?
G = C14.1452+ (1+4) order 448 = 26·7
54th non-split extension by C14 of 2+ (1+4) acting via 2+ (1+4)/C4○D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C14.1452+ (1+4), (C7×D4)⋊19D4, (C2×D4)⋊44D14, (C2×Q8)⋊33D14, C28⋊7D4⋊39C2, D4⋊10(C7⋊D4), C7⋊11(D4⋊5D4), (D4×Dic7)⋊42C2, C28.266(C2×D4), (C22×C4)⋊30D14, D14⋊12(C4○D4), C23⋊D14⋊31C2, D14⋊C4⋊37C22, D14⋊3Q8⋊44C2, (D4×C14)⋊59C22, C4⋊Dic7⋊46C22, (Q8×C14)⋊40C22, Dic7⋊D4⋊44C2, C28.23D4⋊31C2, (C2×C28).652C23, (C2×C14).315C24, Dic7⋊C4⋊40C22, (C22×C28)⋊24C22, (C4×Dic7)⋊44C22, C14.165(C22×D4), C2.69(D4⋊8D14), C23.D7⋊41C22, (C2×D28).184C22, (C23×D7).79C22, C22.324(C23×D7), C23.211(C22×D7), C23.23D14⋊31C2, (C22×C14).241C23, (C2×Dic7).162C23, (C22×Dic7)⋊36C22, (C22×D7).137C23, (C2×D4×D7)⋊26C2, (C2×C4○D4)⋊7D7, (C14×C4○D4)⋊7C2, (C4×C7⋊D4)⋊29C2, C4.72(C2×C7⋊D4), (C2×D14⋊C4)⋊44C2, C2.104(D7×C4○D4), (C2×C14).81(C2×D4), C22.5(C2×C7⋊D4), C14.216(C2×C4○D4), (C2×C7⋊D4)⋊30C22, (C2×C4×D7).168C22, C2.38(C22×C7⋊D4), (C2×C4).250(C22×D7), SmallGroup(448,1282)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1716 in 334 conjugacy classes, 113 normal (43 characteristic)
C1, C2 [×3], C2 [×9], C4 [×2], C4 [×8], C22, C22 [×4], C22 [×25], C7, C2×C4 [×2], C2×C4 [×2], C2×C4 [×15], D4 [×4], D4 [×14], Q8 [×2], C23, C23 [×2], C23 [×13], D7 [×4], C14 [×3], C14 [×5], C42, C22⋊C4 [×12], C4⋊C4 [×4], C22×C4, C22×C4 [×2], C22×C4 [×3], C2×D4, C2×D4 [×2], C2×D4 [×10], C2×Q8, C4○D4 [×4], C24 [×2], Dic7 [×5], C28 [×2], C28 [×3], D14 [×2], D14 [×16], C2×C14, C2×C14 [×4], C2×C14 [×7], C2×C22⋊C4 [×2], C4×D4 [×2], C22≀C2 [×2], C4⋊D4 [×3], C22⋊Q8, C22.D4 [×2], C4.4D4, C22×D4, C2×C4○D4, C4×D7 [×2], D28 [×2], C2×Dic7 [×3], C2×Dic7 [×2], C2×Dic7 [×2], C7⋊D4 [×8], C2×C28 [×2], C2×C28 [×2], C2×C28 [×6], C7×D4 [×4], C7×D4 [×4], C7×Q8 [×2], C22×D7, C22×D7 [×2], C22×D7 [×10], C22×C14, C22×C14 [×2], D4⋊5D4, C4×Dic7, Dic7⋊C4, Dic7⋊C4 [×2], C4⋊Dic7, D14⋊C4, D14⋊C4 [×8], C23.D7, C23.D7 [×2], C2×C4×D7, C2×D28, D4×D7 [×4], C22×Dic7 [×2], C2×C7⋊D4, C2×C7⋊D4 [×4], C22×C28, C22×C28 [×2], D4×C14, D4×C14 [×2], Q8×C14, C7×C4○D4 [×4], C23×D7 [×2], C2×D14⋊C4 [×2], C4×C7⋊D4, C23.23D14 [×2], C28⋊7D4, D4×Dic7, C23⋊D14 [×2], Dic7⋊D4 [×2], D14⋊3Q8, C28.23D4, C2×D4×D7, C14×C4○D4, C14.1452+ (1+4)
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D7, C2×D4 [×6], C4○D4 [×2], C24, D14 [×7], C22×D4, C2×C4○D4, 2+ (1+4), C7⋊D4 [×4], C22×D7 [×7], D4⋊5D4, C2×C7⋊D4 [×6], C23×D7, D7×C4○D4, D4⋊8D14, C22×C7⋊D4, C14.1452+ (1+4)
Generators and relations
G = < a,b,c,d,e | a14=b4=c2=1, d2=b2, e2=a7, ab=ba, ac=ca, ad=da, eae-1=a-1, cbc=b-1, bd=db, ebe-1=a7b, cd=dc, ce=ec, ede-1=a7b2d >
►On 112 points
(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)
(1 112 45 57)(2 99 46 58)(3 100 47 59)(4 101 48 60)(5 102 49 61)(6 103 50 62)(7 104 51 63)(8 105 52 64)(9 106 53 65)(10 107 54 66)(11 108 55 67)(12 109 56 68)(13 110 43 69)(14 111 44 70)(15 86 74 42)(16 87 75 29)(17 88 76 30)(18 89 77 31)(19 90 78 32)(20 91 79 33)(21 92 80 34)(22 93 81 35)(23 94 82 36)(24 95 83 37)(25 96 84 38)(26 97 71 39)(27 98 72 40)(28 85 73 41)
(1 84)(2 71)(3 72)(4 73)(5 74)(6 75)(7 76)(8 77)(9 78)(10 79)(11 80)(12 81)(13 82)(14 83)(15 49)(16 50)(17 51)(18 52)(19 53)(20 54)(21 55)(22 56)(23 43)(24 44)(25 45)(26 46)(27 47)(28 48)(29 62)(30 63)(31 64)(32 65)(33 66)(34 67)(35 68)(36 69)(37 70)(38 57)(39 58)(40 59)(41 60)(42 61)(85 101)(86 102)(87 103)(88 104)(89 105)(90 106)(91 107)(92 108)(93 109)(94 110)(95 111)(96 112)(97 99)(98 100)
(1 57 45 112)(2 58 46 99)(3 59 47 100)(4 60 48 101)(5 61 49 102)(6 62 50 103)(7 63 51 104)(8 64 52 105)(9 65 53 106)(10 66 54 107)(11 67 55 108)(12 68 56 109)(13 69 43 110)(14 70 44 111)(15 86 74 42)(16 87 75 29)(17 88 76 30)(18 89 77 31)(19 90 78 32)(20 91 79 33)(21 92 80 34)(22 93 81 35)(23 94 82 36)(24 95 83 37)(25 96 84 38)(26 97 71 39)(27 98 72 40)(28 85 73 41)
(1 31 8 38)(2 30 9 37)(3 29 10 36)(4 42 11 35)(5 41 12 34)(6 40 13 33)(7 39 14 32)(15 101 22 108)(16 100 23 107)(17 99 24 106)(18 112 25 105)(19 111 26 104)(20 110 27 103)(21 109 28 102)(43 91 50 98)(44 90 51 97)(45 89 52 96)(46 88 53 95)(47 87 54 94)(48 86 55 93)(49 85 56 92)(57 84 64 77)(58 83 65 76)(59 82 66 75)(60 81 67 74)(61 80 68 73)(62 79 69 72)(63 78 70 71)
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), (1,112,45,57)(2,99,46,58)(3,100,47,59)(4,101,48,60)(5,102,49,61)(6,103,50,62)(7,104,51,63)(8,105,52,64)(9,106,53,65)(10,107,54,66)(11,108,55,67)(12,109,56,68)(13,110,43,69)(14,111,44,70)(15,86,74,42)(16,87,75,29)(17,88,76,30)(18,89,77,31)(19,90,78,32)(20,91,79,33)(21,92,80,34)(22,93,81,35)(23,94,82,36)(24,95,83,37)(25,96,84,38)(26,97,71,39)(27,98,72,40)(28,85,73,41), (1,84)(2,71)(3,72)(4,73)(5,74)(6,75)(7,76)(8,77)(9,78)(10,79)(11,80)(12,81)(13,82)(14,83)(15,49)(16,50)(17,51)(18,52)(19,53)(20,54)(21,55)(22,56)(23,43)(24,44)(25,45)(26,46)(27,47)(28,48)(29,62)(30,63)(31,64)(32,65)(33,66)(34,67)(35,68)(36,69)(37,70)(38,57)(39,58)(40,59)(41,60)(42,61)(85,101)(86,102)(87,103)(88,104)(89,105)(90,106)(91,107)(92,108)(93,109)(94,110)(95,111)(96,112)(97,99)(98,100), (1,57,45,112)(2,58,46,99)(3,59,47,100)(4,60,48,101)(5,61,49,102)(6,62,50,103)(7,63,51,104)(8,64,52,105)(9,65,53,106)(10,66,54,107)(11,67,55,108)(12,68,56,109)(13,69,43,110)(14,70,44,111)(15,86,74,42)(16,87,75,29)(17,88,76,30)(18,89,77,31)(19,90,78,32)(20,91,79,33)(21,92,80,34)(22,93,81,35)(23,94,82,36)(24,95,83,37)(25,96,84,38)(26,97,71,39)(27,98,72,40)(28,85,73,41), (1,31,8,38)(2,30,9,37)(3,29,10,36)(4,42,11,35)(5,41,12,34)(6,40,13,33)(7,39,14,32)(15,101,22,108)(16,100,23,107)(17,99,24,106)(18,112,25,105)(19,111,26,104)(20,110,27,103)(21,109,28,102)(43,91,50,98)(44,90,51,97)(45,89,52,96)(46,88,53,95)(47,87,54,94)(48,86,55,93)(49,85,56,92)(57,84,64,77)(58,83,65,76)(59,82,66,75)(60,81,67,74)(61,80,68,73)(62,79,69,72)(63,78,70,71)>;
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), (1,112,45,57)(2,99,46,58)(3,100,47,59)(4,101,48,60)(5,102,49,61)(6,103,50,62)(7,104,51,63)(8,105,52,64)(9,106,53,65)(10,107,54,66)(11,108,55,67)(12,109,56,68)(13,110,43,69)(14,111,44,70)(15,86,74,42)(16,87,75,29)(17,88,76,30)(18,89,77,31)(19,90,78,32)(20,91,79,33)(21,92,80,34)(22,93,81,35)(23,94,82,36)(24,95,83,37)(25,96,84,38)(26,97,71,39)(27,98,72,40)(28,85,73,41), (1,84)(2,71)(3,72)(4,73)(5,74)(6,75)(7,76)(8,77)(9,78)(10,79)(11,80)(12,81)(13,82)(14,83)(15,49)(16,50)(17,51)(18,52)(19,53)(20,54)(21,55)(22,56)(23,43)(24,44)(25,45)(26,46)(27,47)(28,48)(29,62)(30,63)(31,64)(32,65)(33,66)(34,67)(35,68)(36,69)(37,70)(38,57)(39,58)(40,59)(41,60)(42,61)(85,101)(86,102)(87,103)(88,104)(89,105)(90,106)(91,107)(92,108)(93,109)(94,110)(95,111)(96,112)(97,99)(98,100), (1,57,45,112)(2,58,46,99)(3,59,47,100)(4,60,48,101)(5,61,49,102)(6,62,50,103)(7,63,51,104)(8,64,52,105)(9,65,53,106)(10,66,54,107)(11,67,55,108)(12,68,56,109)(13,69,43,110)(14,70,44,111)(15,86,74,42)(16,87,75,29)(17,88,76,30)(18,89,77,31)(19,90,78,32)(20,91,79,33)(21,92,80,34)(22,93,81,35)(23,94,82,36)(24,95,83,37)(25,96,84,38)(26,97,71,39)(27,98,72,40)(28,85,73,41), (1,31,8,38)(2,30,9,37)(3,29,10,36)(4,42,11,35)(5,41,12,34)(6,40,13,33)(7,39,14,32)(15,101,22,108)(16,100,23,107)(17,99,24,106)(18,112,25,105)(19,111,26,104)(20,110,27,103)(21,109,28,102)(43,91,50,98)(44,90,51,97)(45,89,52,96)(46,88,53,95)(47,87,54,94)(48,86,55,93)(49,85,56,92)(57,84,64,77)(58,83,65,76)(59,82,66,75)(60,81,67,74)(61,80,68,73)(62,79,69,72)(63,78,70,71) );
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)], [(1,112,45,57),(2,99,46,58),(3,100,47,59),(4,101,48,60),(5,102,49,61),(6,103,50,62),(7,104,51,63),(8,105,52,64),(9,106,53,65),(10,107,54,66),(11,108,55,67),(12,109,56,68),(13,110,43,69),(14,111,44,70),(15,86,74,42),(16,87,75,29),(17,88,76,30),(18,89,77,31),(19,90,78,32),(20,91,79,33),(21,92,80,34),(22,93,81,35),(23,94,82,36),(24,95,83,37),(25,96,84,38),(26,97,71,39),(27,98,72,40),(28,85,73,41)], [(1,84),(2,71),(3,72),(4,73),(5,74),(6,75),(7,76),(8,77),(9,78),(10,79),(11,80),(12,81),(13,82),(14,83),(15,49),(16,50),(17,51),(18,52),(19,53),(20,54),(21,55),(22,56),(23,43),(24,44),(25,45),(26,46),(27,47),(28,48),(29,62),(30,63),(31,64),(32,65),(33,66),(34,67),(35,68),(36,69),(37,70),(38,57),(39,58),(40,59),(41,60),(42,61),(85,101),(86,102),(87,103),(88,104),(89,105),(90,106),(91,107),(92,108),(93,109),(94,110),(95,111),(96,112),(97,99),(98,100)], [(1,57,45,112),(2,58,46,99),(3,59,47,100),(4,60,48,101),(5,61,49,102),(6,62,50,103),(7,63,51,104),(8,64,52,105),(9,65,53,106),(10,66,54,107),(11,67,55,108),(12,68,56,109),(13,69,43,110),(14,70,44,111),(15,86,74,42),(16,87,75,29),(17,88,76,30),(18,89,77,31),(19,90,78,32),(20,91,79,33),(21,92,80,34),(22,93,81,35),(23,94,82,36),(24,95,83,37),(25,96,84,38),(26,97,71,39),(27,98,72,40),(28,85,73,41)], [(1,31,8,38),(2,30,9,37),(3,29,10,36),(4,42,11,35),(5,41,12,34),(6,40,13,33),(7,39,14,32),(15,101,22,108),(16,100,23,107),(17,99,24,106),(18,112,25,105),(19,111,26,104),(20,110,27,103),(21,109,28,102),(43,91,50,98),(44,90,51,97),(45,89,52,96),(46,88,53,95),(47,87,54,94),(48,86,55,93),(49,85,56,92),(57,84,64,77),(58,83,65,76),(59,82,66,75),(60,81,67,74),(61,80,68,73),(62,79,69,72),(63,78,70,71)])
Matrix representation ►G ⊆ GL4(𝔽29) generated by
| 28 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 1 | 8 |
| 0 | 0 | 12 | 10 |
| 12 | 0 | 0 | 0 |
| 0 | 17 | 0 | 0 |
| 0 | 0 | 20 | 16 |
| 0 | 0 | 24 | 9 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 17 | 0 | 0 | 0 |
| 0 | 17 | 0 | 0 |
| 0 | 0 | 20 | 16 |
| 0 | 0 | 24 | 9 |
| 0 | 12 | 0 | 0 |
| 12 | 0 | 0 | 0 |
| 0 | 0 | 9 | 1 |
| 0 | 0 | 5 | 20 |
G:=sub<GL(4,GF(29))| [28,0,0,0,0,28,0,0,0,0,1,12,0,0,8,10],[12,0,0,0,0,17,0,0,0,0,20,24,0,0,16,9],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[17,0,0,0,0,17,0,0,0,0,20,24,0,0,16,9],[0,12,0,0,12,0,0,0,0,0,9,5,0,0,1,20] >;
85 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 2L | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14AA | 28A | ··· | 28L | 28M | ··· | 28AD |
| order | 1 | 2 | 2 | 2 | 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 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 14 | 14 | 28 | 28 | 2 | 2 | 2 | 2 | 4 | 4 | 14 | 14 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
85 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D7 | C4○D4 | D14 | D14 | D14 | C7⋊D4 | 2+ (1+4) | D7×C4○D4 | D4⋊8D14 |
| kernel | C14.1452+ (1+4) | C2×D14⋊C4 | C4×C7⋊D4 | C23.23D14 | C28⋊7D4 | D4×Dic7 | C23⋊D14 | Dic7⋊D4 | D14⋊3Q8 | C28.23D4 | C2×D4×D7 | C14×C4○D4 | C7×D4 | C2×C4○D4 | D14 | C22×C4 | C2×D4 | C2×Q8 | D4 | C14 | C2 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 4 | 3 | 4 | 9 | 9 | 3 | 24 | 1 | 6 | 6 |
In GAP, Magma, Sage, TeX
C_{14}._{145}2_+^{(1+4)} % in TeX
G:=Group("C14.145ES+(2,2)"); // GroupNames label
G:=SmallGroup(448,1282);
// by ID
G=gap.SmallGroup(448,1282);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,387,675,136,18822]);
// 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,a*d=d*a,e*a*e^-1=a^-1,c*b*c=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