metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42:10D14, C14.962+ 1+4, C4:C4:44D14, (C4xD28):8C2, (C4xC28):6C22, D14:Q8:4C2, C4.D28:6C2, C42:D7:1C2, C42:2D7:3C2, D14:C4:40C22, D14.5D4:4C2, C22:D28.1C2, C42:C2:11D7, (C2xC14).71C24, C4:Dic7:56C22, C22:C4.95D14, D14.15(C4oD4), Dic7:4D4:43C2, C2.8(D4:8D14), (C2xC28).146C23, Dic7:C4:32C22, C22:Dic14:4C2, (C4xDic7):50C22, (C2xDic14):5C22, (C2xD28).23C22, (C22xC4).192D14, C7:2(C22.45C24), C23.D7.4C22, C22.18(C4oD28), (C23xD7).37C22, (C22xD7).21C23, C23.159(C22xD7), C22.100(C23xD7), C23.23D14:27C2, (C22xC28).435C22, (C22xC14).141C23, (C2xDic7).198C23, (C22xDic7).88C22, C4:C4:D7:4C2, C2.10(D7xC4oD4), (C2xC4xD7):45C22, (C2xD14:C4):40C2, (C7xC4:C4):54C22, (D7xC22:C4):26C2, C2.30(C2xC4oD28), C14.28(C2xC4oD4), (C2xC7:D4).9C22, (C7xC42:C2):13C2, (C2xC14).41(C4oD4), (C2xC4).274(C22xD7), (C7xC22:C4).138C22, SmallGroup(448,980)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42:10D14
G = < a,b,c,d | a4=b4=c14=d2=1, ab=ba, cac-1=dad=ab2, bc=cb, dbd=a2b, dcd=c-1 >
Subgroups: 1300 in 248 conjugacy classes, 97 normal (91 characteristic)
C1, C2, C2, C4, C22, C22, C22, C7, C2xC4, C2xC4, D4, Q8, C23, C23, D7, C14, C14, C42, C42, C22:C4, C22:C4, C4:C4, C4:C4, C22xC4, C22xC4, C2xD4, C2xQ8, C24, Dic7, C28, D14, D14, C2xC14, C2xC14, C2xC14, C2xC22:C4, C42:C2, C42:C2, C4xD4, C22wrC2, C22:Q8, C22.D4, C4.4D4, C42:2C2, Dic14, C4xD7, D28, C2xDic7, C2xDic7, C7:D4, C2xC28, C2xC28, C22xD7, C22xD7, C22xC14, C22.45C24, C4xDic7, Dic7:C4, C4:Dic7, D14:C4, C23.D7, C4xC28, C7xC22:C4, C7xC4:C4, C2xDic14, C2xC4xD7, C2xD28, C22xDic7, C2xC7:D4, C22xC28, C23xD7, C42:D7, C4xD28, C4.D28, C42:2D7, C22:Dic14, D7xC22:C4, Dic7:4D4, C22:D28, D14.5D4, D14:Q8, C4:C4:D7, C2xD14:C4, C23.23D14, C7xC42:C2, C42:10D14
Quotients: C1, C2, C22, C23, D7, C4oD4, C24, D14, C2xC4oD4, 2+ 1+4, C22xD7, C22.45C24, C4oD28, C23xD7, C2xC4oD28, D7xC4oD4, D4:8D14, C42:10D14
►On 112 points
(1 72 19 69)(2 80 20 63)(3 74 21 57)(4 82 15 65)(5 76 16 59)(6 84 17 67)(7 78 18 61)(8 75 22 58)(9 83 23 66)(10 77 24 60)(11 71 25 68)(12 79 26 62)(13 73 27 70)(14 81 28 64)(29 112 43 96)(30 106 44 90)(31 100 45 98)(32 108 46 92)(33 102 47 86)(34 110 48 94)(35 104 49 88)(36 103 55 87)(37 111 56 95)(38 105 50 89)(39 99 51 97)(40 107 52 91)(41 101 53 85)(42 109 54 93)
(1 42 12 33)(2 36 13 34)(3 37 14 35)(4 38 8 29)(5 39 9 30)(6 40 10 31)(7 41 11 32)(15 50 22 43)(16 51 23 44)(17 52 24 45)(18 53 25 46)(19 54 26 47)(20 55 27 48)(21 56 28 49)(57 95 64 88)(58 96 65 89)(59 97 66 90)(60 98 67 91)(61 85 68 92)(62 86 69 93)(63 87 70 94)(71 108 78 101)(72 109 79 102)(73 110 80 103)(74 111 81 104)(75 112 82 105)(76 99 83 106)(77 100 84 107)
(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 22)(16 28)(17 27)(18 26)(19 25)(20 24)(21 23)(29 50)(30 56)(31 55)(32 54)(33 53)(34 52)(35 51)(36 45)(37 44)(38 43)(39 49)(40 48)(41 47)(42 46)(57 59)(60 70)(61 69)(62 68)(63 67)(64 66)(71 79)(72 78)(73 77)(74 76)(80 84)(81 83)(85 109)(86 108)(87 107)(88 106)(89 105)(90 104)(91 103)(92 102)(93 101)(94 100)(95 99)(96 112)(97 111)(98 110)
G:=sub<Sym(112)| (1,72,19,69)(2,80,20,63)(3,74,21,57)(4,82,15,65)(5,76,16,59)(6,84,17,67)(7,78,18,61)(8,75,22,58)(9,83,23,66)(10,77,24,60)(11,71,25,68)(12,79,26,62)(13,73,27,70)(14,81,28,64)(29,112,43,96)(30,106,44,90)(31,100,45,98)(32,108,46,92)(33,102,47,86)(34,110,48,94)(35,104,49,88)(36,103,55,87)(37,111,56,95)(38,105,50,89)(39,99,51,97)(40,107,52,91)(41,101,53,85)(42,109,54,93), (1,42,12,33)(2,36,13,34)(3,37,14,35)(4,38,8,29)(5,39,9,30)(6,40,10,31)(7,41,11,32)(15,50,22,43)(16,51,23,44)(17,52,24,45)(18,53,25,46)(19,54,26,47)(20,55,27,48)(21,56,28,49)(57,95,64,88)(58,96,65,89)(59,97,66,90)(60,98,67,91)(61,85,68,92)(62,86,69,93)(63,87,70,94)(71,108,78,101)(72,109,79,102)(73,110,80,103)(74,111,81,104)(75,112,82,105)(76,99,83,106)(77,100,84,107), (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)(15,22)(16,28)(17,27)(18,26)(19,25)(20,24)(21,23)(29,50)(30,56)(31,55)(32,54)(33,53)(34,52)(35,51)(36,45)(37,44)(38,43)(39,49)(40,48)(41,47)(42,46)(57,59)(60,70)(61,69)(62,68)(63,67)(64,66)(71,79)(72,78)(73,77)(74,76)(80,84)(81,83)(85,109)(86,108)(87,107)(88,106)(89,105)(90,104)(91,103)(92,102)(93,101)(94,100)(95,99)(96,112)(97,111)(98,110)>;
G:=Group( (1,72,19,69)(2,80,20,63)(3,74,21,57)(4,82,15,65)(5,76,16,59)(6,84,17,67)(7,78,18,61)(8,75,22,58)(9,83,23,66)(10,77,24,60)(11,71,25,68)(12,79,26,62)(13,73,27,70)(14,81,28,64)(29,112,43,96)(30,106,44,90)(31,100,45,98)(32,108,46,92)(33,102,47,86)(34,110,48,94)(35,104,49,88)(36,103,55,87)(37,111,56,95)(38,105,50,89)(39,99,51,97)(40,107,52,91)(41,101,53,85)(42,109,54,93), (1,42,12,33)(2,36,13,34)(3,37,14,35)(4,38,8,29)(5,39,9,30)(6,40,10,31)(7,41,11,32)(15,50,22,43)(16,51,23,44)(17,52,24,45)(18,53,25,46)(19,54,26,47)(20,55,27,48)(21,56,28,49)(57,95,64,88)(58,96,65,89)(59,97,66,90)(60,98,67,91)(61,85,68,92)(62,86,69,93)(63,87,70,94)(71,108,78,101)(72,109,79,102)(73,110,80,103)(74,111,81,104)(75,112,82,105)(76,99,83,106)(77,100,84,107), (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)(15,22)(16,28)(17,27)(18,26)(19,25)(20,24)(21,23)(29,50)(30,56)(31,55)(32,54)(33,53)(34,52)(35,51)(36,45)(37,44)(38,43)(39,49)(40,48)(41,47)(42,46)(57,59)(60,70)(61,69)(62,68)(63,67)(64,66)(71,79)(72,78)(73,77)(74,76)(80,84)(81,83)(85,109)(86,108)(87,107)(88,106)(89,105)(90,104)(91,103)(92,102)(93,101)(94,100)(95,99)(96,112)(97,111)(98,110) );
G=PermutationGroup([[(1,72,19,69),(2,80,20,63),(3,74,21,57),(4,82,15,65),(5,76,16,59),(6,84,17,67),(7,78,18,61),(8,75,22,58),(9,83,23,66),(10,77,24,60),(11,71,25,68),(12,79,26,62),(13,73,27,70),(14,81,28,64),(29,112,43,96),(30,106,44,90),(31,100,45,98),(32,108,46,92),(33,102,47,86),(34,110,48,94),(35,104,49,88),(36,103,55,87),(37,111,56,95),(38,105,50,89),(39,99,51,97),(40,107,52,91),(41,101,53,85),(42,109,54,93)], [(1,42,12,33),(2,36,13,34),(3,37,14,35),(4,38,8,29),(5,39,9,30),(6,40,10,31),(7,41,11,32),(15,50,22,43),(16,51,23,44),(17,52,24,45),(18,53,25,46),(19,54,26,47),(20,55,27,48),(21,56,28,49),(57,95,64,88),(58,96,65,89),(59,97,66,90),(60,98,67,91),(61,85,68,92),(62,86,69,93),(63,87,70,94),(71,108,78,101),(72,109,79,102),(73,110,80,103),(74,111,81,104),(75,112,82,105),(76,99,83,106),(77,100,84,107)], [(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),(15,22),(16,28),(17,27),(18,26),(19,25),(20,24),(21,23),(29,50),(30,56),(31,55),(32,54),(33,53),(34,52),(35,51),(36,45),(37,44),(38,43),(39,49),(40,48),(41,47),(42,46),(57,59),(60,70),(61,69),(62,68),(63,67),(64,66),(71,79),(72,78),(73,77),(74,76),(80,84),(81,83),(85,109),(86,108),(87,107),(88,106),(89,105),(90,104),(91,103),(92,102),(93,101),(94,100),(95,99),(96,112),(97,111),(98,110)]])
85 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28L | 28M | ··· | 28AP |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 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 | 14 | 14 | 28 | 28 | 2 | ··· | 2 | 4 | 4 | 4 | 14 | 14 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
85 irreducible representations
| dim | 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 | D7 | C4oD4 | C4oD4 | D14 | D14 | D14 | D14 | C4oD28 | 2+ 1+4 | D7xC4oD4 | D4:8D14 |
| kernel | C42:10D14 | C42:D7 | C4xD28 | C4.D28 | C42:2D7 | C22:Dic14 | D7xC22:C4 | Dic7:4D4 | C22:D28 | D14.5D4 | D14:Q8 | C4:C4:D7 | C2xD14:C4 | C23.23D14 | C7xC42:C2 | C42:C2 | D14 | C2xC14 | C42 | C22:C4 | C4:C4 | C22xC4 | C22 | C14 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 3 | 4 | 4 | 6 | 6 | 6 | 3 | 24 | 1 | 6 | 6 |
Matrix representation of C42:10D14 ►in GL4(F29) generated by
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 27 |
| 0 | 0 | 0 | 28 |
| 18 | 2 | 0 | 0 |
| 27 | 11 | 0 | 0 |
| 0 | 0 | 17 | 0 |
| 0 | 0 | 0 | 17 |
| 11 | 25 | 0 | 0 |
| 4 | 25 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 28 |
| 25 | 11 | 0 | 0 |
| 25 | 4 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 28 | 1 |
G:=sub<GL(4,GF(29))| [12,0,0,0,0,12,0,0,0,0,1,0,0,0,27,28],[18,27,0,0,2,11,0,0,0,0,17,0,0,0,0,17],[11,4,0,0,25,25,0,0,0,0,1,1,0,0,0,28],[25,25,0,0,11,4,0,0,0,0,28,28,0,0,0,1] >;
C42:10D14 in GAP, Magma, Sage, TeX
C_4^2\rtimes_{10}D_{14} % in TeX
G:=Group("C4^2:10D14"); // GroupNames label
G:=SmallGroup(448,980);
// by ID
G=gap.SmallGroup(448,980);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,387,100,675,136,18822]);
// 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*b^2,b*c=c*b,d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations