?
G = C14.1462+ (1+4) order 448 = 26·7
55th non-split extension by C14 of 2+ (1+4) acting via 2+ (1+4)/C4○D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C14.1462+ (1+4), (C2×C28)⋊16D4, (C2×D4)⋊45D14, (C2×Q8)⋊34D14, C28⋊D4⋊30C2, C28⋊7D4⋊48C2, C28.430(C2×D4), (C22×C4)⋊31D14, C23⋊D14⋊32C2, D14⋊C4⋊38C22, (D4×C14)⋊48C22, (C22×D28)⋊21C2, C4⋊Dic7⋊66C22, (Q8×C14)⋊41C22, C28.23D4⋊32C2, (C2×C14).316C24, (C2×C28).653C23, (C22×C28)⋊32C22, C7⋊7(C22.29C24), (C4×Dic7)⋊45C22, C14.166(C22×D4), C2.70(D4⋊8D14), (C2×D28).281C22, (C23×D7).80C22, C23.212(C22×D7), C22.325(C23×D7), C23.21D14⋊39C2, (C22×C14).242C23, (C2×Dic7).163C23, (C22×D7).138C23, C23.D7.136C22, (C2×C4○D4)⋊8D7, (C14×C4○D4)⋊8C2, (C2×C4)⋊7(C7⋊D4), C4.33(C2×C7⋊D4), (C2×C14).82(C2×D4), (C2×C7⋊D4)⋊31C22, C2.39(C22×C7⋊D4), C22.24(C2×C7⋊D4), (C2×C4).251(C22×D7), SmallGroup(448,1283)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1876 in 334 conjugacy classes, 111 normal (23 characteristic)
C1, C2, C2 [×2], C2 [×8], C4 [×4], C4 [×6], C22, C22 [×2], C22 [×28], C7, C2×C4 [×2], C2×C4 [×6], C2×C4 [×8], D4 [×22], Q8 [×2], C23, C23 [×2], C23 [×12], D7 [×4], C14, C14 [×2], C14 [×4], C42 [×2], C22⋊C4 [×10], C4⋊C4 [×2], C22×C4, C22×C4 [×2], C2×D4, C2×D4 [×2], C2×D4 [×16], C2×Q8, C4○D4 [×4], C24 [×2], Dic7 [×4], C28 [×4], C28 [×2], D14 [×20], C2×C14, C2×C14 [×2], C2×C14 [×8], C42⋊C2, C22≀C2 [×4], C4⋊D4 [×4], C4.4D4 [×2], C4⋊1D4 [×2], C22×D4, C2×C4○D4, D28 [×8], C2×Dic7 [×4], C7⋊D4 [×8], C2×C28 [×2], C2×C28 [×6], C2×C28 [×4], C7×D4 [×6], C7×Q8 [×2], C22×D7 [×4], C22×D7 [×8], C22×C14, C22×C14 [×2], C22.29C24, C4×Dic7 [×2], C4⋊Dic7 [×2], D14⋊C4 [×8], C23.D7 [×2], C2×D28 [×4], C2×D28 [×4], C2×C7⋊D4 [×8], C22×C28, C22×C28 [×2], D4×C14, D4×C14 [×2], Q8×C14, C7×C4○D4 [×4], C23×D7 [×2], C23.21D14, C28⋊7D4 [×4], C23⋊D14 [×4], C28⋊D4 [×2], C28.23D4 [×2], C22×D28, C14×C4○D4, C14.1462+ (1+4)
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D7, C2×D4 [×6], C24, D14 [×7], C22×D4, 2+ (1+4) [×2], C7⋊D4 [×4], C22×D7 [×7], C22.29C24, C2×C7⋊D4 [×6], C23×D7, D4⋊8D14 [×2], C22×C7⋊D4, C14.1462+ (1+4)
Generators and relations
G = < a,b,c,d,e | a14=b4=e2=1, c2=a7, d2=a7b2, ab=ba, cac-1=dad-1=a-1, ae=ea, cbc-1=a7b-1, dbd-1=a7b, be=eb, cd=dc, 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 70 25 105)(2 57 26 106)(3 58 27 107)(4 59 28 108)(5 60 15 109)(6 61 16 110)(7 62 17 111)(8 63 18 112)(9 64 19 99)(10 65 20 100)(11 66 21 101)(12 67 22 102)(13 68 23 103)(14 69 24 104)(29 45 90 83)(30 46 91 84)(31 47 92 71)(32 48 93 72)(33 49 94 73)(34 50 95 74)(35 51 96 75)(36 52 97 76)(37 53 98 77)(38 54 85 78)(39 55 86 79)(40 56 87 80)(41 43 88 81)(42 44 89 82)
(1 105 8 112)(2 104 9 111)(3 103 10 110)(4 102 11 109)(5 101 12 108)(6 100 13 107)(7 99 14 106)(15 66 22 59)(16 65 23 58)(17 64 24 57)(18 63 25 70)(19 62 26 69)(20 61 27 68)(21 60 28 67)(29 49 36 56)(30 48 37 55)(31 47 38 54)(32 46 39 53)(33 45 40 52)(34 44 41 51)(35 43 42 50)(71 85 78 92)(72 98 79 91)(73 97 80 90)(74 96 81 89)(75 95 82 88)(76 94 83 87)(77 93 84 86)
(1 71 18 54)(2 84 19 53)(3 83 20 52)(4 82 21 51)(5 81 22 50)(6 80 23 49)(7 79 24 48)(8 78 25 47)(9 77 26 46)(10 76 27 45)(11 75 28 44)(12 74 15 43)(13 73 16 56)(14 72 17 55)(29 107 97 65)(30 106 98 64)(31 105 85 63)(32 104 86 62)(33 103 87 61)(34 102 88 60)(35 101 89 59)(36 100 90 58)(37 99 91 57)(38 112 92 70)(39 111 93 69)(40 110 94 68)(41 109 95 67)(42 108 96 66)
(1 54)(2 55)(3 56)(4 43)(5 44)(6 45)(7 46)(8 47)(9 48)(10 49)(11 50)(12 51)(13 52)(14 53)(15 82)(16 83)(17 84)(18 71)(19 72)(20 73)(21 74)(22 75)(23 76)(24 77)(25 78)(26 79)(27 80)(28 81)(29 110)(30 111)(31 112)(32 99)(33 100)(34 101)(35 102)(36 103)(37 104)(38 105)(39 106)(40 107)(41 108)(42 109)(57 86)(58 87)(59 88)(60 89)(61 90)(62 91)(63 92)(64 93)(65 94)(66 95)(67 96)(68 97)(69 98)(70 85)
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), (1,70,25,105)(2,57,26,106)(3,58,27,107)(4,59,28,108)(5,60,15,109)(6,61,16,110)(7,62,17,111)(8,63,18,112)(9,64,19,99)(10,65,20,100)(11,66,21,101)(12,67,22,102)(13,68,23,103)(14,69,24,104)(29,45,90,83)(30,46,91,84)(31,47,92,71)(32,48,93,72)(33,49,94,73)(34,50,95,74)(35,51,96,75)(36,52,97,76)(37,53,98,77)(38,54,85,78)(39,55,86,79)(40,56,87,80)(41,43,88,81)(42,44,89,82), (1,105,8,112)(2,104,9,111)(3,103,10,110)(4,102,11,109)(5,101,12,108)(6,100,13,107)(7,99,14,106)(15,66,22,59)(16,65,23,58)(17,64,24,57)(18,63,25,70)(19,62,26,69)(20,61,27,68)(21,60,28,67)(29,49,36,56)(30,48,37,55)(31,47,38,54)(32,46,39,53)(33,45,40,52)(34,44,41,51)(35,43,42,50)(71,85,78,92)(72,98,79,91)(73,97,80,90)(74,96,81,89)(75,95,82,88)(76,94,83,87)(77,93,84,86), (1,71,18,54)(2,84,19,53)(3,83,20,52)(4,82,21,51)(5,81,22,50)(6,80,23,49)(7,79,24,48)(8,78,25,47)(9,77,26,46)(10,76,27,45)(11,75,28,44)(12,74,15,43)(13,73,16,56)(14,72,17,55)(29,107,97,65)(30,106,98,64)(31,105,85,63)(32,104,86,62)(33,103,87,61)(34,102,88,60)(35,101,89,59)(36,100,90,58)(37,99,91,57)(38,112,92,70)(39,111,93,69)(40,110,94,68)(41,109,95,67)(42,108,96,66), (1,54)(2,55)(3,56)(4,43)(5,44)(6,45)(7,46)(8,47)(9,48)(10,49)(11,50)(12,51)(13,52)(14,53)(15,82)(16,83)(17,84)(18,71)(19,72)(20,73)(21,74)(22,75)(23,76)(24,77)(25,78)(26,79)(27,80)(28,81)(29,110)(30,111)(31,112)(32,99)(33,100)(34,101)(35,102)(36,103)(37,104)(38,105)(39,106)(40,107)(41,108)(42,109)(57,86)(58,87)(59,88)(60,89)(61,90)(62,91)(63,92)(64,93)(65,94)(66,95)(67,96)(68,97)(69,98)(70,85)>;
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), (1,70,25,105)(2,57,26,106)(3,58,27,107)(4,59,28,108)(5,60,15,109)(6,61,16,110)(7,62,17,111)(8,63,18,112)(9,64,19,99)(10,65,20,100)(11,66,21,101)(12,67,22,102)(13,68,23,103)(14,69,24,104)(29,45,90,83)(30,46,91,84)(31,47,92,71)(32,48,93,72)(33,49,94,73)(34,50,95,74)(35,51,96,75)(36,52,97,76)(37,53,98,77)(38,54,85,78)(39,55,86,79)(40,56,87,80)(41,43,88,81)(42,44,89,82), (1,105,8,112)(2,104,9,111)(3,103,10,110)(4,102,11,109)(5,101,12,108)(6,100,13,107)(7,99,14,106)(15,66,22,59)(16,65,23,58)(17,64,24,57)(18,63,25,70)(19,62,26,69)(20,61,27,68)(21,60,28,67)(29,49,36,56)(30,48,37,55)(31,47,38,54)(32,46,39,53)(33,45,40,52)(34,44,41,51)(35,43,42,50)(71,85,78,92)(72,98,79,91)(73,97,80,90)(74,96,81,89)(75,95,82,88)(76,94,83,87)(77,93,84,86), (1,71,18,54)(2,84,19,53)(3,83,20,52)(4,82,21,51)(5,81,22,50)(6,80,23,49)(7,79,24,48)(8,78,25,47)(9,77,26,46)(10,76,27,45)(11,75,28,44)(12,74,15,43)(13,73,16,56)(14,72,17,55)(29,107,97,65)(30,106,98,64)(31,105,85,63)(32,104,86,62)(33,103,87,61)(34,102,88,60)(35,101,89,59)(36,100,90,58)(37,99,91,57)(38,112,92,70)(39,111,93,69)(40,110,94,68)(41,109,95,67)(42,108,96,66), (1,54)(2,55)(3,56)(4,43)(5,44)(6,45)(7,46)(8,47)(9,48)(10,49)(11,50)(12,51)(13,52)(14,53)(15,82)(16,83)(17,84)(18,71)(19,72)(20,73)(21,74)(22,75)(23,76)(24,77)(25,78)(26,79)(27,80)(28,81)(29,110)(30,111)(31,112)(32,99)(33,100)(34,101)(35,102)(36,103)(37,104)(38,105)(39,106)(40,107)(41,108)(42,109)(57,86)(58,87)(59,88)(60,89)(61,90)(62,91)(63,92)(64,93)(65,94)(66,95)(67,96)(68,97)(69,98)(70,85) );
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)], [(1,70,25,105),(2,57,26,106),(3,58,27,107),(4,59,28,108),(5,60,15,109),(6,61,16,110),(7,62,17,111),(8,63,18,112),(9,64,19,99),(10,65,20,100),(11,66,21,101),(12,67,22,102),(13,68,23,103),(14,69,24,104),(29,45,90,83),(30,46,91,84),(31,47,92,71),(32,48,93,72),(33,49,94,73),(34,50,95,74),(35,51,96,75),(36,52,97,76),(37,53,98,77),(38,54,85,78),(39,55,86,79),(40,56,87,80),(41,43,88,81),(42,44,89,82)], [(1,105,8,112),(2,104,9,111),(3,103,10,110),(4,102,11,109),(5,101,12,108),(6,100,13,107),(7,99,14,106),(15,66,22,59),(16,65,23,58),(17,64,24,57),(18,63,25,70),(19,62,26,69),(20,61,27,68),(21,60,28,67),(29,49,36,56),(30,48,37,55),(31,47,38,54),(32,46,39,53),(33,45,40,52),(34,44,41,51),(35,43,42,50),(71,85,78,92),(72,98,79,91),(73,97,80,90),(74,96,81,89),(75,95,82,88),(76,94,83,87),(77,93,84,86)], [(1,71,18,54),(2,84,19,53),(3,83,20,52),(4,82,21,51),(5,81,22,50),(6,80,23,49),(7,79,24,48),(8,78,25,47),(9,77,26,46),(10,76,27,45),(11,75,28,44),(12,74,15,43),(13,73,16,56),(14,72,17,55),(29,107,97,65),(30,106,98,64),(31,105,85,63),(32,104,86,62),(33,103,87,61),(34,102,88,60),(35,101,89,59),(36,100,90,58),(37,99,91,57),(38,112,92,70),(39,111,93,69),(40,110,94,68),(41,109,95,67),(42,108,96,66)], [(1,54),(2,55),(3,56),(4,43),(5,44),(6,45),(7,46),(8,47),(9,48),(10,49),(11,50),(12,51),(13,52),(14,53),(15,82),(16,83),(17,84),(18,71),(19,72),(20,73),(21,74),(22,75),(23,76),(24,77),(25,78),(26,79),(27,80),(28,81),(29,110),(30,111),(31,112),(32,99),(33,100),(34,101),(35,102),(36,103),(37,104),(38,105),(39,106),(40,107),(41,108),(42,109),(57,86),(58,87),(59,88),(60,89),(61,90),(62,91),(63,92),(64,93),(65,94),(66,95),(67,96),(68,97),(69,98),(70,85)])
Matrix representation ►G ⊆ GL8(𝔽29)
| 26 | 21 | 0 | 0 | 0 | 0 | 0 | 0 |
| 8 | 21 | 0 | 0 | 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 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 28 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 25 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 0 | 0 | 13 |
| 0 | 0 | 0 | 0 | 4 | 0 | 1 | 13 |
| 0 | 0 | 0 | 0 | 2 | 28 | 0 | 1 |
| 0 | 0 | 0 | 0 | 7 | 0 | 0 | 16 |
| 21 | 26 | 0 | 0 | 0 | 0 | 0 | 0 |
| 21 | 8 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 0 | 0 | 13 |
| 0 | 0 | 0 | 0 | 4 | 0 | 1 | 13 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 |
| 0 | 0 | 0 | 0 | 25 | 0 | 0 | 16 |
| 21 | 26 | 0 | 0 | 0 | 0 | 0 | 0 |
| 21 | 8 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 22 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 27 | 1 | 0 | 28 |
| 0 | 0 | 0 | 0 | 4 | 16 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 25 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 |
| 0 | 0 | 0 | 0 | 4 | 16 | 1 | 0 |
G:=sub<GL(8,GF(29))| [26,8,0,0,0,0,0,0,21,21,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,13,25,0,0,0,0,0,0,13,16,0,0,0,0,0,0,0,0,13,4,2,7,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,13,13,1,16],[21,21,0,0,0,0,0,0,26,8,0,0,0,0,0,0,0,0,13,7,0,0,0,0,0,0,13,16,0,0,0,0,0,0,0,0,13,4,0,25,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,13,13,1,16],[21,21,0,0,0,0,0,0,26,8,0,0,0,0,0,0,0,0,16,22,0,0,0,0,0,0,16,13,0,0,0,0,0,0,0,0,16,7,27,4,0,0,0,0,13,13,1,16,0,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,4,0,0,0,0,0,0,16,13,0,0,0,0,0,0,0,0,16,25,0,4,0,0,0,0,13,13,1,16,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;
82 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 | ··· | 14AA | 28A | ··· | 28L | 28M | ··· | 28AD |
| 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 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | 4 | 4 | 28 | 28 | 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 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D7 | D14 | D14 | D14 | C7⋊D4 | 2+ (1+4) | D4⋊8D14 |
| kernel | C14.1462+ (1+4) | C23.21D14 | C28⋊7D4 | C23⋊D14 | C28⋊D4 | C28.23D4 | C22×D28 | C14×C4○D4 | C2×C28 | C2×C4○D4 | C22×C4 | C2×D4 | C2×Q8 | C2×C4 | C14 | C2 |
| # reps | 1 | 1 | 4 | 4 | 2 | 2 | 1 | 1 | 4 | 3 | 9 | 9 | 3 | 24 | 2 | 12 |
In GAP, Magma, Sage, TeX
C_{14}._{146}2_+^{(1+4)} % in TeX
G:=Group("C14.146ES+(2,2)"); // GroupNames label
G:=SmallGroup(448,1283);
// by ID
G=gap.SmallGroup(448,1283);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,758,184,675,570,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^14=b^4=e^2=1,c^2=a^7,d^2=a^7*b^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,a*e=e*a,c*b*c^-1=a^7*b^-1,d*b*d^-1=a^7*b,b*e=e*b,c*d=d*c,e*c*e=a^7*c,e*d*e=a^7*b^2*d>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀