?
G = C14.562+ (1+4) order 448 = 26·7
56th non-split extension by C14 of 2+ (1+4) acting via 2+ (1+4)/C2×D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C14.562+ (1+4), C4⋊C4⋊13D14, (C2×Q8)⋊7D14, C22⋊Q8⋊21D7, C28⋊1D4⋊27C2, C28⋊7D4⋊46C2, (C2×D28)⋊8C22, C22⋊D28⋊17C2, D14⋊C4⋊33C22, (C2×C28).64C23, C4⋊Dic7⋊14C22, C22⋊C4.64D14, (Q8×C14)⋊10C22, Dic7⋊4D4⋊15C2, D14.5D4⋊22C2, C28.23D4⋊17C2, (C2×C14).188C24, Dic7⋊C4⋊20C22, C7⋊6(C22.32C24), (C4×Dic7)⋊30C22, (C22×C4).250D14, C2.38(D4⋊8D14), C2.58(D4⋊6D14), C22.4(Q8⋊2D7), (C23×D7).55C22, (C22×D7).79C23, C23.196(C22×D7), C22.209(C23×D7), (C22×C28).316C22, (C22×C14).216C23, (C2×Dic7).241C23, (C22×Dic7).124C22, (C2×C4×D7)⋊19C22, (C2×D14⋊C4)⋊27C2, C4⋊C4⋊D7⋊23C2, (C7×C4⋊C4)⋊22C22, (C7×C22⋊Q8)⋊24C2, C14.116(C2×C4○D4), C2.20(C2×Q8⋊2D7), (C2×C14).28(C4○D4), (C2×C4).185(C22×D7), (C2×C7⋊D4).40C22, (C7×C22⋊C4).43C22, SmallGroup(448,1097)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1436 in 250 conjugacy classes, 95 normal (31 characteristic)
C1, C2 [×3], C2 [×6], C4 [×10], C22, C22 [×2], C22 [×18], C7, C2×C4 [×2], C2×C4 [×4], C2×C4 [×8], D4 [×9], Q8, C23, C23 [×8], D7 [×4], C14 [×3], C14 [×2], C42 [×2], C22⋊C4 [×2], C22⋊C4 [×12], C4⋊C4, C4⋊C4 [×2], C4⋊C4 [×3], C22×C4, C22×C4 [×3], C2×D4 [×7], C2×Q8, C24, Dic7 [×4], C28 [×6], D14 [×16], C2×C14, C2×C14 [×2], C2×C14 [×2], C2×C22⋊C4, C4×D4 [×2], C22≀C2 [×2], C4⋊D4 [×3], C22⋊Q8, C22.D4 [×2], C4.4D4 [×2], C42⋊2C2 [×2], C4×D7 [×2], D28 [×5], C2×Dic7 [×4], C2×Dic7, C7⋊D4 [×4], C2×C28 [×2], C2×C28 [×4], C2×C28, C7×Q8, C22×D7 [×4], C22×D7 [×4], C22×C14, C22.32C24, C4×Dic7 [×2], Dic7⋊C4 [×2], C4⋊Dic7, D14⋊C4 [×12], C7×C22⋊C4 [×2], C7×C4⋊C4, C7×C4⋊C4 [×2], C2×C4×D7 [×2], C2×D28, C2×D28 [×4], C22×Dic7, C2×C7⋊D4 [×2], C22×C28, Q8×C14, C23×D7, Dic7⋊4D4 [×2], C22⋊D28 [×2], D14.5D4 [×2], C28⋊1D4 [×2], C4⋊C4⋊D7 [×2], C2×D14⋊C4, C28⋊7D4, C28.23D4 [×2], C7×C22⋊Q8, C14.562+ (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, Q8⋊2D7 [×2], C23×D7, D4⋊6D14, C2×Q8⋊2D7, D4⋊8D14, C14.562+ (1+4)
Generators and relations
G = < a,b,c,d,e | a14=b4=c2=1, d2=b2, e2=a7, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=a7b-1, bd=db, ebe-1=a7b, cd=dc, ce=ec, ede-1=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 57 20 81)(2 58 21 82)(3 59 22 83)(4 60 23 84)(5 61 24 71)(6 62 25 72)(7 63 26 73)(8 64 27 74)(9 65 28 75)(10 66 15 76)(11 67 16 77)(12 68 17 78)(13 69 18 79)(14 70 19 80)(29 88 50 112)(30 89 51 99)(31 90 52 100)(32 91 53 101)(33 92 54 102)(34 93 55 103)(35 94 56 104)(36 95 43 105)(37 96 44 106)(38 97 45 107)(39 98 46 108)(40 85 47 109)(41 86 48 110)(42 87 49 111)
(1 8)(2 9)(3 10)(4 11)(5 12)(6 13)(7 14)(15 22)(16 23)(17 24)(18 25)(19 26)(20 27)(21 28)(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 81)(58 82)(59 83)(60 84)(61 71)(62 72)(63 73)(64 74)(65 75)(66 76)(67 77)(68 78)(69 79)(70 80)(85 109)(86 110)(87 111)(88 112)(89 99)(90 100)(91 101)(92 102)(93 103)(94 104)(95 105)(96 106)(97 107)(98 108)
(1 56 20 35)(2 55 21 34)(3 54 22 33)(4 53 23 32)(5 52 24 31)(6 51 25 30)(7 50 26 29)(8 49 27 42)(9 48 28 41)(10 47 15 40)(11 46 16 39)(12 45 17 38)(13 44 18 37)(14 43 19 36)(57 104 81 94)(58 103 82 93)(59 102 83 92)(60 101 84 91)(61 100 71 90)(62 99 72 89)(63 112 73 88)(64 111 74 87)(65 110 75 86)(66 109 76 85)(67 108 77 98)(68 107 78 97)(69 106 79 96)(70 105 80 95)
(1 42 8 35)(2 29 9 36)(3 30 10 37)(4 31 11 38)(5 32 12 39)(6 33 13 40)(7 34 14 41)(15 44 22 51)(16 45 23 52)(17 46 24 53)(18 47 25 54)(19 48 26 55)(20 49 27 56)(21 50 28 43)(57 94 64 87)(58 95 65 88)(59 96 66 89)(60 97 67 90)(61 98 68 91)(62 85 69 92)(63 86 70 93)(71 108 78 101)(72 109 79 102)(73 110 80 103)(74 111 81 104)(75 112 82 105)(76 99 83 106)(77 100 84 107)
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), (1,57,20,81)(2,58,21,82)(3,59,22,83)(4,60,23,84)(5,61,24,71)(6,62,25,72)(7,63,26,73)(8,64,27,74)(9,65,28,75)(10,66,15,76)(11,67,16,77)(12,68,17,78)(13,69,18,79)(14,70,19,80)(29,88,50,112)(30,89,51,99)(31,90,52,100)(32,91,53,101)(33,92,54,102)(34,93,55,103)(35,94,56,104)(36,95,43,105)(37,96,44,106)(38,97,45,107)(39,98,46,108)(40,85,47,109)(41,86,48,110)(42,87,49,111), (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28)(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)(57,81)(58,82)(59,83)(60,84)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80)(85,109)(86,110)(87,111)(88,112)(89,99)(90,100)(91,101)(92,102)(93,103)(94,104)(95,105)(96,106)(97,107)(98,108), (1,56,20,35)(2,55,21,34)(3,54,22,33)(4,53,23,32)(5,52,24,31)(6,51,25,30)(7,50,26,29)(8,49,27,42)(9,48,28,41)(10,47,15,40)(11,46,16,39)(12,45,17,38)(13,44,18,37)(14,43,19,36)(57,104,81,94)(58,103,82,93)(59,102,83,92)(60,101,84,91)(61,100,71,90)(62,99,72,89)(63,112,73,88)(64,111,74,87)(65,110,75,86)(66,109,76,85)(67,108,77,98)(68,107,78,97)(69,106,79,96)(70,105,80,95), (1,42,8,35)(2,29,9,36)(3,30,10,37)(4,31,11,38)(5,32,12,39)(6,33,13,40)(7,34,14,41)(15,44,22,51)(16,45,23,52)(17,46,24,53)(18,47,25,54)(19,48,26,55)(20,49,27,56)(21,50,28,43)(57,94,64,87)(58,95,65,88)(59,96,66,89)(60,97,67,90)(61,98,68,91)(62,85,69,92)(63,86,70,93)(71,108,78,101)(72,109,79,102)(73,110,80,103)(74,111,81,104)(75,112,82,105)(76,99,83,106)(77,100,84,107)>;
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), (1,57,20,81)(2,58,21,82)(3,59,22,83)(4,60,23,84)(5,61,24,71)(6,62,25,72)(7,63,26,73)(8,64,27,74)(9,65,28,75)(10,66,15,76)(11,67,16,77)(12,68,17,78)(13,69,18,79)(14,70,19,80)(29,88,50,112)(30,89,51,99)(31,90,52,100)(32,91,53,101)(33,92,54,102)(34,93,55,103)(35,94,56,104)(36,95,43,105)(37,96,44,106)(38,97,45,107)(39,98,46,108)(40,85,47,109)(41,86,48,110)(42,87,49,111), (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28)(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)(57,81)(58,82)(59,83)(60,84)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80)(85,109)(86,110)(87,111)(88,112)(89,99)(90,100)(91,101)(92,102)(93,103)(94,104)(95,105)(96,106)(97,107)(98,108), (1,56,20,35)(2,55,21,34)(3,54,22,33)(4,53,23,32)(5,52,24,31)(6,51,25,30)(7,50,26,29)(8,49,27,42)(9,48,28,41)(10,47,15,40)(11,46,16,39)(12,45,17,38)(13,44,18,37)(14,43,19,36)(57,104,81,94)(58,103,82,93)(59,102,83,92)(60,101,84,91)(61,100,71,90)(62,99,72,89)(63,112,73,88)(64,111,74,87)(65,110,75,86)(66,109,76,85)(67,108,77,98)(68,107,78,97)(69,106,79,96)(70,105,80,95), (1,42,8,35)(2,29,9,36)(3,30,10,37)(4,31,11,38)(5,32,12,39)(6,33,13,40)(7,34,14,41)(15,44,22,51)(16,45,23,52)(17,46,24,53)(18,47,25,54)(19,48,26,55)(20,49,27,56)(21,50,28,43)(57,94,64,87)(58,95,65,88)(59,96,66,89)(60,97,67,90)(61,98,68,91)(62,85,69,92)(63,86,70,93)(71,108,78,101)(72,109,79,102)(73,110,80,103)(74,111,81,104)(75,112,82,105)(76,99,83,106)(77,100,84,107) );
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)], [(1,57,20,81),(2,58,21,82),(3,59,22,83),(4,60,23,84),(5,61,24,71),(6,62,25,72),(7,63,26,73),(8,64,27,74),(9,65,28,75),(10,66,15,76),(11,67,16,77),(12,68,17,78),(13,69,18,79),(14,70,19,80),(29,88,50,112),(30,89,51,99),(31,90,52,100),(32,91,53,101),(33,92,54,102),(34,93,55,103),(35,94,56,104),(36,95,43,105),(37,96,44,106),(38,97,45,107),(39,98,46,108),(40,85,47,109),(41,86,48,110),(42,87,49,111)], [(1,8),(2,9),(3,10),(4,11),(5,12),(6,13),(7,14),(15,22),(16,23),(17,24),(18,25),(19,26),(20,27),(21,28),(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),(57,81),(58,82),(59,83),(60,84),(61,71),(62,72),(63,73),(64,74),(65,75),(66,76),(67,77),(68,78),(69,79),(70,80),(85,109),(86,110),(87,111),(88,112),(89,99),(90,100),(91,101),(92,102),(93,103),(94,104),(95,105),(96,106),(97,107),(98,108)], [(1,56,20,35),(2,55,21,34),(3,54,22,33),(4,53,23,32),(5,52,24,31),(6,51,25,30),(7,50,26,29),(8,49,27,42),(9,48,28,41),(10,47,15,40),(11,46,16,39),(12,45,17,38),(13,44,18,37),(14,43,19,36),(57,104,81,94),(58,103,82,93),(59,102,83,92),(60,101,84,91),(61,100,71,90),(62,99,72,89),(63,112,73,88),(64,111,74,87),(65,110,75,86),(66,109,76,85),(67,108,77,98),(68,107,78,97),(69,106,79,96),(70,105,80,95)], [(1,42,8,35),(2,29,9,36),(3,30,10,37),(4,31,11,38),(5,32,12,39),(6,33,13,40),(7,34,14,41),(15,44,22,51),(16,45,23,52),(17,46,24,53),(18,47,25,54),(19,48,26,55),(20,49,27,56),(21,50,28,43),(57,94,64,87),(58,95,65,88),(59,96,66,89),(60,97,67,90),(61,98,68,91),(62,85,69,92),(63,86,70,93),(71,108,78,101),(72,109,79,102),(73,110,80,103),(74,111,81,104),(75,112,82,105),(76,99,83,106),(77,100,84,107)])
Matrix representation ►G ⊆ GL6(𝔽29)
| 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 | 0 | 0 | 0 | 10 |
| 0 | 0 | 0 | 0 | 26 | 3 |
| 28 | 2 | 0 | 0 | 0 | 0 |
| 28 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 3 | 12 |
| 0 | 0 | 0 | 28 | 5 | 12 |
| 0 | 0 | 1 | 28 | 1 | 0 |
| 0 | 0 | 2 | 22 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 28 | 1 | 28 | 0 |
| 0 | 0 | 27 | 7 | 0 | 28 |
| 17 | 0 | 0 | 0 | 0 | 0 |
| 0 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 14 | 0 | 0 |
| 0 | 0 | 19 | 20 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 9 |
| 0 | 0 | 0 | 0 | 3 | 28 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 12 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 14 | 0 | 0 |
| 0 | 0 | 15 | 20 | 0 | 0 |
| 0 | 0 | 0 | 0 | 24 | 14 |
| 0 | 0 | 0 | 0 | 19 | 5 |
G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,10,19,0,0,0,0,10,22,0,0,0,0,0,0,0,26,0,0,0,0,10,3],[28,28,0,0,0,0,2,1,0,0,0,0,0,0,28,0,1,2,0,0,0,28,28,22,0,0,3,5,1,0,0,0,12,12,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,28,27,0,0,0,1,1,7,0,0,0,0,28,0,0,0,0,0,0,28],[17,0,0,0,0,0,0,17,0,0,0,0,0,0,9,19,0,0,0,0,14,20,0,0,0,0,0,0,1,3,0,0,0,0,9,28],[12,12,0,0,0,0,0,17,0,0,0,0,0,0,9,15,0,0,0,0,14,20,0,0,0,0,0,0,24,19,0,0,0,0,14,5] >;
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28L | 28M | ··· | 28X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | 28 | 28 | 4 | ··· | 4 | 14 | 14 | 14 | 14 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
64 irreducible representations
| dim | 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 | D7 | C4○D4 | D14 | D14 | D14 | D14 | 2+ (1+4) | Q8⋊2D7 | D4⋊6D14 | D4⋊8D14 |
| kernel | C14.562+ (1+4) | Dic7⋊4D4 | C22⋊D28 | D14.5D4 | C28⋊1D4 | C4⋊C4⋊D7 | C2×D14⋊C4 | C28⋊7D4 | C28.23D4 | C7×C22⋊Q8 | C22⋊Q8 | C2×C14 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×Q8 | C14 | C22 | C2 | C2 |
| # reps | 1 | 2 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 1 | 3 | 4 | 6 | 9 | 3 | 3 | 2 | 6 | 6 | 6 |
In GAP, Magma, Sage, TeX
C_{14}._{56}2_+^{(1+4)} % in TeX
G:=Group("C14.56ES+(2,2)"); // GroupNames label
G:=SmallGroup(448,1097);
// by ID
G=gap.SmallGroup(448,1097);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,758,219,184,675,297,136,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^14=b^4=c^2=1,d^2=b^2,e^2=a^7,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,b*d=d*b,e*b*e^-1=a^7*b,c*d=d*c,c*e=e*c,e*d*e^-1=a^7*b^2*d>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀