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, Q8⋊5(C22×D7), (C7×Q8)⋊5C23, 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
Generators and relations for C2×D4⋊D14
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 >
Subgroups: 1556 in 298 conjugacy classes, 111 normal (27 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, D7, C14, C14, C14, C2×C8, M4(2), D8, SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4○D4, C24, C28, C28, C28, D14, C2×C14, C2×C14, C2×C14, C2×M4(2), C2×D8, C2×SD16, C8⋊C22, C22×D4, C2×C4○D4, C7⋊C8, D28, D28, C2×C28, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C7×Q8, C22×D7, C22×C14, C22×C14, C2×C8⋊C22, C2×C7⋊C8, C4.Dic7, D4⋊D7, Q8⋊D7, C2×D28, C2×D28, C22×C28, C22×C28, D4×C14, D4×C14, Q8×C14, C7×C4○D4, C7×C4○D4, C23×D7, C2×C4.Dic7, C2×D4⋊D7, C2×Q8⋊D7, D4⋊D14, C22×D28, C14×C4○D4, C2×D4⋊D14
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C24, D14, C8⋊C22, C22×D4, C7⋊D4, C22×D7, C2×C8⋊C22, C2×C7⋊D4, C23×D7, D4⋊D14, C22×C7⋊D4, C2×D4⋊D14
►On 112 points
Generators in S112
(1 8)(2 9)(3 10)(4 11)(5 12)(6 13)(7 14)(15 77)(16 78)(17 79)(18 80)(19 81)(20 82)(21 83)(22 84)(23 71)(24 72)(25 73)(26 74)(27 75)(28 76)(29 36)(30 37)(31 38)(32 39)(33 40)(34 41)(35 42)(43 89)(44 90)(45 91)(46 92)(47 93)(48 94)(49 95)(50 96)(51 97)(52 98)(53 85)(54 86)(55 87)(56 88)(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 32 60 108)(2 33 61 109)(3 34 62 110)(4 35 63 111)(5 36 64 112)(6 37 65 99)(7 38 66 100)(8 39 67 101)(9 40 68 102)(10 41 69 103)(11 42 70 104)(12 29 57 105)(13 30 58 106)(14 31 59 107)(15 44 84 97)(16 45 71 98)(17 46 72 85)(18 47 73 86)(19 48 74 87)(20 49 75 88)(21 50 76 89)(22 51 77 90)(23 52 78 91)(24 53 79 92)(25 54 80 93)(26 55 81 94)(27 56 82 95)(28 43 83 96)
(1 72)(2 18)(3 74)(4 20)(5 76)(6 22)(7 78)(8 24)(9 80)(10 26)(11 82)(12 28)(13 84)(14 16)(15 58)(17 60)(19 62)(21 64)(23 66)(25 68)(27 70)(29 96)(30 44)(31 98)(32 46)(33 86)(34 48)(35 88)(36 50)(37 90)(38 52)(39 92)(40 54)(41 94)(42 56)(43 105)(45 107)(47 109)(49 111)(51 99)(53 101)(55 103)(57 83)(59 71)(61 73)(63 75)(65 77)(67 79)(69 81)(85 108)(87 110)(89 112)(91 100)(93 102)(95 104)(97 106)
(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 86)(16 85)(17 98)(18 97)(19 96)(20 95)(21 94)(22 93)(23 92)(24 91)(25 90)(26 89)(27 88)(28 87)(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 74)(44 73)(45 72)(46 71)(47 84)(48 83)(49 82)(50 81)(51 80)(52 79)(53 78)(54 77)(55 76)(56 75)(57 62)(58 61)(59 60)(63 70)(64 69)(65 68)(66 67)
G:=sub<Sym(112)| (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(15,77)(16,78)(17,79)(18,80)(19,81)(20,82)(21,83)(22,84)(23,71)(24,72)(25,73)(26,74)(27,75)(28,76)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(43,89)(44,90)(45,91)(46,92)(47,93)(48,94)(49,95)(50,96)(51,97)(52,98)(53,85)(54,86)(55,87)(56,88)(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), (1,32,60,108)(2,33,61,109)(3,34,62,110)(4,35,63,111)(5,36,64,112)(6,37,65,99)(7,38,66,100)(8,39,67,101)(9,40,68,102)(10,41,69,103)(11,42,70,104)(12,29,57,105)(13,30,58,106)(14,31,59,107)(15,44,84,97)(16,45,71,98)(17,46,72,85)(18,47,73,86)(19,48,74,87)(20,49,75,88)(21,50,76,89)(22,51,77,90)(23,52,78,91)(24,53,79,92)(25,54,80,93)(26,55,81,94)(27,56,82,95)(28,43,83,96), (1,72)(2,18)(3,74)(4,20)(5,76)(6,22)(7,78)(8,24)(9,80)(10,26)(11,82)(12,28)(13,84)(14,16)(15,58)(17,60)(19,62)(21,64)(23,66)(25,68)(27,70)(29,96)(30,44)(31,98)(32,46)(33,86)(34,48)(35,88)(36,50)(37,90)(38,52)(39,92)(40,54)(41,94)(42,56)(43,105)(45,107)(47,109)(49,111)(51,99)(53,101)(55,103)(57,83)(59,71)(61,73)(63,75)(65,77)(67,79)(69,81)(85,108)(87,110)(89,112)(91,100)(93,102)(95,104)(97,106), (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)(15,86)(16,85)(17,98)(18,97)(19,96)(20,95)(21,94)(22,93)(23,92)(24,91)(25,90)(26,89)(27,88)(28,87)(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)(43,74)(44,73)(45,72)(46,71)(47,84)(48,83)(49,82)(50,81)(51,80)(52,79)(53,78)(54,77)(55,76)(56,75)(57,62)(58,61)(59,60)(63,70)(64,69)(65,68)(66,67)>;
G:=Group( (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(15,77)(16,78)(17,79)(18,80)(19,81)(20,82)(21,83)(22,84)(23,71)(24,72)(25,73)(26,74)(27,75)(28,76)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(43,89)(44,90)(45,91)(46,92)(47,93)(48,94)(49,95)(50,96)(51,97)(52,98)(53,85)(54,86)(55,87)(56,88)(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), (1,32,60,108)(2,33,61,109)(3,34,62,110)(4,35,63,111)(5,36,64,112)(6,37,65,99)(7,38,66,100)(8,39,67,101)(9,40,68,102)(10,41,69,103)(11,42,70,104)(12,29,57,105)(13,30,58,106)(14,31,59,107)(15,44,84,97)(16,45,71,98)(17,46,72,85)(18,47,73,86)(19,48,74,87)(20,49,75,88)(21,50,76,89)(22,51,77,90)(23,52,78,91)(24,53,79,92)(25,54,80,93)(26,55,81,94)(27,56,82,95)(28,43,83,96), (1,72)(2,18)(3,74)(4,20)(5,76)(6,22)(7,78)(8,24)(9,80)(10,26)(11,82)(12,28)(13,84)(14,16)(15,58)(17,60)(19,62)(21,64)(23,66)(25,68)(27,70)(29,96)(30,44)(31,98)(32,46)(33,86)(34,48)(35,88)(36,50)(37,90)(38,52)(39,92)(40,54)(41,94)(42,56)(43,105)(45,107)(47,109)(49,111)(51,99)(53,101)(55,103)(57,83)(59,71)(61,73)(63,75)(65,77)(67,79)(69,81)(85,108)(87,110)(89,112)(91,100)(93,102)(95,104)(97,106), (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)(15,86)(16,85)(17,98)(18,97)(19,96)(20,95)(21,94)(22,93)(23,92)(24,91)(25,90)(26,89)(27,88)(28,87)(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)(43,74)(44,73)(45,72)(46,71)(47,84)(48,83)(49,82)(50,81)(51,80)(52,79)(53,78)(54,77)(55,76)(56,75)(57,62)(58,61)(59,60)(63,70)(64,69)(65,68)(66,67) );
G=PermutationGroup([[(1,8),(2,9),(3,10),(4,11),(5,12),(6,13),(7,14),(15,77),(16,78),(17,79),(18,80),(19,81),(20,82),(21,83),(22,84),(23,71),(24,72),(25,73),(26,74),(27,75),(28,76),(29,36),(30,37),(31,38),(32,39),(33,40),(34,41),(35,42),(43,89),(44,90),(45,91),(46,92),(47,93),(48,94),(49,95),(50,96),(51,97),(52,98),(53,85),(54,86),(55,87),(56,88),(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)], [(1,32,60,108),(2,33,61,109),(3,34,62,110),(4,35,63,111),(5,36,64,112),(6,37,65,99),(7,38,66,100),(8,39,67,101),(9,40,68,102),(10,41,69,103),(11,42,70,104),(12,29,57,105),(13,30,58,106),(14,31,59,107),(15,44,84,97),(16,45,71,98),(17,46,72,85),(18,47,73,86),(19,48,74,87),(20,49,75,88),(21,50,76,89),(22,51,77,90),(23,52,78,91),(24,53,79,92),(25,54,80,93),(26,55,81,94),(27,56,82,95),(28,43,83,96)], [(1,72),(2,18),(3,74),(4,20),(5,76),(6,22),(7,78),(8,24),(9,80),(10,26),(11,82),(12,28),(13,84),(14,16),(15,58),(17,60),(19,62),(21,64),(23,66),(25,68),(27,70),(29,96),(30,44),(31,98),(32,46),(33,86),(34,48),(35,88),(36,50),(37,90),(38,52),(39,92),(40,54),(41,94),(42,56),(43,105),(45,107),(47,109),(49,111),(51,99),(53,101),(55,103),(57,83),(59,71),(61,73),(63,75),(65,77),(67,79),(69,81),(85,108),(87,110),(89,112),(91,100),(93,102),(95,104),(97,106)], [(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),(15,86),(16,85),(17,98),(18,97),(19,96),(20,95),(21,94),(22,93),(23,92),(24,91),(25,90),(26,89),(27,88),(28,87),(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),(43,74),(44,73),(45,72),(46,71),(47,84),(48,83),(49,82),(50,81),(51,80),(52,79),(53,78),(54,77),(55,76),(56,75),(57,62),(58,61),(59,60),(63,70),(64,69),(65,68),(66,67)]])
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 |
Matrix representation of C2×D4⋊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 | 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] >;
C2×D4⋊D14 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