?
G = C14.612+ (1+4) order 448 = 26·7
61st non-split extension by C14 of 2+ (1+4) acting via 2+ (1+4)/C2×D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C14.612+ (1+4), C4⋊C4⋊29D14, C28⋊7D4⋊21C2, C28⋊2D4⋊30C2, C22⋊C4⋊16D14, (C2×D4).98D14, (C22×C4)⋊24D14, D28⋊C4⋊33C2, D14⋊D4⋊32C2, C23⋊D14⋊17C2, C22⋊D28⋊21C2, D14⋊C4⋊28C22, D14⋊2Q8⋊31C2, D14.6(C4○D4), C4⋊Dic7⋊15C22, C22.D4⋊6D7, D14.5D4⋊29C2, (C2×C28).179C23, (C2×C14).201C24, Dic7⋊C4⋊54C22, (C22×C28)⋊18C22, C7⋊7(C22.32C24), (C4×Dic7)⋊33C22, (C2×D28).32C22, C2.63(D4⋊6D14), C2.42(D4⋊8D14), C23.D7⋊53C22, Dic7.D4⋊32C2, (C2×Dic14)⋊29C22, (D4×C14).139C22, C23.D14⋊30C2, (C23×D7).58C22, (C22×D7).85C23, C22.222(C23×D7), C23.129(C22×D7), (C22×C14).221C23, (C2×Dic7).244C23, (C4×C7⋊D4)⋊6C2, C2.63(D7×C4○D4), (C2×C4×D7)⋊23C22, C4⋊C4⋊D7⋊27C2, (C7×C4⋊C4)⋊27C22, (D7×C22⋊C4)⋊13C2, C14.175(C2×C4○D4), (C2×C7⋊D4)⋊20C22, (C2×C4).64(C22×D7), (C7×C22⋊C4)⋊23C22, (C7×C22.D4)⋊9C2, SmallGroup(448,1110)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1388 in 250 conjugacy classes, 93 normal (91 characteristic)
C1, C2 [×3], C2 [×6], C4 [×10], C22, C22 [×20], C7, C2×C4 [×5], C2×C4 [×9], D4 [×9], Q8, C23 [×2], C23 [×7], D7 [×4], C14 [×3], C14 [×2], C42 [×2], C22⋊C4 [×3], C22⋊C4 [×11], C4⋊C4 [×2], C4⋊C4 [×4], C22×C4, C22×C4 [×3], C2×D4, C2×D4 [×6], C2×Q8, C24, Dic7 [×5], C28 [×5], D14 [×2], D14 [×12], C2×C14, C2×C14 [×6], C2×C22⋊C4, C4×D4 [×2], C22≀C2 [×2], C4⋊D4 [×3], C22⋊Q8, C22.D4, C22.D4, C4.4D4 [×2], C42⋊2C2 [×2], Dic14, C4×D7 [×3], D28 [×3], C2×Dic7 [×5], C7⋊D4 [×5], C2×C28 [×5], C2×C28, C7×D4, C22×D7 [×3], C22×D7 [×4], C22×C14 [×2], C22.32C24, C4×Dic7 [×2], Dic7⋊C4 [×2], C4⋊Dic7 [×2], D14⋊C4 [×8], C23.D7 [×3], C7×C22⋊C4 [×3], C7×C4⋊C4 [×2], C2×Dic14, C2×C4×D7 [×3], C2×D28 [×2], C2×C7⋊D4 [×4], C22×C28, D4×C14, C23×D7, C23.D14, D7×C22⋊C4, C22⋊D28, D14⋊D4, Dic7.D4 [×2], D28⋊C4, D14.5D4, D14⋊2Q8, C4⋊C4⋊D7, C4×C7⋊D4, C28⋊7D4, C23⋊D14, C28⋊2D4, C7×C22.D4, C14.612+ (1+4)
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D7, C4○D4 [×2], C24, D14 [×7], C2×C4○D4, 2+ (1+4) [×2], C22×D7 [×7], C22.32C24, C23×D7, D4⋊6D14, D7×C4○D4, D4⋊8D14, C14.612+ (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=a7b-1, dbd-1=ebe=a7b, 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 68 26 77)(2 69 27 78)(3 70 28 79)(4 57 15 80)(5 58 16 81)(6 59 17 82)(7 60 18 83)(8 61 19 84)(9 62 20 71)(10 63 21 72)(11 64 22 73)(12 65 23 74)(13 66 24 75)(14 67 25 76)(29 97 53 106)(30 98 54 107)(31 85 55 108)(32 86 56 109)(33 87 43 110)(34 88 44 111)(35 89 45 112)(36 90 46 99)(37 91 47 100)(38 92 48 101)(39 93 49 102)(40 94 50 103)(41 95 51 104)(42 96 52 105)
(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 73)(58 74)(59 75)(60 76)(61 77)(62 78)(63 79)(64 80)(65 81)(66 82)(67 83)(68 84)(69 71)(70 72)(85 108)(86 109)(87 110)(88 111)(89 112)(90 99)(91 100)(92 101)(93 102)(94 103)(95 104)(96 105)(97 106)(98 107)
(1 45 19 42)(2 44 20 41)(3 43 21 40)(4 56 22 39)(5 55 23 38)(6 54 24 37)(7 53 25 36)(8 52 26 35)(9 51 27 34)(10 50 28 33)(11 49 15 32)(12 48 16 31)(13 47 17 30)(14 46 18 29)(57 102 73 86)(58 101 74 85)(59 100 75 98)(60 99 76 97)(61 112 77 96)(62 111 78 95)(63 110 79 94)(64 109 80 93)(65 108 81 92)(66 107 82 91)(67 106 83 90)(68 105 84 89)(69 104 71 88)(70 103 72 87)
(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 48)(16 49)(17 50)(18 51)(19 52)(20 53)(21 54)(22 55)(23 56)(24 43)(25 44)(26 45)(27 46)(28 47)(57 85)(58 86)(59 87)(60 88)(61 89)(62 90)(63 91)(64 92)(65 93)(66 94)(67 95)(68 96)(69 97)(70 98)(71 99)(72 100)(73 101)(74 102)(75 103)(76 104)(77 105)(78 106)(79 107)(80 108)(81 109)(82 110)(83 111)(84 112)
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), (1,68,26,77)(2,69,27,78)(3,70,28,79)(4,57,15,80)(5,58,16,81)(6,59,17,82)(7,60,18,83)(8,61,19,84)(9,62,20,71)(10,63,21,72)(11,64,22,73)(12,65,23,74)(13,66,24,75)(14,67,25,76)(29,97,53,106)(30,98,54,107)(31,85,55,108)(32,86,56,109)(33,87,43,110)(34,88,44,111)(35,89,45,112)(36,90,46,99)(37,91,47,100)(38,92,48,101)(39,93,49,102)(40,94,50,103)(41,95,51,104)(42,96,52,105), (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)(57,73)(58,74)(59,75)(60,76)(61,77)(62,78)(63,79)(64,80)(65,81)(66,82)(67,83)(68,84)(69,71)(70,72)(85,108)(86,109)(87,110)(88,111)(89,112)(90,99)(91,100)(92,101)(93,102)(94,103)(95,104)(96,105)(97,106)(98,107), (1,45,19,42)(2,44,20,41)(3,43,21,40)(4,56,22,39)(5,55,23,38)(6,54,24,37)(7,53,25,36)(8,52,26,35)(9,51,27,34)(10,50,28,33)(11,49,15,32)(12,48,16,31)(13,47,17,30)(14,46,18,29)(57,102,73,86)(58,101,74,85)(59,100,75,98)(60,99,76,97)(61,112,77,96)(62,111,78,95)(63,110,79,94)(64,109,80,93)(65,108,81,92)(66,107,82,91)(67,106,83,90)(68,105,84,89)(69,104,71,88)(70,103,72,87), (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)(15,48)(16,49)(17,50)(18,51)(19,52)(20,53)(21,54)(22,55)(23,56)(24,43)(25,44)(26,45)(27,46)(28,47)(57,85)(58,86)(59,87)(60,88)(61,89)(62,90)(63,91)(64,92)(65,93)(66,94)(67,95)(68,96)(69,97)(70,98)(71,99)(72,100)(73,101)(74,102)(75,103)(76,104)(77,105)(78,106)(79,107)(80,108)(81,109)(82,110)(83,111)(84,112)>;
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), (1,68,26,77)(2,69,27,78)(3,70,28,79)(4,57,15,80)(5,58,16,81)(6,59,17,82)(7,60,18,83)(8,61,19,84)(9,62,20,71)(10,63,21,72)(11,64,22,73)(12,65,23,74)(13,66,24,75)(14,67,25,76)(29,97,53,106)(30,98,54,107)(31,85,55,108)(32,86,56,109)(33,87,43,110)(34,88,44,111)(35,89,45,112)(36,90,46,99)(37,91,47,100)(38,92,48,101)(39,93,49,102)(40,94,50,103)(41,95,51,104)(42,96,52,105), (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)(57,73)(58,74)(59,75)(60,76)(61,77)(62,78)(63,79)(64,80)(65,81)(66,82)(67,83)(68,84)(69,71)(70,72)(85,108)(86,109)(87,110)(88,111)(89,112)(90,99)(91,100)(92,101)(93,102)(94,103)(95,104)(96,105)(97,106)(98,107), (1,45,19,42)(2,44,20,41)(3,43,21,40)(4,56,22,39)(5,55,23,38)(6,54,24,37)(7,53,25,36)(8,52,26,35)(9,51,27,34)(10,50,28,33)(11,49,15,32)(12,48,16,31)(13,47,17,30)(14,46,18,29)(57,102,73,86)(58,101,74,85)(59,100,75,98)(60,99,76,97)(61,112,77,96)(62,111,78,95)(63,110,79,94)(64,109,80,93)(65,108,81,92)(66,107,82,91)(67,106,83,90)(68,105,84,89)(69,104,71,88)(70,103,72,87), (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)(15,48)(16,49)(17,50)(18,51)(19,52)(20,53)(21,54)(22,55)(23,56)(24,43)(25,44)(26,45)(27,46)(28,47)(57,85)(58,86)(59,87)(60,88)(61,89)(62,90)(63,91)(64,92)(65,93)(66,94)(67,95)(68,96)(69,97)(70,98)(71,99)(72,100)(73,101)(74,102)(75,103)(76,104)(77,105)(78,106)(79,107)(80,108)(81,109)(82,110)(83,111)(84,112) );
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)], [(1,68,26,77),(2,69,27,78),(3,70,28,79),(4,57,15,80),(5,58,16,81),(6,59,17,82),(7,60,18,83),(8,61,19,84),(9,62,20,71),(10,63,21,72),(11,64,22,73),(12,65,23,74),(13,66,24,75),(14,67,25,76),(29,97,53,106),(30,98,54,107),(31,85,55,108),(32,86,56,109),(33,87,43,110),(34,88,44,111),(35,89,45,112),(36,90,46,99),(37,91,47,100),(38,92,48,101),(39,93,49,102),(40,94,50,103),(41,95,51,104),(42,96,52,105)], [(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),(57,73),(58,74),(59,75),(60,76),(61,77),(62,78),(63,79),(64,80),(65,81),(66,82),(67,83),(68,84),(69,71),(70,72),(85,108),(86,109),(87,110),(88,111),(89,112),(90,99),(91,100),(92,101),(93,102),(94,103),(95,104),(96,105),(97,106),(98,107)], [(1,45,19,42),(2,44,20,41),(3,43,21,40),(4,56,22,39),(5,55,23,38),(6,54,24,37),(7,53,25,36),(8,52,26,35),(9,51,27,34),(10,50,28,33),(11,49,15,32),(12,48,16,31),(13,47,17,30),(14,46,18,29),(57,102,73,86),(58,101,74,85),(59,100,75,98),(60,99,76,97),(61,112,77,96),(62,111,78,95),(63,110,79,94),(64,109,80,93),(65,108,81,92),(66,107,82,91),(67,106,83,90),(68,105,84,89),(69,104,71,88),(70,103,72,87)], [(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),(15,48),(16,49),(17,50),(18,51),(19,52),(20,53),(21,54),(22,55),(23,56),(24,43),(25,44),(26,45),(27,46),(28,47),(57,85),(58,86),(59,87),(60,88),(61,89),(62,90),(63,91),(64,92),(65,93),(66,94),(67,95),(68,96),(69,97),(70,98),(71,99),(72,100),(73,101),(74,102),(75,103),(76,104),(77,105),(78,106),(79,107),(80,108),(81,109),(82,110),(83,111),(84,112)])
Matrix representation ►G ⊆ GL6(𝔽29)
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 19 | 19 | 0 | 0 |
| 0 | 0 | 10 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 19 | 19 |
| 0 | 0 | 0 | 0 | 10 | 7 |
| 17 | 0 | 0 | 0 | 0 | 0 |
| 23 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 6 | 13 | 17 |
| 0 | 0 | 23 | 21 | 12 | 16 |
| 0 | 0 | 8 | 6 | 21 | 23 |
| 0 | 0 | 23 | 21 | 6 | 8 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 15 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 28 | 0 |
| 0 | 0 | 0 | 1 | 0 | 28 |
| 23 | 24 | 0 | 0 | 0 | 0 |
| 7 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 2 | 0 |
| 0 | 0 | 22 | 1 | 14 | 27 |
| 0 | 0 | 28 | 0 | 1 | 0 |
| 0 | 0 | 22 | 1 | 7 | 28 |
| 23 | 24 | 0 | 0 | 0 | 0 |
| 7 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 27 | 0 |
| 0 | 0 | 0 | 1 | 0 | 27 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,19,10,0,0,0,0,19,7,0,0,0,0,0,0,19,10,0,0,0,0,19,7],[17,23,0,0,0,0,0,12,0,0,0,0,0,0,8,23,8,23,0,0,6,21,6,21,0,0,13,12,21,6,0,0,17,16,23,8],[1,15,0,0,0,0,0,28,0,0,0,0,0,0,1,0,1,0,0,0,0,1,0,1,0,0,0,0,28,0,0,0,0,0,0,28],[23,7,0,0,0,0,24,6,0,0,0,0,0,0,28,22,28,22,0,0,0,1,0,1,0,0,2,14,1,7,0,0,0,27,0,28],[23,7,0,0,0,0,24,6,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,27,0,28,0,0,0,0,27,0,28] >;
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14O | 14P | 14Q | 14R | 28A | ··· | 28L | 28M | ··· | 28U |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | 14 | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 14 | 14 | 28 | 28 | 2 | 2 | 4 | 4 | 4 | 4 | 14 | 14 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | ··· | 8 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D7 | C4○D4 | D14 | D14 | D14 | D14 | 2+ (1+4) | D4⋊6D14 | D7×C4○D4 | D4⋊8D14 |
| kernel | C14.612+ (1+4) | C23.D14 | D7×C22⋊C4 | C22⋊D28 | D14⋊D4 | Dic7.D4 | D28⋊C4 | D14.5D4 | D14⋊2Q8 | C4⋊C4⋊D7 | C4×C7⋊D4 | C28⋊7D4 | C23⋊D14 | C28⋊2D4 | C7×C22.D4 | C22.D4 | D14 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C14 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 4 | 9 | 6 | 3 | 3 | 2 | 6 | 6 | 6 |
In GAP, Magma, Sage, TeX
C_{14}._{61}2_+^{(1+4)} % in TeX
G:=Group("C14.61ES+(2,2)"); // GroupNames label
G:=SmallGroup(448,1110);
// by ID
G=gap.SmallGroup(448,1110);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,219,184,675,570,18822]);
// 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=a^7*b^-1,d*b*d^-1=e*b*e=a^7*b,d*c*d^-1=e*c*e=a^7*c,e*d*e=a^7*b^2*d>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀