metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊26D14, C14.762+ 1+4, (C4×D7)⋊5D4, C4⋊1D4⋊5D7, C4.34(D4×D7), (C2×D4)⋊12D14, C28.65(C2×D4), C28⋊D4⋊26C2, (C4×C28)⋊26C22, D14.47(C2×D4), C23⋊D14⋊26C2, D14⋊C4⋊34C22, C4.D28⋊25C2, (D4×C14)⋊32C22, C42⋊D7⋊23C2, Dic7.52(C2×D4), C14.93(C22×D4), Dic7⋊D4⋊36C2, C28.17D4⋊26C2, (C2×C28).635C23, (C2×C14).259C24, Dic7⋊C4⋊71C22, C7⋊5(C22.29C24), (C4×Dic7)⋊39C22, C23.D7⋊36C22, C2.80(D4⋊6D14), C23.65(C22×D7), (C2×Dic14)⋊34C22, (C2×D28).170C22, (C22×C14).73C23, (C23×D7).72C22, C22.280(C23×D7), (C2×Dic7).134C23, (C22×Dic7)⋊29C22, (C22×D7).227C23, (C2×D4×D7)⋊19C2, C2.66(C2×D4×D7), (C7×C4⋊1D4)⋊6C2, (C2×D4⋊2D7)⋊20C2, (C2×C7⋊D4)⋊26C22, (C2×C4×D7).138C22, (C2×C4).213(C22×D7), SmallGroup(448,1168)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42⋊26D14
G = < a,b,c,d | a4=b4=c14=d2=1, ab=ba, cac-1=dad=a-1, cbc-1=b-1, dbd=a2b-1, dcd=c-1 >
Subgroups: 1900 in 334 conjugacy classes, 103 normal (29 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, D7, C14, C14, C14, C42, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, Dic7, Dic7, C28, C28, D14, D14, C2×C14, C2×C14, C42⋊C2, C22≀C2, C4⋊D4, C4.4D4, C4⋊1D4, C4⋊1D4, C22×D4, C2×C4○D4, Dic14, C4×D7, D28, C2×Dic7, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×D4, C22×D7, C22×D7, C22×D7, C22×C14, C22.29C24, C4×Dic7, Dic7⋊C4, D14⋊C4, C23.D7, C4×C28, C2×Dic14, C2×C4×D7, C2×D28, D4×D7, D4⋊2D7, C22×Dic7, C2×C7⋊D4, D4×C14, D4×C14, C23×D7, C42⋊D7, C4.D28, C28.17D4, C23⋊D14, Dic7⋊D4, C28⋊D4, C7×C4⋊1D4, C2×D4×D7, C2×D4⋊2D7, C42⋊26D14
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C24, D14, C22×D4, 2+ 1+4, C22×D7, C22.29C24, D4×D7, C23×D7, C2×D4×D7, D4⋊6D14, C42⋊26D14
►On 112 points
(1 62 19 69)(2 70 20 63)(3 64 21 57)(4 58 15 65)(5 66 16 59)(6 60 17 67)(7 68 18 61)(8 95 28 88)(9 89 22 96)(10 97 23 90)(11 91 24 98)(12 85 25 92)(13 93 26 86)(14 87 27 94)(29 79 45 110)(30 111 46 80)(31 81 47 112)(32 99 48 82)(33 83 49 100)(34 101 50 84)(35 71 51 102)(36 103 52 72)(37 73 53 104)(38 105 54 74)(39 75 55 106)(40 107 56 76)(41 77 43 108)(42 109 44 78)
(1 34 12 41)(2 42 13 35)(3 36 14 29)(4 30 8 37)(5 38 9 31)(6 32 10 39)(7 40 11 33)(15 46 28 53)(16 54 22 47)(17 48 23 55)(18 56 24 49)(19 50 25 43)(20 44 26 51)(21 52 27 45)(57 72 94 110)(58 111 95 73)(59 74 96 112)(60 99 97 75)(61 76 98 100)(62 101 85 77)(63 78 86 102)(64 103 87 79)(65 80 88 104)(66 105 89 81)(67 82 90 106)(68 107 91 83)(69 84 92 108)(70 109 93 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 11)(2 10)(3 9)(4 8)(5 14)(6 13)(7 12)(15 28)(16 27)(17 26)(18 25)(19 24)(20 23)(21 22)(29 47)(30 46)(31 45)(32 44)(33 43)(34 56)(35 55)(36 54)(37 53)(38 52)(39 51)(40 50)(41 49)(42 48)(57 89)(58 88)(59 87)(60 86)(61 85)(62 98)(63 97)(64 96)(65 95)(66 94)(67 93)(68 92)(69 91)(70 90)(71 75)(72 74)(76 84)(77 83)(78 82)(79 81)(99 109)(100 108)(101 107)(102 106)(103 105)(110 112)
G:=sub<Sym(112)| (1,62,19,69)(2,70,20,63)(3,64,21,57)(4,58,15,65)(5,66,16,59)(6,60,17,67)(7,68,18,61)(8,95,28,88)(9,89,22,96)(10,97,23,90)(11,91,24,98)(12,85,25,92)(13,93,26,86)(14,87,27,94)(29,79,45,110)(30,111,46,80)(31,81,47,112)(32,99,48,82)(33,83,49,100)(34,101,50,84)(35,71,51,102)(36,103,52,72)(37,73,53,104)(38,105,54,74)(39,75,55,106)(40,107,56,76)(41,77,43,108)(42,109,44,78), (1,34,12,41)(2,42,13,35)(3,36,14,29)(4,30,8,37)(5,38,9,31)(6,32,10,39)(7,40,11,33)(15,46,28,53)(16,54,22,47)(17,48,23,55)(18,56,24,49)(19,50,25,43)(20,44,26,51)(21,52,27,45)(57,72,94,110)(58,111,95,73)(59,74,96,112)(60,99,97,75)(61,76,98,100)(62,101,85,77)(63,78,86,102)(64,103,87,79)(65,80,88,104)(66,105,89,81)(67,82,90,106)(68,107,91,83)(69,84,92,108)(70,109,93,71), (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), (1,11)(2,10)(3,9)(4,8)(5,14)(6,13)(7,12)(15,28)(16,27)(17,26)(18,25)(19,24)(20,23)(21,22)(29,47)(30,46)(31,45)(32,44)(33,43)(34,56)(35,55)(36,54)(37,53)(38,52)(39,51)(40,50)(41,49)(42,48)(57,89)(58,88)(59,87)(60,86)(61,85)(62,98)(63,97)(64,96)(65,95)(66,94)(67,93)(68,92)(69,91)(70,90)(71,75)(72,74)(76,84)(77,83)(78,82)(79,81)(99,109)(100,108)(101,107)(102,106)(103,105)(110,112)>;
G:=Group( (1,62,19,69)(2,70,20,63)(3,64,21,57)(4,58,15,65)(5,66,16,59)(6,60,17,67)(7,68,18,61)(8,95,28,88)(9,89,22,96)(10,97,23,90)(11,91,24,98)(12,85,25,92)(13,93,26,86)(14,87,27,94)(29,79,45,110)(30,111,46,80)(31,81,47,112)(32,99,48,82)(33,83,49,100)(34,101,50,84)(35,71,51,102)(36,103,52,72)(37,73,53,104)(38,105,54,74)(39,75,55,106)(40,107,56,76)(41,77,43,108)(42,109,44,78), (1,34,12,41)(2,42,13,35)(3,36,14,29)(4,30,8,37)(5,38,9,31)(6,32,10,39)(7,40,11,33)(15,46,28,53)(16,54,22,47)(17,48,23,55)(18,56,24,49)(19,50,25,43)(20,44,26,51)(21,52,27,45)(57,72,94,110)(58,111,95,73)(59,74,96,112)(60,99,97,75)(61,76,98,100)(62,101,85,77)(63,78,86,102)(64,103,87,79)(65,80,88,104)(66,105,89,81)(67,82,90,106)(68,107,91,83)(69,84,92,108)(70,109,93,71), (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), (1,11)(2,10)(3,9)(4,8)(5,14)(6,13)(7,12)(15,28)(16,27)(17,26)(18,25)(19,24)(20,23)(21,22)(29,47)(30,46)(31,45)(32,44)(33,43)(34,56)(35,55)(36,54)(37,53)(38,52)(39,51)(40,50)(41,49)(42,48)(57,89)(58,88)(59,87)(60,86)(61,85)(62,98)(63,97)(64,96)(65,95)(66,94)(67,93)(68,92)(69,91)(70,90)(71,75)(72,74)(76,84)(77,83)(78,82)(79,81)(99,109)(100,108)(101,107)(102,106)(103,105)(110,112) );
G=PermutationGroup([[(1,62,19,69),(2,70,20,63),(3,64,21,57),(4,58,15,65),(5,66,16,59),(6,60,17,67),(7,68,18,61),(8,95,28,88),(9,89,22,96),(10,97,23,90),(11,91,24,98),(12,85,25,92),(13,93,26,86),(14,87,27,94),(29,79,45,110),(30,111,46,80),(31,81,47,112),(32,99,48,82),(33,83,49,100),(34,101,50,84),(35,71,51,102),(36,103,52,72),(37,73,53,104),(38,105,54,74),(39,75,55,106),(40,107,56,76),(41,77,43,108),(42,109,44,78)], [(1,34,12,41),(2,42,13,35),(3,36,14,29),(4,30,8,37),(5,38,9,31),(6,32,10,39),(7,40,11,33),(15,46,28,53),(16,54,22,47),(17,48,23,55),(18,56,24,49),(19,50,25,43),(20,44,26,51),(21,52,27,45),(57,72,94,110),(58,111,95,73),(59,74,96,112),(60,99,97,75),(61,76,98,100),(62,101,85,77),(63,78,86,102),(64,103,87,79),(65,80,88,104),(66,105,89,81),(67,82,90,106),(68,107,91,83),(69,84,92,108),(70,109,93,71)], [(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)], [(1,11),(2,10),(3,9),(4,8),(5,14),(6,13),(7,12),(15,28),(16,27),(17,26),(18,25),(19,24),(20,23),(21,22),(29,47),(30,46),(31,45),(32,44),(33,43),(34,56),(35,55),(36,54),(37,53),(38,52),(39,51),(40,50),(41,49),(42,48),(57,89),(58,88),(59,87),(60,86),(61,85),(62,98),(63,97),(64,96),(65,95),(66,94),(67,93),(68,92),(69,91),(70,90),(71,75),(72,74),(76,84),(77,83),(78,82),(79,81),(99,109),(100,108),(101,107),(102,106),(103,105),(110,112)]])
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14U | 28A | ··· | 28R |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 14 | 14 | 28 | 28 | 2 | 2 | 4 | 4 | 14 | 14 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | ··· | 4 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D7 | D14 | D14 | 2+ 1+4 | D4×D7 | D4⋊6D14 |
| kernel | C42⋊26D14 | C42⋊D7 | C4.D28 | C28.17D4 | C23⋊D14 | Dic7⋊D4 | C28⋊D4 | C7×C4⋊1D4 | C2×D4×D7 | C2×D4⋊2D7 | C4×D7 | C4⋊1D4 | C42 | C2×D4 | C14 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 1 | 1 | 1 | 1 | 4 | 3 | 3 | 18 | 2 | 6 | 12 |
Matrix representation of C42⋊26D14 ►in GL6(𝔽29)
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 2 |
| 0 | 0 | 0 | 0 | 27 | 18 |
| 0 | 0 | 18 | 27 | 0 | 0 |
| 0 | 0 | 2 | 11 | 0 | 0 |
| 1 | 2 | 0 | 0 | 0 | 0 |
| 28 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 28 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 25 | 25 | 0 | 0 |
| 0 | 0 | 4 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 4 |
| 0 | 0 | 0 | 0 | 25 | 18 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 28 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 4 | 0 | 0 |
| 0 | 0 | 18 | 25 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 4 |
| 0 | 0 | 0 | 0 | 18 | 25 |
G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,18,2,0,0,0,0,27,11,0,0,11,27,0,0,0,0,2,18,0,0],[1,28,0,0,0,0,2,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,1,0,0,0,0,0,0,1,0,0],[1,28,0,0,0,0,0,28,0,0,0,0,0,0,25,4,0,0,0,0,25,11,0,0,0,0,0,0,4,25,0,0,0,0,4,18],[1,28,0,0,0,0,0,28,0,0,0,0,0,0,4,18,0,0,0,0,4,25,0,0,0,0,0,0,4,18,0,0,0,0,4,25] >;
C42⋊26D14 in GAP, Magma, Sage, TeX
C_4^2\rtimes_{26}D_{14} % in TeX
G:=Group("C4^2:26D14"); // GroupNames label
G:=SmallGroup(448,1168);
// by ID
G=gap.SmallGroup(448,1168);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,219,675,570,297,136,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^14=d^2=1,a*b=b*a,c*a*c^-1=d*a*d=a^-1,c*b*c^-1=b^-1,d*b*d=a^2*b^-1,d*c*d=c^-1>;
// generators/relations