metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42:9D14, C14.952+ 1+4, (C2xC4):5D28, C4:C4:43D14, (C2xC28):11D4, C28:4D4:3C2, (C4xC28):1C22, C4.71(C2xD28), C22:D28:4C2, C4:D28:11C2, D14:C4:3C22, C4.D28:3C2, C28.287(C2xD4), (C2xD28):5C22, C42:C2:9D7, (C22xD28):14C2, (C2xC14).69C24, C22:C4.93D14, C2.15(C22xD28), C14.13(C22xD4), C22.20(C2xD28), C2.7(D4:8D14), (C2xC28).144C23, C7:1(C22.29C24), (C22xC4).190D14, C22.98(C23xD7), (C2xDic14):51C22, (C2xDic7).23C23, (C22xD7).19C23, (C23xD7).36C22, C23.157(C22xD7), (C22xC28).229C22, (C22xC14).139C23, (C2xC4xD7):1C22, (C2xC4oD28):18C2, (C7xC4:C4):53C22, (C2xC14).50(C2xD4), (C7xC42:C2):11C2, (C2xC4).149(C22xD7), (C2xC7:D4).100C22, (C7xC22:C4).101C22, SmallGroup(448,978)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42:9D14
G = < a,b,c,d | a4=b4=c14=d2=1, ab=ba, cac-1=ab2, dad=a-1, bc=cb, dbd=b-1, dcd=c-1 >
Subgroups: 2164 in 334 conjugacy classes, 111 normal (21 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C7, C2xC4, C2xC4, C2xC4, D4, Q8, C23, C23, D7, C14, C14, C14, C42, C22:C4, C22:C4, C4:C4, C22xC4, C22xC4, C2xD4, C2xQ8, C4oD4, C24, Dic7, C28, C28, D14, C2xC14, C2xC14, C2xC14, C42:C2, C22wrC2, C4:D4, C4.4D4, C4:1D4, C22xD4, C2xC4oD4, Dic14, C4xD7, D28, C2xDic7, C7:D4, C2xC28, C2xC28, C22xD7, C22xD7, C22xC14, C22.29C24, D14:C4, C4xC28, C7xC22:C4, C7xC4:C4, C2xDic14, C2xC4xD7, C2xD28, C2xD28, C2xD28, C4oD28, C2xC7:D4, C22xC28, C23xD7, C28:4D4, C4.D28, C22:D28, C4:D28, C7xC42:C2, C22xD28, C2xC4oD28, C42:9D14
Quotients: C1, C2, C22, D4, C23, D7, C2xD4, C24, D14, C22xD4, 2+ 1+4, D28, C22xD7, C22.29C24, C2xD28, C23xD7, C22xD28, D4:8D14, C42:9D14
►On 112 points
Generators in S112
(1 87 26 102)(2 95 27 110)(3 89 28 104)(4 97 22 112)(5 91 23 106)(6 85 24 100)(7 93 25 108)(8 90 15 105)(9 98 16 99)(10 92 17 107)(11 86 18 101)(12 94 19 109)(13 88 20 103)(14 96 21 111)(29 61 54 73)(30 69 55 81)(31 63 56 75)(32 57 50 83)(33 65 51 77)(34 59 52 71)(35 67 53 79)(36 68 48 80)(37 62 49 74)(38 70 43 82)(39 64 44 76)(40 58 45 84)(41 66 46 78)(42 60 47 72)
(1 56 12 43)(2 50 13 44)(3 51 14 45)(4 52 8 46)(5 53 9 47)(6 54 10 48)(7 55 11 49)(15 41 22 34)(16 42 23 35)(17 36 24 29)(18 37 25 30)(19 38 26 31)(20 39 27 32)(21 40 28 33)(57 103 64 110)(58 104 65 111)(59 105 66 112)(60 106 67 99)(61 107 68 100)(62 108 69 101)(63 109 70 102)(71 90 78 97)(72 91 79 98)(73 92 80 85)(74 93 81 86)(75 94 82 87)(76 95 83 88)(77 96 84 89)
(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 7)(2 6)(3 5)(9 14)(10 13)(11 12)(16 21)(17 20)(18 19)(23 28)(24 27)(25 26)(29 39)(30 38)(31 37)(32 36)(33 42)(34 41)(35 40)(43 55)(44 54)(45 53)(46 52)(47 51)(48 50)(49 56)(57 80)(58 79)(59 78)(60 77)(61 76)(62 75)(63 74)(64 73)(65 72)(66 71)(67 84)(68 83)(69 82)(70 81)(85 110)(86 109)(87 108)(88 107)(89 106)(90 105)(91 104)(92 103)(93 102)(94 101)(95 100)(96 99)(97 112)(98 111)
G:=sub<Sym(112)| (1,87,26,102)(2,95,27,110)(3,89,28,104)(4,97,22,112)(5,91,23,106)(6,85,24,100)(7,93,25,108)(8,90,15,105)(9,98,16,99)(10,92,17,107)(11,86,18,101)(12,94,19,109)(13,88,20,103)(14,96,21,111)(29,61,54,73)(30,69,55,81)(31,63,56,75)(32,57,50,83)(33,65,51,77)(34,59,52,71)(35,67,53,79)(36,68,48,80)(37,62,49,74)(38,70,43,82)(39,64,44,76)(40,58,45,84)(41,66,46,78)(42,60,47,72), (1,56,12,43)(2,50,13,44)(3,51,14,45)(4,52,8,46)(5,53,9,47)(6,54,10,48)(7,55,11,49)(15,41,22,34)(16,42,23,35)(17,36,24,29)(18,37,25,30)(19,38,26,31)(20,39,27,32)(21,40,28,33)(57,103,64,110)(58,104,65,111)(59,105,66,112)(60,106,67,99)(61,107,68,100)(62,108,69,101)(63,109,70,102)(71,90,78,97)(72,91,79,98)(73,92,80,85)(74,93,81,86)(75,94,82,87)(76,95,83,88)(77,96,84,89), (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), (1,7)(2,6)(3,5)(9,14)(10,13)(11,12)(16,21)(17,20)(18,19)(23,28)(24,27)(25,26)(29,39)(30,38)(31,37)(32,36)(33,42)(34,41)(35,40)(43,55)(44,54)(45,53)(46,52)(47,51)(48,50)(49,56)(57,80)(58,79)(59,78)(60,77)(61,76)(62,75)(63,74)(64,73)(65,72)(66,71)(67,84)(68,83)(69,82)(70,81)(85,110)(86,109)(87,108)(88,107)(89,106)(90,105)(91,104)(92,103)(93,102)(94,101)(95,100)(96,99)(97,112)(98,111)>;
G:=Group( (1,87,26,102)(2,95,27,110)(3,89,28,104)(4,97,22,112)(5,91,23,106)(6,85,24,100)(7,93,25,108)(8,90,15,105)(9,98,16,99)(10,92,17,107)(11,86,18,101)(12,94,19,109)(13,88,20,103)(14,96,21,111)(29,61,54,73)(30,69,55,81)(31,63,56,75)(32,57,50,83)(33,65,51,77)(34,59,52,71)(35,67,53,79)(36,68,48,80)(37,62,49,74)(38,70,43,82)(39,64,44,76)(40,58,45,84)(41,66,46,78)(42,60,47,72), (1,56,12,43)(2,50,13,44)(3,51,14,45)(4,52,8,46)(5,53,9,47)(6,54,10,48)(7,55,11,49)(15,41,22,34)(16,42,23,35)(17,36,24,29)(18,37,25,30)(19,38,26,31)(20,39,27,32)(21,40,28,33)(57,103,64,110)(58,104,65,111)(59,105,66,112)(60,106,67,99)(61,107,68,100)(62,108,69,101)(63,109,70,102)(71,90,78,97)(72,91,79,98)(73,92,80,85)(74,93,81,86)(75,94,82,87)(76,95,83,88)(77,96,84,89), (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), (1,7)(2,6)(3,5)(9,14)(10,13)(11,12)(16,21)(17,20)(18,19)(23,28)(24,27)(25,26)(29,39)(30,38)(31,37)(32,36)(33,42)(34,41)(35,40)(43,55)(44,54)(45,53)(46,52)(47,51)(48,50)(49,56)(57,80)(58,79)(59,78)(60,77)(61,76)(62,75)(63,74)(64,73)(65,72)(66,71)(67,84)(68,83)(69,82)(70,81)(85,110)(86,109)(87,108)(88,107)(89,106)(90,105)(91,104)(92,103)(93,102)(94,101)(95,100)(96,99)(97,112)(98,111) );
G=PermutationGroup([[(1,87,26,102),(2,95,27,110),(3,89,28,104),(4,97,22,112),(5,91,23,106),(6,85,24,100),(7,93,25,108),(8,90,15,105),(9,98,16,99),(10,92,17,107),(11,86,18,101),(12,94,19,109),(13,88,20,103),(14,96,21,111),(29,61,54,73),(30,69,55,81),(31,63,56,75),(32,57,50,83),(33,65,51,77),(34,59,52,71),(35,67,53,79),(36,68,48,80),(37,62,49,74),(38,70,43,82),(39,64,44,76),(40,58,45,84),(41,66,46,78),(42,60,47,72)], [(1,56,12,43),(2,50,13,44),(3,51,14,45),(4,52,8,46),(5,53,9,47),(6,54,10,48),(7,55,11,49),(15,41,22,34),(16,42,23,35),(17,36,24,29),(18,37,25,30),(19,38,26,31),(20,39,27,32),(21,40,28,33),(57,103,64,110),(58,104,65,111),(59,105,66,112),(60,106,67,99),(61,107,68,100),(62,108,69,101),(63,109,70,102),(71,90,78,97),(72,91,79,98),(73,92,80,85),(74,93,81,86),(75,94,82,87),(76,95,83,88),(77,96,84,89)], [(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)], [(1,7),(2,6),(3,5),(9,14),(10,13),(11,12),(16,21),(17,20),(18,19),(23,28),(24,27),(25,26),(29,39),(30,38),(31,37),(32,36),(33,42),(34,41),(35,40),(43,55),(44,54),(45,53),(46,52),(47,51),(48,50),(49,56),(57,80),(58,79),(59,78),(60,77),(61,76),(62,75),(63,74),(64,73),(65,72),(66,71),(67,84),(68,83),(69,82),(70,81),(85,110),(86,109),(87,108),(88,107),(89,106),(90,105),(91,104),(92,103),(93,102),(94,101),(95,100),(96,99),(97,112),(98,111)]])
82 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | ··· | 2K | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28L | 28M | ··· | 28AP |
| order | 1 | 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 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 28 | ··· | 28 | 2 | 2 | 2 | 2 | 4 | 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 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D7 | D14 | D14 | D14 | D14 | D28 | 2+ 1+4 | D4:8D14 |
| kernel | C42:9D14 | C28:4D4 | C4.D28 | C22:D28 | C4:D28 | C7xC42:C2 | C22xD28 | C2xC4oD28 | C2xC28 | C42:C2 | C42 | C22:C4 | C4:C4 | C22xC4 | C2xC4 | C14 | C2 |
| # reps | 1 | 2 | 2 | 4 | 4 | 1 | 1 | 1 | 4 | 3 | 6 | 6 | 6 | 3 | 24 | 2 | 12 |
Matrix representation of C42:9D14 ►in GL6(F29)
| 28 | 28 | 0 | 0 | 0 | 0 |
| 2 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 21 | 23 | 2 | 0 |
| 0 | 0 | 6 | 8 | 0 | 2 |
| 0 | 0 | 1 | 0 | 8 | 6 |
| 0 | 0 | 0 | 1 | 23 | 21 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 21 | 23 | 0 | 0 |
| 0 | 0 | 6 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 21 | 23 |
| 0 | 0 | 0 | 0 | 6 | 8 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 10 | 0 | 0 |
| 0 | 0 | 19 | 22 | 0 | 0 |
| 0 | 0 | 20 | 9 | 19 | 19 |
| 0 | 0 | 20 | 25 | 10 | 7 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 27 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 10 | 0 | 0 |
| 0 | 0 | 22 | 19 | 0 | 0 |
| 0 | 0 | 9 | 20 | 10 | 10 |
| 0 | 0 | 25 | 20 | 22 | 19 |
G:=sub<GL(6,GF(29))| [28,2,0,0,0,0,28,1,0,0,0,0,0,0,21,6,1,0,0,0,23,8,0,1,0,0,2,0,8,23,0,0,0,2,6,21],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,21,6,0,0,0,0,23,8,0,0,0,0,0,0,21,6,0,0,0,0,23,8],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,10,19,20,20,0,0,10,22,9,25,0,0,0,0,19,10,0,0,0,0,19,7],[1,27,0,0,0,0,0,28,0,0,0,0,0,0,10,22,9,25,0,0,10,19,20,20,0,0,0,0,10,22,0,0,0,0,10,19] >;
C42:9D14 in GAP, Magma, Sage, TeX
C_4^2\rtimes_9D_{14} % in TeX
G:=Group("C4^2:9D14"); // GroupNames label
G:=SmallGroup(448,978);
// by ID
G=gap.SmallGroup(448,978);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,758,184,675,570,80,18822]);
// 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*b^2,d*a*d=a^-1,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations