?
G = C14.622+ (1+4) order 448 = 26·7
62nd non-split extension by C14 of 2+ (1+4) acting via 2+ (1+4)/C2×D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C14.622+ (1+4), C4⋊C4⋊31D14, (C22×C4)⋊6D14, C22⋊C4⋊18D14, (C2×D4).99D14, D14⋊Q8⋊30C2, C23⋊D14.3C2, C4⋊Dic7⋊40C22, C22.D4⋊8D7, D14.22(C4○D4), D14.D4⋊32C2, C28.17D4⋊21C2, (C2×C14).203C24, (C2×C28).599C23, Dic7⋊C4⋊66C22, (C22×C28)⋊39C22, (C4×Dic7)⋊56C22, C2.64(D4⋊6D14), C23.D7⋊54C22, D14⋊C4.131C22, C7⋊7(C22.45C24), (C2×Dic14)⋊31C22, (D4×C14).141C22, C23.D14⋊31C2, (C23×D7).60C22, C23.130(C22×D7), C22.224(C23×D7), (C22×C14).223C23, (C2×Dic7).105C23, (C22×D7).209C23, (C4×C7⋊D4)⋊48C2, C2.65(D7×C4○D4), C4⋊C4⋊7D7⋊33C2, (C7×C4⋊C4)⋊29C22, (D7×C22⋊C4)⋊15C2, C14.177(C2×C4○D4), (C2×C4×D7).112C22, (C2×C4).65(C22×D7), (C7×C22⋊C4)⋊25C22, (C7×C22.D4)⋊11C2, (C2×C7⋊D4).135C22, SmallGroup(448,1112)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1228 in 248 conjugacy classes, 95 normal (27 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×11], C22, C22 [×18], C7, C2×C4, C2×C4 [×4], C2×C4 [×13], D4 [×5], Q8, C23 [×2], C23 [×7], D7 [×4], C14, C14 [×2], C14 [×2], C42 [×3], C22⋊C4, C22⋊C4 [×2], C22⋊C4 [×11], C4⋊C4 [×2], C4⋊C4 [×6], C22×C4, C22×C4 [×4], C2×D4, C2×D4 [×2], C2×Q8, C24, Dic7 [×6], C28 [×5], D14 [×4], D14 [×8], C2×C14, C2×C14 [×6], C2×C22⋊C4 [×2], C42⋊C2 [×2], C4×D4 [×2], C22≀C2, C22⋊Q8 [×2], C22.D4, C22.D4 [×2], C4.4D4, C42⋊2C2 [×2], Dic14, C4×D7 [×6], C2×Dic7 [×6], C7⋊D4 [×4], C2×C28, C2×C28 [×4], C2×C28, C7×D4, C22×D7 [×2], C22×D7 [×5], C22×C14 [×2], C22.45C24, C4×Dic7, C4×Dic7 [×2], Dic7⋊C4 [×4], C4⋊Dic7 [×2], D14⋊C4 [×6], C23.D7, C23.D7 [×4], C7×C22⋊C4, C7×C22⋊C4 [×2], C7×C4⋊C4 [×2], C2×Dic14, C2×C4×D7 [×4], C2×C7⋊D4 [×2], C22×C28, D4×C14, C23×D7, C23.D14 [×2], D7×C22⋊C4 [×2], D14.D4 [×2], C4⋊C4⋊7D7 [×2], D14⋊Q8 [×2], C4×C7⋊D4 [×2], C28.17D4, C23⋊D14, C7×C22.D4, C14.622+ (1+4)
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D7, C4○D4 [×4], C24, D14 [×7], C2×C4○D4 [×2], 2+ (1+4), C22×D7 [×7], C22.45C24, C23×D7, D4⋊6D14, D7×C4○D4 [×2], C14.622+ (1+4)
Generators and relations
G = < a,b,c,d,e | a14=b4=c2=e2=1, d2=b2, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=a7b-1, dbd-1=ebe=a7b, dcd-1=ece=a7c, ede=b2d >
►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 60 26 84)(2 61 27 71)(3 62 28 72)(4 63 15 73)(5 64 16 74)(6 65 17 75)(7 66 18 76)(8 67 19 77)(9 68 20 78)(10 69 21 79)(11 70 22 80)(12 57 23 81)(13 58 24 82)(14 59 25 83)(29 90 56 111)(30 91 43 112)(31 92 44 99)(32 93 45 100)(33 94 46 101)(34 95 47 102)(35 96 48 103)(36 97 49 104)(37 98 50 105)(38 85 51 106)(39 86 52 107)(40 87 53 108)(41 88 54 109)(42 89 55 110)
(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 74)(58 75)(59 76)(60 77)(61 78)(62 79)(63 80)(64 81)(65 82)(66 83)(67 84)(68 71)(69 72)(70 73)(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 56 26 29)(2 55 27 42)(3 54 28 41)(4 53 15 40)(5 52 16 39)(6 51 17 38)(7 50 18 37)(8 49 19 36)(9 48 20 35)(10 47 21 34)(11 46 22 33)(12 45 23 32)(13 44 24 31)(14 43 25 30)(57 107 81 86)(58 106 82 85)(59 105 83 98)(60 104 84 97)(61 103 71 96)(62 102 72 95)(63 101 73 94)(64 100 74 93)(65 99 75 92)(66 112 76 91)(67 111 77 90)(68 110 78 89)(69 109 79 88)(70 108 80 87)
(1 29)(2 30)(3 31)(4 32)(5 33)(6 34)(7 35)(8 36)(9 37)(10 38)(11 39)(12 40)(13 41)(14 42)(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 94)(58 95)(59 96)(60 97)(61 98)(62 85)(63 86)(64 87)(65 88)(66 89)(67 90)(68 91)(69 92)(70 93)(71 105)(72 106)(73 107)(74 108)(75 109)(76 110)(77 111)(78 112)(79 99)(80 100)(81 101)(82 102)(83 103)(84 104)
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), (1,60,26,84)(2,61,27,71)(3,62,28,72)(4,63,15,73)(5,64,16,74)(6,65,17,75)(7,66,18,76)(8,67,19,77)(9,68,20,78)(10,69,21,79)(11,70,22,80)(12,57,23,81)(13,58,24,82)(14,59,25,83)(29,90,56,111)(30,91,43,112)(31,92,44,99)(32,93,45,100)(33,94,46,101)(34,95,47,102)(35,96,48,103)(36,97,49,104)(37,98,50,105)(38,85,51,106)(39,86,52,107)(40,87,53,108)(41,88,54,109)(42,89,55,110), (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)(57,74)(58,75)(59,76)(60,77)(61,78)(62,79)(63,80)(64,81)(65,82)(66,83)(67,84)(68,71)(69,72)(70,73)(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), (1,56,26,29)(2,55,27,42)(3,54,28,41)(4,53,15,40)(5,52,16,39)(6,51,17,38)(7,50,18,37)(8,49,19,36)(9,48,20,35)(10,47,21,34)(11,46,22,33)(12,45,23,32)(13,44,24,31)(14,43,25,30)(57,107,81,86)(58,106,82,85)(59,105,83,98)(60,104,84,97)(61,103,71,96)(62,102,72,95)(63,101,73,94)(64,100,74,93)(65,99,75,92)(66,112,76,91)(67,111,77,90)(68,110,78,89)(69,109,79,88)(70,108,80,87), (1,29)(2,30)(3,31)(4,32)(5,33)(6,34)(7,35)(8,36)(9,37)(10,38)(11,39)(12,40)(13,41)(14,42)(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)(57,94)(58,95)(59,96)(60,97)(61,98)(62,85)(63,86)(64,87)(65,88)(66,89)(67,90)(68,91)(69,92)(70,93)(71,105)(72,106)(73,107)(74,108)(75,109)(76,110)(77,111)(78,112)(79,99)(80,100)(81,101)(82,102)(83,103)(84,104)>;
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), (1,60,26,84)(2,61,27,71)(3,62,28,72)(4,63,15,73)(5,64,16,74)(6,65,17,75)(7,66,18,76)(8,67,19,77)(9,68,20,78)(10,69,21,79)(11,70,22,80)(12,57,23,81)(13,58,24,82)(14,59,25,83)(29,90,56,111)(30,91,43,112)(31,92,44,99)(32,93,45,100)(33,94,46,101)(34,95,47,102)(35,96,48,103)(36,97,49,104)(37,98,50,105)(38,85,51,106)(39,86,52,107)(40,87,53,108)(41,88,54,109)(42,89,55,110), (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)(57,74)(58,75)(59,76)(60,77)(61,78)(62,79)(63,80)(64,81)(65,82)(66,83)(67,84)(68,71)(69,72)(70,73)(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), (1,56,26,29)(2,55,27,42)(3,54,28,41)(4,53,15,40)(5,52,16,39)(6,51,17,38)(7,50,18,37)(8,49,19,36)(9,48,20,35)(10,47,21,34)(11,46,22,33)(12,45,23,32)(13,44,24,31)(14,43,25,30)(57,107,81,86)(58,106,82,85)(59,105,83,98)(60,104,84,97)(61,103,71,96)(62,102,72,95)(63,101,73,94)(64,100,74,93)(65,99,75,92)(66,112,76,91)(67,111,77,90)(68,110,78,89)(69,109,79,88)(70,108,80,87), (1,29)(2,30)(3,31)(4,32)(5,33)(6,34)(7,35)(8,36)(9,37)(10,38)(11,39)(12,40)(13,41)(14,42)(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)(57,94)(58,95)(59,96)(60,97)(61,98)(62,85)(63,86)(64,87)(65,88)(66,89)(67,90)(68,91)(69,92)(70,93)(71,105)(72,106)(73,107)(74,108)(75,109)(76,110)(77,111)(78,112)(79,99)(80,100)(81,101)(82,102)(83,103)(84,104) );
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)], [(1,60,26,84),(2,61,27,71),(3,62,28,72),(4,63,15,73),(5,64,16,74),(6,65,17,75),(7,66,18,76),(8,67,19,77),(9,68,20,78),(10,69,21,79),(11,70,22,80),(12,57,23,81),(13,58,24,82),(14,59,25,83),(29,90,56,111),(30,91,43,112),(31,92,44,99),(32,93,45,100),(33,94,46,101),(34,95,47,102),(35,96,48,103),(36,97,49,104),(37,98,50,105),(38,85,51,106),(39,86,52,107),(40,87,53,108),(41,88,54,109),(42,89,55,110)], [(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),(57,74),(58,75),(59,76),(60,77),(61,78),(62,79),(63,80),(64,81),(65,82),(66,83),(67,84),(68,71),(69,72),(70,73),(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)], [(1,56,26,29),(2,55,27,42),(3,54,28,41),(4,53,15,40),(5,52,16,39),(6,51,17,38),(7,50,18,37),(8,49,19,36),(9,48,20,35),(10,47,21,34),(11,46,22,33),(12,45,23,32),(13,44,24,31),(14,43,25,30),(57,107,81,86),(58,106,82,85),(59,105,83,98),(60,104,84,97),(61,103,71,96),(62,102,72,95),(63,101,73,94),(64,100,74,93),(65,99,75,92),(66,112,76,91),(67,111,77,90),(68,110,78,89),(69,109,79,88),(70,108,80,87)], [(1,29),(2,30),(3,31),(4,32),(5,33),(6,34),(7,35),(8,36),(9,37),(10,38),(11,39),(12,40),(13,41),(14,42),(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),(57,94),(58,95),(59,96),(60,97),(61,98),(62,85),(63,86),(64,87),(65,88),(66,89),(67,90),(68,91),(69,92),(70,93),(71,105),(72,106),(73,107),(74,108),(75,109),(76,110),(77,111),(78,112),(79,99),(80,100),(81,101),(82,102),(83,103),(84,104)])
Matrix representation ►G ⊆ GL6(𝔽29)
| 8 | 25 | 0 | 0 | 0 | 0 |
| 24 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 17 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 18 | 17 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 16 | 28 |
| 0 | 18 | 0 | 0 | 0 | 0 |
| 21 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 28 |
| 0 | 0 | 0 | 0 | 7 | 21 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 28 |
| 0 | 0 | 0 | 0 | 5 | 21 |
G:=sub<GL(6,GF(29))| [8,24,0,0,0,0,25,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,17,0,0,0,0,12,0,0,0,0,0,0,0,12,18,0,0,0,0,0,17],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,1,16,0,0,0,0,0,28],[0,21,0,0,0,0,18,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,8,7,0,0,0,0,28,21],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,8,5,0,0,0,0,28,21] >;
67 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 | 4M | 4N | 4O | 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 | 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 | 14 | 14 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 14 | 14 | 14 | 14 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | ··· | 8 |
67 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D7 | C4○D4 | D14 | D14 | D14 | D14 | 2+ (1+4) | D4⋊6D14 | D7×C4○D4 |
| kernel | C14.622+ (1+4) | C23.D14 | D7×C22⋊C4 | D14.D4 | C4⋊C4⋊7D7 | D14⋊Q8 | C4×C7⋊D4 | C28.17D4 | C23⋊D14 | C7×C22.D4 | C22.D4 | D14 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C14 | C2 | C2 |
| # reps | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 3 | 8 | 9 | 6 | 3 | 3 | 1 | 6 | 12 |
In GAP, Magma, Sage, TeX
C_{14}._{62}2_+^{(1+4)} % in TeX
G:=Group("C14.62ES+(2,2)"); // GroupNames label
G:=SmallGroup(448,1112);
// by ID
G=gap.SmallGroup(448,1112);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,219,184,1571,570,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^14=b^4=c^2=e^2=1,d^2=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=b^2*d>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀