metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊1D14, M4(2)⋊16D14, C4≀C2⋊5D7, (D4×D7)⋊4C4, (Q8×D7)⋊4C4, D4⋊2D7⋊4C4, Q8⋊2D7⋊4C4, (C4×D7).48D4, D28.5(C2×C4), D4.11(C4×D7), C4.201(D4×D7), Q8.11(C4×D7), Dic14⋊C4⋊4C2, D28⋊4C4⋊6C2, (C4×C28)⋊10C22, C4○D4.19D14, C28.360(C2×D4), C42⋊D7⋊9C2, (D7×M4(2))⋊9C2, C22.28(D4×D7), D4⋊2Dic7⋊2C2, C28.18(C22×C4), Dic14.5(C2×C4), (C2×Dic7).37D4, (C4×Dic7)⋊3C22, (C22×D7).23D4, C4.Dic7⋊3C22, (C2×C28).261C23, D14.8(C22⋊C4), C7⋊1(C42⋊C22), C4○D28.10C22, (C7×M4(2))⋊14C22, Dic7.14(C22⋊C4), (C7×C4≀C2)⋊6C2, C4.18(C2×C4×D7), (D7×C4○D4).2C2, (C4×D7).5(C2×C4), (C7×D4).5(C2×C4), (C7×Q8).5(C2×C4), (C2×C14).25(C2×D4), C2.26(D7×C22⋊C4), (C2×C4×D7).29C22, C14.25(C2×C22⋊C4), (C7×C4○D4).2C22, (C2×C4).368(C22×D7), SmallGroup(448,355)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42⋊D14
G = < a,b,c,d | a4=b4=c14=d2=1, cac-1=ab=ba, dad=ab-1, cbc-1=dbd=b-1, dcd=c-1 >
Subgroups: 828 in 154 conjugacy classes, 51 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, D7, C14, C14, C42, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), M4(2), C22×C4, C2×D4, C2×Q8, C4○D4, C4○D4, Dic7, Dic7, C28, C28, D14, D14, C2×C14, C2×C14, C4≀C2, C4≀C2, C42⋊C2, C2×M4(2), C2×C4○D4, C7⋊C8, C56, Dic14, Dic14, C4×D7, C4×D7, D28, D28, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C22×D7, C22×D7, C42⋊C22, C8×D7, C8⋊D7, C4.Dic7, C4×Dic7, Dic7⋊C4, D14⋊C4, C4×C28, C7×M4(2), C2×C4×D7, C2×C4×D7, C4○D28, C4○D28, D4×D7, D4×D7, D4⋊2D7, D4⋊2D7, Q8×D7, Q8⋊2D7, C7×C4○D4, Dic14⋊C4, D28⋊4C4, D4⋊2Dic7, C7×C4≀C2, C42⋊D7, D7×M4(2), D7×C4○D4, C42⋊D14
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C22⋊C4, C22×C4, C2×D4, D14, C2×C22⋊C4, C4×D7, C22×D7, C42⋊C22, C2×C4×D7, D4×D7, D7×C22⋊C4, C42⋊D14
►On 112 points
(1 83 108 85)(2 109)(3 71 110 87)(4 111)(5 73 112 89)(6 99)(7 75 100 91)(8 101)(9 77 102 93)(10 103)(11 79 104 95)(12 105)(13 81 106 97)(14 107)(16 35 43 69)(18 37 45 57)(20 39 47 59)(22 41 49 61)(24 29 51 63)(26 31 53 65)(28 33 55 67)(72 88)(74 90)(76 92)(78 94)(80 96)(82 98)(84 86)
(1 83 108 85)(2 86 109 84)(3 71 110 87)(4 88 111 72)(5 73 112 89)(6 90 99 74)(7 75 100 91)(8 92 101 76)(9 77 102 93)(10 94 103 78)(11 79 104 95)(12 96 105 80)(13 81 106 97)(14 98 107 82)(15 34 56 68)(16 69 43 35)(17 36 44 70)(18 57 45 37)(19 38 46 58)(20 59 47 39)(21 40 48 60)(22 61 49 41)(23 42 50 62)(24 63 51 29)(25 30 52 64)(26 65 53 31)(27 32 54 66)(28 67 55 33)
(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 60)(2 59)(3 58)(4 57)(5 70)(6 69)(7 68)(8 67)(9 66)(10 65)(11 64)(12 63)(13 62)(14 61)(15 91)(16 90)(17 89)(18 88)(19 87)(20 86)(21 85)(22 98)(23 97)(24 96)(25 95)(26 94)(27 93)(28 92)(29 105)(30 104)(31 103)(32 102)(33 101)(34 100)(35 99)(36 112)(37 111)(38 110)(39 109)(40 108)(41 107)(42 106)(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)
G:=sub<Sym(112)| (1,83,108,85)(2,109)(3,71,110,87)(4,111)(5,73,112,89)(6,99)(7,75,100,91)(8,101)(9,77,102,93)(10,103)(11,79,104,95)(12,105)(13,81,106,97)(14,107)(16,35,43,69)(18,37,45,57)(20,39,47,59)(22,41,49,61)(24,29,51,63)(26,31,53,65)(28,33,55,67)(72,88)(74,90)(76,92)(78,94)(80,96)(82,98)(84,86), (1,83,108,85)(2,86,109,84)(3,71,110,87)(4,88,111,72)(5,73,112,89)(6,90,99,74)(7,75,100,91)(8,92,101,76)(9,77,102,93)(10,94,103,78)(11,79,104,95)(12,96,105,80)(13,81,106,97)(14,98,107,82)(15,34,56,68)(16,69,43,35)(17,36,44,70)(18,57,45,37)(19,38,46,58)(20,59,47,39)(21,40,48,60)(22,61,49,41)(23,42,50,62)(24,63,51,29)(25,30,52,64)(26,65,53,31)(27,32,54,66)(28,67,55,33), (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), (1,60)(2,59)(3,58)(4,57)(5,70)(6,69)(7,68)(8,67)(9,66)(10,65)(11,64)(12,63)(13,62)(14,61)(15,91)(16,90)(17,89)(18,88)(19,87)(20,86)(21,85)(22,98)(23,97)(24,96)(25,95)(26,94)(27,93)(28,92)(29,105)(30,104)(31,103)(32,102)(33,101)(34,100)(35,99)(36,112)(37,111)(38,110)(39,109)(40,108)(41,107)(42,106)(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)>;
G:=Group( (1,83,108,85)(2,109)(3,71,110,87)(4,111)(5,73,112,89)(6,99)(7,75,100,91)(8,101)(9,77,102,93)(10,103)(11,79,104,95)(12,105)(13,81,106,97)(14,107)(16,35,43,69)(18,37,45,57)(20,39,47,59)(22,41,49,61)(24,29,51,63)(26,31,53,65)(28,33,55,67)(72,88)(74,90)(76,92)(78,94)(80,96)(82,98)(84,86), (1,83,108,85)(2,86,109,84)(3,71,110,87)(4,88,111,72)(5,73,112,89)(6,90,99,74)(7,75,100,91)(8,92,101,76)(9,77,102,93)(10,94,103,78)(11,79,104,95)(12,96,105,80)(13,81,106,97)(14,98,107,82)(15,34,56,68)(16,69,43,35)(17,36,44,70)(18,57,45,37)(19,38,46,58)(20,59,47,39)(21,40,48,60)(22,61,49,41)(23,42,50,62)(24,63,51,29)(25,30,52,64)(26,65,53,31)(27,32,54,66)(28,67,55,33), (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), (1,60)(2,59)(3,58)(4,57)(5,70)(6,69)(7,68)(8,67)(9,66)(10,65)(11,64)(12,63)(13,62)(14,61)(15,91)(16,90)(17,89)(18,88)(19,87)(20,86)(21,85)(22,98)(23,97)(24,96)(25,95)(26,94)(27,93)(28,92)(29,105)(30,104)(31,103)(32,102)(33,101)(34,100)(35,99)(36,112)(37,111)(38,110)(39,109)(40,108)(41,107)(42,106)(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) );
G=PermutationGroup([[(1,83,108,85),(2,109),(3,71,110,87),(4,111),(5,73,112,89),(6,99),(7,75,100,91),(8,101),(9,77,102,93),(10,103),(11,79,104,95),(12,105),(13,81,106,97),(14,107),(16,35,43,69),(18,37,45,57),(20,39,47,59),(22,41,49,61),(24,29,51,63),(26,31,53,65),(28,33,55,67),(72,88),(74,90),(76,92),(78,94),(80,96),(82,98),(84,86)], [(1,83,108,85),(2,86,109,84),(3,71,110,87),(4,88,111,72),(5,73,112,89),(6,90,99,74),(7,75,100,91),(8,92,101,76),(9,77,102,93),(10,94,103,78),(11,79,104,95),(12,96,105,80),(13,81,106,97),(14,98,107,82),(15,34,56,68),(16,69,43,35),(17,36,44,70),(18,57,45,37),(19,38,46,58),(20,59,47,39),(21,40,48,60),(22,61,49,41),(23,42,50,62),(24,63,51,29),(25,30,52,64),(26,65,53,31),(27,32,54,66),(28,67,55,33)], [(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)], [(1,60),(2,59),(3,58),(4,57),(5,70),(6,69),(7,68),(8,67),(9,66),(10,65),(11,64),(12,63),(13,62),(14,61),(15,91),(16,90),(17,89),(18,88),(19,87),(20,86),(21,85),(22,98),(23,97),(24,96),(25,95),(26,94),(27,93),(28,92),(29,105),(30,104),(31,103),(32,102),(33,101),(34,100),(35,99),(36,112),(37,111),(38,110),(39,109),(40,108),(41,107),(42,106),(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)]])
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | 14B | 14C | 14D | 14E | 14F | 14G | 14H | 14I | 28A | ··· | 28F | 28G | ··· | 28U | 28V | 28W | 28X | 56A | ··· | 56F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | 14 | 14 | 14 | 14 | 14 | 14 | 14 | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 28 | 28 | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 4 | 14 | 14 | 28 | 1 | 1 | 2 | 4 | 4 | 4 | 14 | 14 | 28 | 28 | 28 | 2 | 2 | 2 | 4 | 4 | 28 | 28 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 8 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | ··· | 8 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | D4 | D4 | D4 | D7 | D14 | D14 | D14 | C4×D7 | C4×D7 | C42⋊C22 | D4×D7 | D4×D7 | C42⋊D14 |
| kernel | C42⋊D14 | Dic14⋊C4 | D28⋊4C4 | D4⋊2Dic7 | C7×C4≀C2 | C42⋊D7 | D7×M4(2) | D7×C4○D4 | D4×D7 | D4⋊2D7 | Q8×D7 | Q8⋊2D7 | C4×D7 | C2×Dic7 | C22×D7 | C4≀C2 | C42 | M4(2) | C4○D4 | D4 | Q8 | C7 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 1 | 1 | 3 | 3 | 3 | 3 | 6 | 6 | 2 | 3 | 3 | 12 |
Matrix representation of C42⋊D14 ►in GL4(𝔽113) generated by
| 98 | 0 | 0 | 0 |
| 19 | 112 | 0 | 0 |
| 72 | 16 | 15 | 0 |
| 32 | 43 | 58 | 1 |
| 98 | 0 | 0 | 0 |
| 73 | 15 | 0 | 0 |
| 40 | 83 | 98 | 0 |
| 36 | 21 | 21 | 15 |
| 99 | 67 | 0 | 0 |
| 38 | 14 | 0 | 0 |
| 27 | 8 | 8 | 76 |
| 26 | 102 | 34 | 105 |
| 92 | 41 | 34 | 97 |
| 63 | 47 | 0 | 54 |
| 22 | 17 | 40 | 100 |
| 45 | 18 | 111 | 47 |
G:=sub<GL(4,GF(113))| [98,19,72,32,0,112,16,43,0,0,15,58,0,0,0,1],[98,73,40,36,0,15,83,21,0,0,98,21,0,0,0,15],[99,38,27,26,67,14,8,102,0,0,8,34,0,0,76,105],[92,63,22,45,41,47,17,18,34,0,40,111,97,54,100,47] >;
C42⋊D14 in GAP, Magma, Sage, TeX
C_4^2\rtimes D_{14} % in TeX
G:=Group("C4^2:D14"); // GroupNames label
G:=SmallGroup(448,355);
// by ID
G=gap.SmallGroup(448,355);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,758,219,58,136,851,438,102,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^14=d^2=1,c*a*c^-1=a*b=b*a,d*a*d=a*b^-1,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations