98),(17,97),(18,96),(19,95),(20,94),(21,93),(22,92),(23,91),(24,90),(25,89),(26,88),(27,87),(28,86),(29,103),(30,102),(31,101),(32,100),(33,99),(34,112),(35,111),(36,110),(37,109),(38,108),(39,107),(40,106),(41,105),(42,104),(43,75),(44,74),(45,73),(46,72),(47,71),(48,84),(49,83),(50,82),(51,81),(52,80),(53,79),(54,78),(55,77),(56,76),(57,66),(58,65),(59,64),(60,63),(61,62),(67,70),(68,69)])
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 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 67 | 0 | 0 |
| 0 | 0 | 46 | 104 | 0 | 0 |
| 0 | 0 | 0 | 0 | 104 | 46 |
| 0 | 0 | 0 | 0 | 67 | 9 |
| 1 | 111 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 104 | 46 |
| 0 | 0 | 0 | 0 | 67 | 9 |
| 0 | 0 | 9 | 67 | 0 | 0 |
| 0 | 0 | 46 | 104 | 0 | 0 |
| 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 | 0 | 0 | 79 | 88 |
| 0 | 0 | 0 | 0 | 25 | 25 |
| 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 | 0 | 0 | 13 | 94 |
| 0 | 0 | 0 | 0 | 104 | 100 |
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],[112,0,0,0,0,0,0,112,0,0,0,0,0,0,9,46,0,0,0,0,67,104,0,0,0,0,0,0,104,67,0,0,0,0,46,9],[1,0,0,0,0,0,111,112,0,0,0,0,0,0,0,0,9,46,0,0,0,0,67,104,0,0,104,67,0,0,0,0,46,9,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,34,88,0,0,0,0,25,88,0,0,0,0,0,0,79,25,0,0,0,0,88,25],[112,112,0,0,0,0,0,1,0,0,0,0,0,0,34,112,0,0,0,0,25,79,0,0,0,0,0,0,13,104,0,0,0,0,94,100] >;
82 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | 4F | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 14J | ··· | 14AA | 28A | ··· | 28L | 28M | ··· | 28AD |
| order | 1 | 2 | 2 | 2 | 2 | 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 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 28 | 28 | 28 | 28 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
82 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 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 | D14 | C7⋊D4 | C7⋊D4 | C8⋊C22 | D4⋊D14 |
| kernel | C2×D4⋊D14 | C2×C4.Dic7 | C2×D4⋊D7 | C2×Q8⋊D7 | D4⋊D14 | C22×D28 | C14×C4○D4 | C2×C28 | C22×C14 | C2×C4○D4 | C22×C4 | C2×D4 | C2×Q8 | C4○D4 | C2×C4 | C23 | C14 | C2 |
| # reps | 1 | 1 | 2 | 2 | 8 | 1 | 1 | 3 | 1 | 3 | 3 | 3 | 3 | 12 | 18 | 6 | 2 | 12 |
In GAP, Magma, Sage, TeX
C_2\times D_4\rtimes D_{14} % in TeX
G:=Group("C2xD4:D14"); // GroupNames label
G:=SmallGroup(448,1273);
// by ID
G=gap.SmallGroup(448,1273);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,675,297,1684,235,102,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=c^2=d^14=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=e*b*e=b^-1,b*d=d*b,d*c*d^-1=b^2*c,e*c*e=b^-1*c,e*d*e=d^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀