?
G = C14.482+ (1+4) order 448 = 26·7
48th non-split extension by C14 of 2+ (1+4) acting via 2+ (1+4)/C2×D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C14.482+ (1+4), C4⋊C4⋊9D14, (C2×D4)⋊11D14, C4⋊D4⋊23D7, C28⋊7D4⋊45C2, C28⋊2D4⋊28C2, C28⋊D4⋊20C2, C22⋊C4⋊13D14, (C22×C4)⋊21D14, D14⋊D4⋊23C2, C22⋊D28⋊14C2, C23⋊D14⋊14C2, D14⋊C4⋊32C22, (D4×C14)⋊16C22, (C2×C28).46C23, C4⋊Dic7⋊13C22, Dic7⋊D4⋊18C2, D14.5D4⋊14C2, (C2×C14).164C24, Dic7⋊C4⋊35C22, (C22×C28)⋊31C22, (C4×Dic7)⋊26C22, C23.D7⋊27C22, C2.32(D4⋊8D14), C2.50(D4⋊6D14), C7⋊2(C22.54C24), (C2×D28).146C22, C22.D28⋊14C2, C23.D14⋊21C2, (C2×Dic7).81C23, (C22×D7).71C23, (C23×D7).51C22, C23.114(C22×D7), C22.185(C23×D7), C23.23D14⋊12C2, (C22×C14).192C23, (C22×Dic7)⋊23C22, (C2×C4×D7)⋊16C22, C4⋊C4⋊D7⋊15C2, (C7×C4⋊D4)⋊26C2, (C7×C4⋊C4)⋊16C22, (C2×C7⋊D4)⋊17C22, (C2×C4).42(C22×D7), (C7×C22⋊C4)⋊18C22, SmallGroup(448,1073)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1452 in 252 conjugacy classes, 91 normal (all characteristic)
C1, C2 [×3], C2 [×6], C4 [×9], C22, C22 [×22], C7, C2×C4 [×4], C2×C4 [×8], D4 [×12], C23 [×3], C23 [×6], D7 [×3], C14 [×3], C14 [×3], C42, C22⋊C4 [×2], C22⋊C4 [×10], C4⋊C4, C4⋊C4 [×5], C22×C4, C22×C4 [×2], C2×D4 [×3], C2×D4 [×9], C24, Dic7 [×5], C28 [×4], D14 [×13], C2×C14, C2×C14 [×9], C22≀C2 [×3], C4⋊D4, C4⋊D4 [×5], C22.D4 [×3], C42⋊2C2 [×2], C4⋊1D4, C4×D7, D28 [×2], C2×Dic7 [×5], C2×Dic7, C7⋊D4 [×7], C2×C28 [×4], C2×C28, C7×D4 [×3], C22×D7 [×3], C22×D7 [×3], C22×C14 [×3], C22.54C24, C4×Dic7, Dic7⋊C4 [×3], C4⋊Dic7 [×2], D14⋊C4 [×7], C23.D7 [×3], C7×C22⋊C4 [×2], C7×C4⋊C4, C2×C4×D7, C2×D28 [×2], C22×Dic7, C2×C7⋊D4 [×7], C22×C28, D4×C14 [×3], C23×D7, C23.D14, C22⋊D28, D14⋊D4, C22.D28, D14.5D4, C4⋊C4⋊D7, C23.23D14, C28⋊7D4, C23⋊D14 [×2], C28⋊2D4, Dic7⋊D4 [×2], C28⋊D4, C7×C4⋊D4, C14.482+ (1+4)
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D7, C24, D14 [×7], 2+ (1+4) [×3], C22×D7 [×7], C22.54C24, C23×D7, D4⋊6D14 [×2], D4⋊8D14, C14.482+ (1+4)
Generators and relations
G = < a,b,c,d,e | a14=b4=c2=e2=1, d2=a7b2, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=b-1, dbd-1=a7b, be=eb, dcd-1=ece=a7c, ede=a7b2d >
►On 112 points
Generators in S112
(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 62 16 81)(2 63 17 82)(3 64 18 83)(4 65 19 84)(5 66 20 71)(6 67 21 72)(7 68 22 73)(8 69 23 74)(9 70 24 75)(10 57 25 76)(11 58 26 77)(12 59 27 78)(13 60 28 79)(14 61 15 80)(29 94 54 108)(30 95 55 109)(31 96 56 110)(32 97 43 111)(33 98 44 112)(34 85 45 99)(35 86 46 100)(36 87 47 101)(37 88 48 102)(38 89 49 103)(39 90 50 104)(40 91 51 105)(41 92 52 106)(42 93 53 107)
(29 36)(30 37)(31 38)(32 39)(33 40)(34 41)(35 42)(43 50)(44 51)(45 52)(46 53)(47 54)(48 55)(49 56)(57 76)(58 77)(59 78)(60 79)(61 80)(62 81)(63 82)(64 83)(65 84)(66 71)(67 72)(68 73)(69 74)(70 75)(85 106)(86 107)(87 108)(88 109)(89 110)(90 111)(91 112)(92 99)(93 100)(94 101)(95 102)(96 103)(97 104)(98 105)
(1 35 23 53)(2 34 24 52)(3 33 25 51)(4 32 26 50)(5 31 27 49)(6 30 28 48)(7 29 15 47)(8 42 16 46)(9 41 17 45)(10 40 18 44)(11 39 19 43)(12 38 20 56)(13 37 21 55)(14 36 22 54)(57 98 83 105)(58 97 84 104)(59 96 71 103)(60 95 72 102)(61 94 73 101)(62 93 74 100)(63 92 75 99)(64 91 76 112)(65 90 77 111)(66 89 78 110)(67 88 79 109)(68 87 80 108)(69 86 81 107)(70 85 82 106)
(1 35)(2 36)(3 37)(4 38)(5 39)(6 40)(7 41)(8 42)(9 29)(10 30)(11 31)(12 32)(13 33)(14 34)(15 45)(16 46)(17 47)(18 48)(19 49)(20 50)(21 51)(22 52)(23 53)(24 54)(25 55)(26 56)(27 43)(28 44)(57 95)(58 96)(59 97)(60 98)(61 85)(62 86)(63 87)(64 88)(65 89)(66 90)(67 91)(68 92)(69 93)(70 94)(71 104)(72 105)(73 106)(74 107)(75 108)(76 109)(77 110)(78 111)(79 112)(80 99)(81 100)(82 101)(83 102)(84 103)
G:=sub<Sym(112)| (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), (1,62,16,81)(2,63,17,82)(3,64,18,83)(4,65,19,84)(5,66,20,71)(6,67,21,72)(7,68,22,73)(8,69,23,74)(9,70,24,75)(10,57,25,76)(11,58,26,77)(12,59,27,78)(13,60,28,79)(14,61,15,80)(29,94,54,108)(30,95,55,109)(31,96,56,110)(32,97,43,111)(33,98,44,112)(34,85,45,99)(35,86,46,100)(36,87,47,101)(37,88,48,102)(38,89,49,103)(39,90,50,104)(40,91,51,105)(41,92,52,106)(42,93,53,107), (29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(43,50)(44,51)(45,52)(46,53)(47,54)(48,55)(49,56)(57,76)(58,77)(59,78)(60,79)(61,80)(62,81)(63,82)(64,83)(65,84)(66,71)(67,72)(68,73)(69,74)(70,75)(85,106)(86,107)(87,108)(88,109)(89,110)(90,111)(91,112)(92,99)(93,100)(94,101)(95,102)(96,103)(97,104)(98,105), (1,35,23,53)(2,34,24,52)(3,33,25,51)(4,32,26,50)(5,31,27,49)(6,30,28,48)(7,29,15,47)(8,42,16,46)(9,41,17,45)(10,40,18,44)(11,39,19,43)(12,38,20,56)(13,37,21,55)(14,36,22,54)(57,98,83,105)(58,97,84,104)(59,96,71,103)(60,95,72,102)(61,94,73,101)(62,93,74,100)(63,92,75,99)(64,91,76,112)(65,90,77,111)(66,89,78,110)(67,88,79,109)(68,87,80,108)(69,86,81,107)(70,85,82,106), (1,35)(2,36)(3,37)(4,38)(5,39)(6,40)(7,41)(8,42)(9,29)(10,30)(11,31)(12,32)(13,33)(14,34)(15,45)(16,46)(17,47)(18,48)(19,49)(20,50)(21,51)(22,52)(23,53)(24,54)(25,55)(26,56)(27,43)(28,44)(57,95)(58,96)(59,97)(60,98)(61,85)(62,86)(63,87)(64,88)(65,89)(66,90)(67,91)(68,92)(69,93)(70,94)(71,104)(72,105)(73,106)(74,107)(75,108)(76,109)(77,110)(78,111)(79,112)(80,99)(81,100)(82,101)(83,102)(84,103)>;
G:=Group( (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), (1,62,16,81)(2,63,17,82)(3,64,18,83)(4,65,19,84)(5,66,20,71)(6,67,21,72)(7,68,22,73)(8,69,23,74)(9,70,24,75)(10,57,25,76)(11,58,26,77)(12,59,27,78)(13,60,28,79)(14,61,15,80)(29,94,54,108)(30,95,55,109)(31,96,56,110)(32,97,43,111)(33,98,44,112)(34,85,45,99)(35,86,46,100)(36,87,47,101)(37,88,48,102)(38,89,49,103)(39,90,50,104)(40,91,51,105)(41,92,52,106)(42,93,53,107), (29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(43,50)(44,51)(45,52)(46,53)(47,54)(48,55)(49,56)(57,76)(58,77)(59,78)(60,79)(61,80)(62,81)(63,82)(64,83)(65,84)(66,71)(67,72)(68,73)(69,74)(70,75)(85,106)(86,107)(87,108)(88,109)(89,110)(90,111)(91,112)(92,99)(93,100)(94,101)(95,102)(96,103)(97,104)(98,105), (1,35,23,53)(2,34,24,52)(3,33,25,51)(4,32,26,50)(5,31,27,49)(6,30,28,48)(7,29,15,47)(8,42,16,46)(9,41,17,45)(10,40,18,44)(11,39,19,43)(12,38,20,56)(13,37,21,55)(14,36,22,54)(57,98,83,105)(58,97,84,104)(59,96,71,103)(60,95,72,102)(61,94,73,101)(62,93,74,100)(63,92,75,99)(64,91,76,112)(65,90,77,111)(66,89,78,110)(67,88,79,109)(68,87,80,108)(69,86,81,107)(70,85,82,106), (1,35)(2,36)(3,37)(4,38)(5,39)(6,40)(7,41)(8,42)(9,29)(10,30)(11,31)(12,32)(13,33)(14,34)(15,45)(16,46)(17,47)(18,48)(19,49)(20,50)(21,51)(22,52)(23,53)(24,54)(25,55)(26,56)(27,43)(28,44)(57,95)(58,96)(59,97)(60,98)(61,85)(62,86)(63,87)(64,88)(65,89)(66,90)(67,91)(68,92)(69,93)(70,94)(71,104)(72,105)(73,106)(74,107)(75,108)(76,109)(77,110)(78,111)(79,112)(80,99)(81,100)(82,101)(83,102)(84,103) );
G=PermutationGroup([(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)], [(1,62,16,81),(2,63,17,82),(3,64,18,83),(4,65,19,84),(5,66,20,71),(6,67,21,72),(7,68,22,73),(8,69,23,74),(9,70,24,75),(10,57,25,76),(11,58,26,77),(12,59,27,78),(13,60,28,79),(14,61,15,80),(29,94,54,108),(30,95,55,109),(31,96,56,110),(32,97,43,111),(33,98,44,112),(34,85,45,99),(35,86,46,100),(36,87,47,101),(37,88,48,102),(38,89,49,103),(39,90,50,104),(40,91,51,105),(41,92,52,106),(42,93,53,107)], [(29,36),(30,37),(31,38),(32,39),(33,40),(34,41),(35,42),(43,50),(44,51),(45,52),(46,53),(47,54),(48,55),(49,56),(57,76),(58,77),(59,78),(60,79),(61,80),(62,81),(63,82),(64,83),(65,84),(66,71),(67,72),(68,73),(69,74),(70,75),(85,106),(86,107),(87,108),(88,109),(89,110),(90,111),(91,112),(92,99),(93,100),(94,101),(95,102),(96,103),(97,104),(98,105)], [(1,35,23,53),(2,34,24,52),(3,33,25,51),(4,32,26,50),(5,31,27,49),(6,30,28,48),(7,29,15,47),(8,42,16,46),(9,41,17,45),(10,40,18,44),(11,39,19,43),(12,38,20,56),(13,37,21,55),(14,36,22,54),(57,98,83,105),(58,97,84,104),(59,96,71,103),(60,95,72,102),(61,94,73,101),(62,93,74,100),(63,92,75,99),(64,91,76,112),(65,90,77,111),(66,89,78,110),(67,88,79,109),(68,87,80,108),(69,86,81,107),(70,85,82,106)], [(1,35),(2,36),(3,37),(4,38),(5,39),(6,40),(7,41),(8,42),(9,29),(10,30),(11,31),(12,32),(13,33),(14,34),(15,45),(16,46),(17,47),(18,48),(19,49),(20,50),(21,51),(22,52),(23,53),(24,54),(25,55),(26,56),(27,43),(28,44),(57,95),(58,96),(59,97),(60,98),(61,85),(62,86),(63,87),(64,88),(65,89),(66,90),(67,91),(68,92),(69,93),(70,94),(71,104),(72,105),(73,106),(74,107),(75,108),(76,109),(77,110),(78,111),(79,112),(80,99),(81,100),(82,101),(83,102),(84,103)])
Matrix representation ►G ⊆ GL8(𝔽29)
| 27 | 13 | 21 | 6 | 0 | 0 | 0 | 0 |
| 8 | 20 | 0 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 26 | 21 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 21 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 21 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 8 | 8 |
| 0 | 0 | 0 | 0 | 0 | 0 | 21 | 3 |
| 11 | 14 | 23 | 26 | 0 | 0 | 0 | 0 |
| 18 | 16 | 11 | 22 | 0 | 0 | 0 | 0 |
| 0 | 1 | 14 | 18 | 0 | 0 | 0 | 0 |
| 28 | 8 | 11 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 27 | 11 |
| 0 | 0 | 0 | 0 | 0 | 0 | 18 | 2 |
| 0 | 0 | 0 | 0 | 27 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 18 | 2 | 0 | 0 |
| 1 | 0 | 28 | 22 | 0 | 0 | 0 | 0 |
| 0 | 1 | 28 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 28 |
| 8 | 18 | 2 | 4 | 0 | 0 | 0 | 0 |
| 27 | 21 | 24 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 24 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 3 | 28 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 28 | 0 | 0 |
| 7 | 26 | 13 | 12 | 0 | 0 | 0 | 0 |
| 16 | 22 | 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 24 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(8,GF(29))| [27,8,0,0,0,0,0,0,13,20,0,0,0,0,0,0,21,0,26,8,0,0,0,0,6,8,21,21,0,0,0,0,0,0,0,0,8,21,0,0,0,0,0,0,8,3,0,0,0,0,0,0,0,0,8,21,0,0,0,0,0,0,8,3],[11,18,0,28,0,0,0,0,14,16,1,8,0,0,0,0,23,11,14,11,0,0,0,0,26,22,18,17,0,0,0,0,0,0,0,0,0,0,27,18,0,0,0,0,0,0,11,2,0,0,0,0,27,18,0,0,0,0,0,0,11,2,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,28,28,28,0,0,0,0,0,22,7,0,28,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28],[8,27,0,0,0,0,0,0,18,21,0,0,0,0,0,0,2,24,5,16,0,0,0,0,4,12,2,24,0,0,0,0,0,0,0,0,0,0,1,3,0,0,0,0,0,0,0,28,0,0,0,0,1,3,0,0,0,0,0,0,0,28,0,0],[7,16,0,0,0,0,0,0,26,22,0,0,0,0,0,0,13,0,5,16,0,0,0,0,12,16,13,24,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0] >;
61 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | ··· | 4I | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14O | 14P | ··· | 14U | 28A | ··· | 28L | 28M | ··· | 28R |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 28 | 28 | 28 | 4 | 4 | 4 | 4 | 28 | ··· | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 | 8 | ··· | 8 |
61 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D7 | D14 | D14 | D14 | D14 | 2+ (1+4) | D4⋊6D14 | D4⋊8D14 |
| kernel | C14.482+ (1+4) | C23.D14 | C22⋊D28 | D14⋊D4 | C22.D28 | D14.5D4 | C4⋊C4⋊D7 | C23.23D14 | C28⋊7D4 | C23⋊D14 | C28⋊2D4 | Dic7⋊D4 | C28⋊D4 | C7×C4⋊D4 | C4⋊D4 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C14 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 1 | 3 | 6 | 3 | 3 | 9 | 3 | 12 | 6 |
In GAP, Magma, Sage, TeX
C_{14}._{48}2_+^{(1+4)} % in TeX
G:=Group("C14.48ES+(2,2)"); // GroupNames label
G:=SmallGroup(448,1073);
// by ID
G=gap.SmallGroup(448,1073);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,758,219,1571,570,297,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^14=b^4=c^2=e^2=1,d^2=a^7*b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,a*e=e*a,c*b*c=b^-1,d*b*d^-1=a^7*b,b*e=e*b,d*c*d^-1=e*c*e=a^7*c,e*d*e=a^7*b^2*d>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀