?
G = C14.532+ (1+4) order 448 = 26·7
53rd non-split extension by C14 of 2+ (1+4) acting via 2+ (1+4)/C2×D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C14.532+ (1+4), C4⋊C4⋊12D14, (C2×Q8)⋊6D14, C22⋊Q8⋊14D7, D28⋊C4⋊28C2, (Q8×C14)⋊9C22, D14⋊C4⋊68C22, D14⋊3Q8⋊19C2, C22⋊D28.2C2, (C2×C28).60C23, C4⋊Dic7⋊37C22, C22⋊C4.62D14, D14.19(C4○D4), D14.5D4⋊19C2, C28.23D4⋊14C2, (C2×C14).181C24, Dic7⋊C4⋊19C22, (C4×Dic7)⋊29C22, (C22×C4).243D14, C2.55(D4⋊6D14), C7⋊5(C22.45C24), (C2×D28).150C22, C22.D28⋊16C2, C23.11D14⋊8C2, C22.9(Q8⋊2D7), (C2×Dic7).92C23, (C23×D7).54C22, (C22×D7).74C23, C22.202(C23×D7), C23.194(C22×D7), (C22×C14).209C23, (C22×C28).381C22, C23.D7.121C22, (C22×Dic7).122C22, (C4×C7⋊D4)⋊57C2, (D7×C22⋊C4)⋊9C2, C2.52(D7×C4○D4), (C2×C4×D7)⋊51C22, (C2×D14⋊C4)⋊37C2, C4⋊C4⋊7D7⋊26C2, C4⋊C4⋊D7⋊17C2, (C7×C4⋊C4)⋊21C22, (C7×C22⋊Q8)⋊17C2, C14.164(C2×C4○D4), C2.18(C2×Q8⋊2D7), (C2×C4).51(C22×D7), (C2×C14).26(C4○D4), (C2×C7⋊D4).128C22, (C7×C22⋊C4).36C22, SmallGroup(448,1090)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1276 in 248 conjugacy classes, 97 normal (91 characteristic)
C1, C2 [×3], C2 [×6], C4 [×11], C22, C22 [×2], C22 [×16], C7, C2×C4 [×6], C2×C4 [×12], D4 [×5], Q8, C23, C23 [×8], D7 [×4], C14 [×3], C14 [×2], C42 [×3], C22⋊C4 [×2], C22⋊C4 [×12], C4⋊C4 [×3], C4⋊C4 [×5], C22×C4, C22×C4 [×4], C2×D4 [×3], C2×Q8, C24, Dic7 [×5], C28 [×6], D14 [×2], D14 [×12], C2×C14, C2×C14 [×2], C2×C14 [×2], C2×C22⋊C4 [×2], C42⋊C2 [×2], C4×D4 [×2], C22≀C2, C22⋊Q8, C22⋊Q8, C22.D4 [×3], C4.4D4, C42⋊2C2 [×2], C4×D7 [×4], D28 [×3], C2×Dic7 [×5], C2×Dic7 [×2], C7⋊D4 [×2], C2×C28 [×6], C2×C28, C7×Q8, C22×D7 [×3], C22×D7 [×5], C22×C14, C22.45C24, C4×Dic7 [×3], Dic7⋊C4 [×3], C4⋊Dic7 [×2], D14⋊C4 [×11], C23.D7, C7×C22⋊C4 [×2], C7×C4⋊C4 [×3], C2×C4×D7 [×3], C2×D28 [×2], C22×Dic7, C2×C7⋊D4, C22×C28, Q8×C14, C23×D7, C23.11D14, D7×C22⋊C4, C22⋊D28, C22.D28, C4⋊C4⋊7D7, D28⋊C4, D14.5D4 [×2], C4⋊C4⋊D7 [×2], C2×D14⋊C4, C4×C7⋊D4, D14⋊3Q8, C28.23D4, C7×C22⋊Q8, C14.532+ (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, Q8⋊2D7 [×2], C23×D7, D4⋊6D14, C2×Q8⋊2D7, D7×C4○D4, C14.532+ (1+4)
Generators and relations
G = < a,b,c,d,e | a14=b4=c2=1, d2=a7b2, e2=a7, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=a7b-1, dbd-1=ebe-1=a7b, cd=dc, ce=ec, ede-1=b2d >
►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 58 23 73)(2 59 24 74)(3 60 25 75)(4 61 26 76)(5 62 27 77)(6 63 28 78)(7 64 15 79)(8 65 16 80)(9 66 17 81)(10 67 18 82)(11 68 19 83)(12 69 20 84)(13 70 21 71)(14 57 22 72)(29 91 47 101)(30 92 48 102)(31 93 49 103)(32 94 50 104)(33 95 51 105)(34 96 52 106)(35 97 53 107)(36 98 54 108)(37 85 55 109)(38 86 56 110)(39 87 43 111)(40 88 44 112)(41 89 45 99)(42 90 46 100)
(57 79)(58 80)(59 81)(60 82)(61 83)(62 84)(63 71)(64 72)(65 73)(66 74)(67 75)(68 76)(69 77)(70 78)(85 102)(86 103)(87 104)(88 105)(89 106)(90 107)(91 108)(92 109)(93 110)(94 111)(95 112)(96 99)(97 100)(98 101)
(1 56 16 31)(2 55 17 30)(3 54 18 29)(4 53 19 42)(5 52 20 41)(6 51 21 40)(7 50 22 39)(8 49 23 38)(9 48 24 37)(10 47 25 36)(11 46 26 35)(12 45 27 34)(13 44 28 33)(14 43 15 32)(57 104 79 87)(58 103 80 86)(59 102 81 85)(60 101 82 98)(61 100 83 97)(62 99 84 96)(63 112 71 95)(64 111 72 94)(65 110 73 93)(66 109 74 92)(67 108 75 91)(68 107 76 90)(69 106 77 89)(70 105 78 88)
(1 31 8 38)(2 32 9 39)(3 33 10 40)(4 34 11 41)(5 35 12 42)(6 36 13 29)(7 37 14 30)(15 55 22 48)(16 56 23 49)(17 43 24 50)(18 44 25 51)(19 45 26 52)(20 46 27 53)(21 47 28 54)(57 85 64 92)(58 86 65 93)(59 87 66 94)(60 88 67 95)(61 89 68 96)(62 90 69 97)(63 91 70 98)(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), (1,58,23,73)(2,59,24,74)(3,60,25,75)(4,61,26,76)(5,62,27,77)(6,63,28,78)(7,64,15,79)(8,65,16,80)(9,66,17,81)(10,67,18,82)(11,68,19,83)(12,69,20,84)(13,70,21,71)(14,57,22,72)(29,91,47,101)(30,92,48,102)(31,93,49,103)(32,94,50,104)(33,95,51,105)(34,96,52,106)(35,97,53,107)(36,98,54,108)(37,85,55,109)(38,86,56,110)(39,87,43,111)(40,88,44,112)(41,89,45,99)(42,90,46,100), (57,79)(58,80)(59,81)(60,82)(61,83)(62,84)(63,71)(64,72)(65,73)(66,74)(67,75)(68,76)(69,77)(70,78)(85,102)(86,103)(87,104)(88,105)(89,106)(90,107)(91,108)(92,109)(93,110)(94,111)(95,112)(96,99)(97,100)(98,101), (1,56,16,31)(2,55,17,30)(3,54,18,29)(4,53,19,42)(5,52,20,41)(6,51,21,40)(7,50,22,39)(8,49,23,38)(9,48,24,37)(10,47,25,36)(11,46,26,35)(12,45,27,34)(13,44,28,33)(14,43,15,32)(57,104,79,87)(58,103,80,86)(59,102,81,85)(60,101,82,98)(61,100,83,97)(62,99,84,96)(63,112,71,95)(64,111,72,94)(65,110,73,93)(66,109,74,92)(67,108,75,91)(68,107,76,90)(69,106,77,89)(70,105,78,88), (1,31,8,38)(2,32,9,39)(3,33,10,40)(4,34,11,41)(5,35,12,42)(6,36,13,29)(7,37,14,30)(15,55,22,48)(16,56,23,49)(17,43,24,50)(18,44,25,51)(19,45,26,52)(20,46,27,53)(21,47,28,54)(57,85,64,92)(58,86,65,93)(59,87,66,94)(60,88,67,95)(61,89,68,96)(62,90,69,97)(63,91,70,98)(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), (1,58,23,73)(2,59,24,74)(3,60,25,75)(4,61,26,76)(5,62,27,77)(6,63,28,78)(7,64,15,79)(8,65,16,80)(9,66,17,81)(10,67,18,82)(11,68,19,83)(12,69,20,84)(13,70,21,71)(14,57,22,72)(29,91,47,101)(30,92,48,102)(31,93,49,103)(32,94,50,104)(33,95,51,105)(34,96,52,106)(35,97,53,107)(36,98,54,108)(37,85,55,109)(38,86,56,110)(39,87,43,111)(40,88,44,112)(41,89,45,99)(42,90,46,100), (57,79)(58,80)(59,81)(60,82)(61,83)(62,84)(63,71)(64,72)(65,73)(66,74)(67,75)(68,76)(69,77)(70,78)(85,102)(86,103)(87,104)(88,105)(89,106)(90,107)(91,108)(92,109)(93,110)(94,111)(95,112)(96,99)(97,100)(98,101), (1,56,16,31)(2,55,17,30)(3,54,18,29)(4,53,19,42)(5,52,20,41)(6,51,21,40)(7,50,22,39)(8,49,23,38)(9,48,24,37)(10,47,25,36)(11,46,26,35)(12,45,27,34)(13,44,28,33)(14,43,15,32)(57,104,79,87)(58,103,80,86)(59,102,81,85)(60,101,82,98)(61,100,83,97)(62,99,84,96)(63,112,71,95)(64,111,72,94)(65,110,73,93)(66,109,74,92)(67,108,75,91)(68,107,76,90)(69,106,77,89)(70,105,78,88), (1,31,8,38)(2,32,9,39)(3,33,10,40)(4,34,11,41)(5,35,12,42)(6,36,13,29)(7,37,14,30)(15,55,22,48)(16,56,23,49)(17,43,24,50)(18,44,25,51)(19,45,26,52)(20,46,27,53)(21,47,28,54)(57,85,64,92)(58,86,65,93)(59,87,66,94)(60,88,67,95)(61,89,68,96)(62,90,69,97)(63,91,70,98)(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)], [(1,58,23,73),(2,59,24,74),(3,60,25,75),(4,61,26,76),(5,62,27,77),(6,63,28,78),(7,64,15,79),(8,65,16,80),(9,66,17,81),(10,67,18,82),(11,68,19,83),(12,69,20,84),(13,70,21,71),(14,57,22,72),(29,91,47,101),(30,92,48,102),(31,93,49,103),(32,94,50,104),(33,95,51,105),(34,96,52,106),(35,97,53,107),(36,98,54,108),(37,85,55,109),(38,86,56,110),(39,87,43,111),(40,88,44,112),(41,89,45,99),(42,90,46,100)], [(57,79),(58,80),(59,81),(60,82),(61,83),(62,84),(63,71),(64,72),(65,73),(66,74),(67,75),(68,76),(69,77),(70,78),(85,102),(86,103),(87,104),(88,105),(89,106),(90,107),(91,108),(92,109),(93,110),(94,111),(95,112),(96,99),(97,100),(98,101)], [(1,56,16,31),(2,55,17,30),(3,54,18,29),(4,53,19,42),(5,52,20,41),(6,51,21,40),(7,50,22,39),(8,49,23,38),(9,48,24,37),(10,47,25,36),(11,46,26,35),(12,45,27,34),(13,44,28,33),(14,43,15,32),(57,104,79,87),(58,103,80,86),(59,102,81,85),(60,101,82,98),(61,100,83,97),(62,99,84,96),(63,112,71,95),(64,111,72,94),(65,110,73,93),(66,109,74,92),(67,108,75,91),(68,107,76,90),(69,106,77,89),(70,105,78,88)], [(1,31,8,38),(2,32,9,39),(3,33,10,40),(4,34,11,41),(5,35,12,42),(6,36,13,29),(7,37,14,30),(15,55,22,48),(16,56,23,49),(17,43,24,50),(18,44,25,51),(19,45,26,52),(20,46,27,53),(21,47,28,54),(57,85,64,92),(58,86,65,93),(59,87,66,94),(60,88,67,95),(61,89,68,96),(62,90,69,97),(63,91,70,98),(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 | 21 | 0 | 0 | 0 | 0 |
| 17 | 19 | 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 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 17 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 27 |
| 0 | 0 | 0 | 0 | 28 | 12 |
| 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 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 17 | 28 |
| 1 | 8 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 1 | 17 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 0 |
| 0 | 0 | 0 | 0 | 28 | 12 |
G:=sub<GL(6,GF(29))| [28,17,0,0,0,0,21,19,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,12,0,0,0,0,0,0,17,0,0,0,0,0,0,17,28,0,0,0,0,27,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,17,0,0,0,0,0,28],[1,0,0,0,0,0,8,28,0,0,0,0,0,0,0,28,0,0,0,0,28,0,0,0,0,0,0,0,12,1,0,0,0,0,0,17],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,28,0,0,0,0,1,0,0,0,0,0,0,0,17,28,0,0,0,0,0,12] >;
67 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | ··· | 4G | 4H | ··· | 4M | 4N | 4O | 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 | 14 | 14 | 28 | 28 | 2 | 2 | 4 | ··· | 4 | 14 | ··· | 14 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
67 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 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 | D7 | C4○D4 | C4○D4 | D14 | D14 | D14 | D14 | 2+ (1+4) | Q8⋊2D7 | D4⋊6D14 | D7×C4○D4 |
| kernel | C14.532+ (1+4) | C23.11D14 | D7×C22⋊C4 | C22⋊D28 | C22.D28 | C4⋊C4⋊7D7 | D28⋊C4 | D14.5D4 | C4⋊C4⋊D7 | C2×D14⋊C4 | C4×C7⋊D4 | D14⋊3Q8 | C28.23D4 | C7×C22⋊Q8 | C22⋊Q8 | D14 | C2×C14 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×Q8 | C14 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 3 | 4 | 4 | 6 | 9 | 3 | 3 | 1 | 6 | 6 | 6 |
In GAP, Magma, Sage, TeX
C_{14}._{53}2_+^{(1+4)} % in TeX
G:=Group("C14.53ES+(2,2)"); // GroupNames label
G:=SmallGroup(448,1090);
// by ID
G=gap.SmallGroup(448,1090);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,219,184,1571,297,136,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^14=b^4=c^2=1,d^2=a^7*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,d*b*d^-1=e*b*e^-1=a^7*b,c*d=d*c,c*e=e*c,e*d*e^-1=b^2*d>;
// generators/relations