metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊17D14, C14.212+ 1+4, C4⋊C4⋊50D14, (C4×D4)⋊22D7, (D4×C28)⋊24C2, C22⋊D28⋊7C2, C28⋊7D4⋊11C2, (C4×C28)⋊28C22, C22⋊C4⋊49D14, (C22×C4)⋊14D14, D14⋊D4⋊10C2, C23⋊D14⋊21C2, D14⋊C4⋊31C22, D14.5D4⋊8C2, (C2×D4).221D14, C4.D28⋊28C2, C42⋊2D7⋊10C2, C4⋊Dic7⋊10C22, Dic7⋊D4⋊27C2, (C2×C14).104C24, (C2×C28).162C23, Dic7⋊C4⋊33C22, (C22×C28)⋊11C22, Dic7.D4⋊9C2, C22⋊Dic14⋊9C2, C7⋊2(C22.32C24), (C2×Dic14)⋊7C22, (C4×Dic7)⋊53C22, (C2×D28).27C22, C22.6(C4○D28), C2.22(D4⋊6D14), C23.D7⋊10C22, C2.17(D4⋊8D14), (D4×C14).308C22, C23.23D14⋊2C2, (C2×Dic7).45C23, (C23×D7).42C22, (C22×D7).38C23, C23.101(C22×D7), C22.129(C23×D7), (C22×C14).174C23, (C22×Dic7).99C22, C4⋊C4⋊D7⋊8C2, (C4×C7⋊D4)⋊46C2, (C2×C4×D7)⋊49C22, (C2×D14⋊C4)⋊35C2, (C7×C4⋊C4)⋊62C22, C2.53(C2×C4○D28), C14.46(C2×C4○D4), (C2×C7⋊D4)⋊5C22, (C2×C14).17(C4○D4), (C7×C22⋊C4)⋊58C22, (C2×C4).162(C22×D7), SmallGroup(448,1013)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42⋊17D14
G = < a,b,c,d | a4=b4=c14=d2=1, ab=ba, cac-1=a-1, dad=a-1b2, bc=cb, dbd=a2b, dcd=c-1 >
Subgroups: 1364 in 250 conjugacy classes, 95 normal (91 characteristic)
C1, C2, C2, C4, C22, C22, C22, C7, C2×C4, C2×C4, D4, Q8, C23, C23, D7, C14, C14, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C24, Dic7, C28, D14, C2×C14, C2×C14, C2×C14, C2×C22⋊C4, C4×D4, C4×D4, C22≀C2, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42⋊2C2, Dic14, C4×D7, D28, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×D4, C22×D7, C22×D7, C22×C14, C22.32C24, C4×Dic7, Dic7⋊C4, C4⋊Dic7, D14⋊C4, C23.D7, C4×C28, C7×C22⋊C4, C7×C4⋊C4, C2×Dic14, C2×C4×D7, C2×D28, C22×Dic7, C2×C7⋊D4, C22×C28, D4×C14, C23×D7, C4.D28, C42⋊2D7, C22⋊Dic14, C22⋊D28, D14⋊D4, Dic7.D4, D14.5D4, C4⋊C4⋊D7, C2×D14⋊C4, C4×C7⋊D4, C23.23D14, C28⋊7D4, C23⋊D14, Dic7⋊D4, D4×C28, C42⋊17D14
Quotients: C1, C2, C22, C23, D7, C4○D4, C24, D14, C2×C4○D4, 2+ 1+4, C22×D7, C22.32C24, C4○D28, C23×D7, C2×C4○D28, D4⋊6D14, D4⋊8D14, C42⋊17D14
►On 112 points
(1 88 58 43)(2 44 59 89)(3 90 60 45)(4 46 61 91)(5 92 62 47)(6 48 63 93)(7 94 64 49)(8 50 65 95)(9 96 66 51)(10 52 67 97)(11 98 68 53)(12 54 69 85)(13 86 70 55)(14 56 57 87)(15 99 40 84)(16 71 41 100)(17 101 42 72)(18 73 29 102)(19 103 30 74)(20 75 31 104)(21 105 32 76)(22 77 33 106)(23 107 34 78)(24 79 35 108)(25 109 36 80)(26 81 37 110)(27 111 38 82)(28 83 39 112)
(1 24 8 17)(2 25 9 18)(3 26 10 19)(4 27 11 20)(5 28 12 21)(6 15 13 22)(7 16 14 23)(29 59 36 66)(30 60 37 67)(31 61 38 68)(32 62 39 69)(33 63 40 70)(34 64 41 57)(35 65 42 58)(43 108 95 72)(44 109 96 73)(45 110 97 74)(46 111 98 75)(47 112 85 76)(48 99 86 77)(49 100 87 78)(50 101 88 79)(51 102 89 80)(52 103 90 81)(53 104 91 82)(54 105 92 83)(55 106 93 84)(56 107 94 71)
(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 29)(16 42)(17 41)(18 40)(19 39)(20 38)(21 37)(22 36)(23 35)(24 34)(25 33)(26 32)(27 31)(28 30)(43 94)(44 93)(45 92)(46 91)(47 90)(48 89)(49 88)(50 87)(51 86)(52 85)(53 98)(54 97)(55 96)(56 95)(57 58)(59 70)(60 69)(61 68)(62 67)(63 66)(64 65)(71 79)(72 78)(73 77)(74 76)(80 84)(81 83)(99 109)(100 108)(101 107)(102 106)(103 105)(110 112)
G:=sub<Sym(112)| (1,88,58,43)(2,44,59,89)(3,90,60,45)(4,46,61,91)(5,92,62,47)(6,48,63,93)(7,94,64,49)(8,50,65,95)(9,96,66,51)(10,52,67,97)(11,98,68,53)(12,54,69,85)(13,86,70,55)(14,56,57,87)(15,99,40,84)(16,71,41,100)(17,101,42,72)(18,73,29,102)(19,103,30,74)(20,75,31,104)(21,105,32,76)(22,77,33,106)(23,107,34,78)(24,79,35,108)(25,109,36,80)(26,81,37,110)(27,111,38,82)(28,83,39,112), (1,24,8,17)(2,25,9,18)(3,26,10,19)(4,27,11,20)(5,28,12,21)(6,15,13,22)(7,16,14,23)(29,59,36,66)(30,60,37,67)(31,61,38,68)(32,62,39,69)(33,63,40,70)(34,64,41,57)(35,65,42,58)(43,108,95,72)(44,109,96,73)(45,110,97,74)(46,111,98,75)(47,112,85,76)(48,99,86,77)(49,100,87,78)(50,101,88,79)(51,102,89,80)(52,103,90,81)(53,104,91,82)(54,105,92,83)(55,106,93,84)(56,107,94,71), (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)(15,29)(16,42)(17,41)(18,40)(19,39)(20,38)(21,37)(22,36)(23,35)(24,34)(25,33)(26,32)(27,31)(28,30)(43,94)(44,93)(45,92)(46,91)(47,90)(48,89)(49,88)(50,87)(51,86)(52,85)(53,98)(54,97)(55,96)(56,95)(57,58)(59,70)(60,69)(61,68)(62,67)(63,66)(64,65)(71,79)(72,78)(73,77)(74,76)(80,84)(81,83)(99,109)(100,108)(101,107)(102,106)(103,105)(110,112)>;
G:=Group( (1,88,58,43)(2,44,59,89)(3,90,60,45)(4,46,61,91)(5,92,62,47)(6,48,63,93)(7,94,64,49)(8,50,65,95)(9,96,66,51)(10,52,67,97)(11,98,68,53)(12,54,69,85)(13,86,70,55)(14,56,57,87)(15,99,40,84)(16,71,41,100)(17,101,42,72)(18,73,29,102)(19,103,30,74)(20,75,31,104)(21,105,32,76)(22,77,33,106)(23,107,34,78)(24,79,35,108)(25,109,36,80)(26,81,37,110)(27,111,38,82)(28,83,39,112), (1,24,8,17)(2,25,9,18)(3,26,10,19)(4,27,11,20)(5,28,12,21)(6,15,13,22)(7,16,14,23)(29,59,36,66)(30,60,37,67)(31,61,38,68)(32,62,39,69)(33,63,40,70)(34,64,41,57)(35,65,42,58)(43,108,95,72)(44,109,96,73)(45,110,97,74)(46,111,98,75)(47,112,85,76)(48,99,86,77)(49,100,87,78)(50,101,88,79)(51,102,89,80)(52,103,90,81)(53,104,91,82)(54,105,92,83)(55,106,93,84)(56,107,94,71), (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)(15,29)(16,42)(17,41)(18,40)(19,39)(20,38)(21,37)(22,36)(23,35)(24,34)(25,33)(26,32)(27,31)(28,30)(43,94)(44,93)(45,92)(46,91)(47,90)(48,89)(49,88)(50,87)(51,86)(52,85)(53,98)(54,97)(55,96)(56,95)(57,58)(59,70)(60,69)(61,68)(62,67)(63,66)(64,65)(71,79)(72,78)(73,77)(74,76)(80,84)(81,83)(99,109)(100,108)(101,107)(102,106)(103,105)(110,112) );
G=PermutationGroup([[(1,88,58,43),(2,44,59,89),(3,90,60,45),(4,46,61,91),(5,92,62,47),(6,48,63,93),(7,94,64,49),(8,50,65,95),(9,96,66,51),(10,52,67,97),(11,98,68,53),(12,54,69,85),(13,86,70,55),(14,56,57,87),(15,99,40,84),(16,71,41,100),(17,101,42,72),(18,73,29,102),(19,103,30,74),(20,75,31,104),(21,105,32,76),(22,77,33,106),(23,107,34,78),(24,79,35,108),(25,109,36,80),(26,81,37,110),(27,111,38,82),(28,83,39,112)], [(1,24,8,17),(2,25,9,18),(3,26,10,19),(4,27,11,20),(5,28,12,21),(6,15,13,22),(7,16,14,23),(29,59,36,66),(30,60,37,67),(31,61,38,68),(32,62,39,69),(33,63,40,70),(34,64,41,57),(35,65,42,58),(43,108,95,72),(44,109,96,73),(45,110,97,74),(46,111,98,75),(47,112,85,76),(48,99,86,77),(49,100,87,78),(50,101,88,79),(51,102,89,80),(52,103,90,81),(53,104,91,82),(54,105,92,83),(55,106,93,84),(56,107,94,71)], [(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),(15,29),(16,42),(17,41),(18,40),(19,39),(20,38),(21,37),(22,36),(23,35),(24,34),(25,33),(26,32),(27,31),(28,30),(43,94),(44,93),(45,92),(46,91),(47,90),(48,89),(49,88),(50,87),(51,86),(52,85),(53,98),(54,97),(55,96),(56,95),(57,58),(59,70),(60,69),(61,68),(62,67),(63,66),(64,65),(71,79),(72,78),(73,77),(74,76),(80,84),(81,83),(99,109),(100,108),(101,107),(102,106),(103,105),(110,112)]])
82 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | ··· | 4L | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14U | 28A | ··· | 28L | 28M | ··· | 28AJ |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 28 | ··· | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
82 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D7 | C4○D4 | D14 | D14 | D14 | D14 | D14 | C4○D28 | 2+ 1+4 | D4⋊6D14 | D4⋊8D14 |
| kernel | C42⋊17D14 | C4.D28 | C42⋊2D7 | C22⋊Dic14 | C22⋊D28 | D14⋊D4 | Dic7.D4 | D14.5D4 | C4⋊C4⋊D7 | C2×D14⋊C4 | C4×C7⋊D4 | C23.23D14 | C28⋊7D4 | C23⋊D14 | Dic7⋊D4 | D4×C28 | C4×D4 | C2×C14 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C22 | C14 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 4 | 3 | 6 | 3 | 6 | 3 | 24 | 2 | 6 | 6 |
Matrix representation of C42⋊17D14 ►in GL6(𝔽29)
| 11 | 24 | 0 | 0 | 0 | 0 |
| 24 | 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 16 | 0 | 4 |
| 0 | 0 | 24 | 20 | 25 | 11 |
| 0 | 0 | 24 | 15 | 24 | 13 |
| 0 | 0 | 14 | 0 | 16 | 5 |
| 17 | 0 | 0 | 0 | 0 | 0 |
| 0 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 19 | 6 | 0 | 0 |
| 0 | 0 | 17 | 10 | 0 | 0 |
| 0 | 0 | 20 | 0 | 8 | 6 |
| 0 | 0 | 26 | 9 | 23 | 21 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 20 | 19 | 0 | 0 |
| 0 | 0 | 20 | 6 | 0 | 0 |
| 0 | 0 | 13 | 26 | 10 | 10 |
| 0 | 0 | 6 | 12 | 19 | 22 |
| 28 | 13 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 22 | 22 | 0 | 0 |
| 0 | 0 | 11 | 7 | 0 | 0 |
| 0 | 0 | 7 | 17 | 19 | 19 |
| 0 | 0 | 1 | 19 | 7 | 10 |
G:=sub<GL(6,GF(29))| [11,24,0,0,0,0,24,18,0,0,0,0,0,0,9,24,24,14,0,0,16,20,15,0,0,0,0,25,24,16,0,0,4,11,13,5],[17,0,0,0,0,0,0,17,0,0,0,0,0,0,19,17,20,26,0,0,6,10,0,9,0,0,0,0,8,23,0,0,0,0,6,21],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,20,20,13,6,0,0,19,6,26,12,0,0,0,0,10,19,0,0,0,0,10,22],[28,0,0,0,0,0,13,1,0,0,0,0,0,0,22,11,7,1,0,0,22,7,17,19,0,0,0,0,19,7,0,0,0,0,19,10] >;
C42⋊17D14 in GAP, Magma, Sage, TeX
C_4^2\rtimes_{17}D_{14} % in TeX
G:=Group("C4^2:17D14"); // GroupNames label
G:=SmallGroup(448,1013);
// by ID
G=gap.SmallGroup(448,1013);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,758,100,675,570,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^14=d^2=1,a*b=b*a,c*a*c^-1=a^-1,d*a*d=a^-1*b^2,b*c=c*b,d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations