?
G = C14.1202+ (1+4) order 448 = 26·7
29th non-split extension by C14 of 2+ (1+4) acting via 2+ (1+4)/C4○D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C14.1202+ (1+4), C4⋊C4⋊14D14, D14.5(C2×D4), (C22×D7)⋊8D4, C28⋊1D4⋊29C2, C22⋊C4⋊14D14, (C22×C4)⋊22D14, C7⋊4(C23⋊3D4), D14⋊D4⋊28C2, C22⋊D28⋊18C2, C23⋊D14⋊15C2, C22.44(D4×D7), D14⋊C4⋊21C22, (C2×D4).162D14, (C22×D28)⋊10C2, (C2×D28)⋊47C22, (C2×C28).70C23, Dic7⋊C4⋊4C22, C14.82(C22×D4), C22.D4⋊3D7, D14.5D4⋊26C2, (C2×C14).197C24, (C22×C28)⋊12C22, (C23×D7)⋊11C22, C2.40(D4⋊8D14), C23.D7⋊29C22, (D4×C14).135C22, C23.23D14⋊7C2, (C22×C14).32C23, C23.200(C22×D7), C22.218(C23×D7), (C2×Dic7).101C23, (C22×D7).205C23, (C2×D4×D7)⋊14C2, C2.55(C2×D4×D7), (C2×C4×D7)⋊20C22, (C7×C4⋊C4)⋊24C22, (D7×C22⋊C4)⋊11C2, (C2×C14).58(C2×D4), (C2×C7⋊D4)⋊18C22, (C7×C22⋊C4)⋊20C22, (C2×C4).189(C22×D7), (C7×C22.D4)⋊5C2, SmallGroup(448,1106)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 2220 in 346 conjugacy classes, 103 normal (39 characteristic)
C1, C2, C2 [×2], C2 [×10], C4 [×8], C22, C22 [×2], C22 [×34], C7, C2×C4, C2×C4 [×4], C2×C4 [×9], D4 [×20], C23 [×2], C23 [×19], D7 [×7], C14, C14 [×2], C14 [×3], C22⋊C4, C22⋊C4 [×2], C22⋊C4 [×9], C4⋊C4 [×2], C4⋊C4 [×2], C22×C4, C22×C4 [×3], C2×D4, C2×D4 [×19], C24 [×3], Dic7 [×3], C28 [×5], D14 [×4], D14 [×25], C2×C14, C2×C14 [×2], C2×C14 [×5], C2×C22⋊C4, C22≀C2 [×4], C4⋊D4 [×4], C22.D4, C22.D4 [×3], C22×D4 [×2], C4×D7 [×4], D28 [×12], C2×Dic7, C2×Dic7 [×2], C7⋊D4 [×6], C2×C28, C2×C28 [×4], C2×C28 [×2], C7×D4 [×2], C22×D7 [×3], C22×D7 [×6], C22×D7 [×10], C22×C14 [×2], C23⋊3D4, Dic7⋊C4 [×2], D14⋊C4 [×8], C23.D7, C7×C22⋊C4, C7×C22⋊C4 [×2], C7×C4⋊C4 [×2], C2×C4×D7, C2×C4×D7 [×2], C2×D28, C2×D28 [×6], C2×D28 [×4], D4×D7 [×4], C2×C7⋊D4 [×2], C2×C7⋊D4 [×2], C22×C28, D4×C14, C23×D7 [×3], D7×C22⋊C4, C22⋊D28, C22⋊D28 [×2], D14⋊D4 [×2], D14.5D4 [×2], C28⋊1D4 [×2], C23.23D14, C23⋊D14, C7×C22.D4, C22×D28, C2×D4×D7, C14.1202+ (1+4)
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D7, C2×D4 [×6], C24, D14 [×7], C22×D4, 2+ (1+4) [×2], C22×D7 [×7], C23⋊3D4, D4×D7 [×2], C23×D7, C2×D4×D7, D4⋊8D14 [×2], C14.1202+ (1+4)
Generators and relations
G = < a,b,c,d,e | a14=b4=c2=e2=1, d2=a7b2, ab=ba, ac=ca, ad=da, eae=a-1, cbc=a7b-1, dbd-1=ebe=a7b, cd=dc, ce=ec, ede=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 95 17 111)(2 96 18 112)(3 97 19 99)(4 98 20 100)(5 85 21 101)(6 86 22 102)(7 87 23 103)(8 88 24 104)(9 89 25 105)(10 90 26 106)(11 91 27 107)(12 92 28 108)(13 93 15 109)(14 94 16 110)(29 73 56 62)(30 74 43 63)(31 75 44 64)(32 76 45 65)(33 77 46 66)(34 78 47 67)(35 79 48 68)(36 80 49 69)(37 81 50 70)(38 82 51 57)(39 83 52 58)(40 84 53 59)(41 71 54 60)(42 72 55 61)
(57 75)(58 76)(59 77)(60 78)(61 79)(62 80)(63 81)(64 82)(65 83)(66 84)(67 71)(68 72)(69 73)(70 74)(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 44 24 38)(2 45 25 39)(3 46 26 40)(4 47 27 41)(5 48 28 42)(6 49 15 29)(7 50 16 30)(8 51 17 31)(9 52 18 32)(10 53 19 33)(11 54 20 34)(12 55 21 35)(13 56 22 36)(14 43 23 37)(57 104 75 95)(58 105 76 96)(59 106 77 97)(60 107 78 98)(61 108 79 85)(62 109 80 86)(63 110 81 87)(64 111 82 88)(65 112 83 89)(66 99 84 90)(67 100 71 91)(68 101 72 92)(69 102 73 93)(70 103 74 94)
(1 31)(2 30)(3 29)(4 42)(5 41)(6 40)(7 39)(8 38)(9 37)(10 36)(11 35)(12 34)(13 33)(14 32)(15 46)(16 45)(17 44)(18 43)(19 56)(20 55)(21 54)(22 53)(23 52)(24 51)(25 50)(26 49)(27 48)(28 47)(57 111)(58 110)(59 109)(60 108)(61 107)(62 106)(63 105)(64 104)(65 103)(66 102)(67 101)(68 100)(69 99)(70 112)(71 92)(72 91)(73 90)(74 89)(75 88)(76 87)(77 86)(78 85)(79 98)(80 97)(81 96)(82 95)(83 94)(84 93)
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), (1,95,17,111)(2,96,18,112)(3,97,19,99)(4,98,20,100)(5,85,21,101)(6,86,22,102)(7,87,23,103)(8,88,24,104)(9,89,25,105)(10,90,26,106)(11,91,27,107)(12,92,28,108)(13,93,15,109)(14,94,16,110)(29,73,56,62)(30,74,43,63)(31,75,44,64)(32,76,45,65)(33,77,46,66)(34,78,47,67)(35,79,48,68)(36,80,49,69)(37,81,50,70)(38,82,51,57)(39,83,52,58)(40,84,53,59)(41,71,54,60)(42,72,55,61), (57,75)(58,76)(59,77)(60,78)(61,79)(62,80)(63,81)(64,82)(65,83)(66,84)(67,71)(68,72)(69,73)(70,74)(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), (1,44,24,38)(2,45,25,39)(3,46,26,40)(4,47,27,41)(5,48,28,42)(6,49,15,29)(7,50,16,30)(8,51,17,31)(9,52,18,32)(10,53,19,33)(11,54,20,34)(12,55,21,35)(13,56,22,36)(14,43,23,37)(57,104,75,95)(58,105,76,96)(59,106,77,97)(60,107,78,98)(61,108,79,85)(62,109,80,86)(63,110,81,87)(64,111,82,88)(65,112,83,89)(66,99,84,90)(67,100,71,91)(68,101,72,92)(69,102,73,93)(70,103,74,94), (1,31)(2,30)(3,29)(4,42)(5,41)(6,40)(7,39)(8,38)(9,37)(10,36)(11,35)(12,34)(13,33)(14,32)(15,46)(16,45)(17,44)(18,43)(19,56)(20,55)(21,54)(22,53)(23,52)(24,51)(25,50)(26,49)(27,48)(28,47)(57,111)(58,110)(59,109)(60,108)(61,107)(62,106)(63,105)(64,104)(65,103)(66,102)(67,101)(68,100)(69,99)(70,112)(71,92)(72,91)(73,90)(74,89)(75,88)(76,87)(77,86)(78,85)(79,98)(80,97)(81,96)(82,95)(83,94)(84,93)>;
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), (1,95,17,111)(2,96,18,112)(3,97,19,99)(4,98,20,100)(5,85,21,101)(6,86,22,102)(7,87,23,103)(8,88,24,104)(9,89,25,105)(10,90,26,106)(11,91,27,107)(12,92,28,108)(13,93,15,109)(14,94,16,110)(29,73,56,62)(30,74,43,63)(31,75,44,64)(32,76,45,65)(33,77,46,66)(34,78,47,67)(35,79,48,68)(36,80,49,69)(37,81,50,70)(38,82,51,57)(39,83,52,58)(40,84,53,59)(41,71,54,60)(42,72,55,61), (57,75)(58,76)(59,77)(60,78)(61,79)(62,80)(63,81)(64,82)(65,83)(66,84)(67,71)(68,72)(69,73)(70,74)(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), (1,44,24,38)(2,45,25,39)(3,46,26,40)(4,47,27,41)(5,48,28,42)(6,49,15,29)(7,50,16,30)(8,51,17,31)(9,52,18,32)(10,53,19,33)(11,54,20,34)(12,55,21,35)(13,56,22,36)(14,43,23,37)(57,104,75,95)(58,105,76,96)(59,106,77,97)(60,107,78,98)(61,108,79,85)(62,109,80,86)(63,110,81,87)(64,111,82,88)(65,112,83,89)(66,99,84,90)(67,100,71,91)(68,101,72,92)(69,102,73,93)(70,103,74,94), (1,31)(2,30)(3,29)(4,42)(5,41)(6,40)(7,39)(8,38)(9,37)(10,36)(11,35)(12,34)(13,33)(14,32)(15,46)(16,45)(17,44)(18,43)(19,56)(20,55)(21,54)(22,53)(23,52)(24,51)(25,50)(26,49)(27,48)(28,47)(57,111)(58,110)(59,109)(60,108)(61,107)(62,106)(63,105)(64,104)(65,103)(66,102)(67,101)(68,100)(69,99)(70,112)(71,92)(72,91)(73,90)(74,89)(75,88)(76,87)(77,86)(78,85)(79,98)(80,97)(81,96)(82,95)(83,94)(84,93) );
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)], [(1,95,17,111),(2,96,18,112),(3,97,19,99),(4,98,20,100),(5,85,21,101),(6,86,22,102),(7,87,23,103),(8,88,24,104),(9,89,25,105),(10,90,26,106),(11,91,27,107),(12,92,28,108),(13,93,15,109),(14,94,16,110),(29,73,56,62),(30,74,43,63),(31,75,44,64),(32,76,45,65),(33,77,46,66),(34,78,47,67),(35,79,48,68),(36,80,49,69),(37,81,50,70),(38,82,51,57),(39,83,52,58),(40,84,53,59),(41,71,54,60),(42,72,55,61)], [(57,75),(58,76),(59,77),(60,78),(61,79),(62,80),(63,81),(64,82),(65,83),(66,84),(67,71),(68,72),(69,73),(70,74),(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)], [(1,44,24,38),(2,45,25,39),(3,46,26,40),(4,47,27,41),(5,48,28,42),(6,49,15,29),(7,50,16,30),(8,51,17,31),(9,52,18,32),(10,53,19,33),(11,54,20,34),(12,55,21,35),(13,56,22,36),(14,43,23,37),(57,104,75,95),(58,105,76,96),(59,106,77,97),(60,107,78,98),(61,108,79,85),(62,109,80,86),(63,110,81,87),(64,111,82,88),(65,112,83,89),(66,99,84,90),(67,100,71,91),(68,101,72,92),(69,102,73,93),(70,103,74,94)], [(1,31),(2,30),(3,29),(4,42),(5,41),(6,40),(7,39),(8,38),(9,37),(10,36),(11,35),(12,34),(13,33),(14,32),(15,46),(16,45),(17,44),(18,43),(19,56),(20,55),(21,54),(22,53),(23,52),(24,51),(25,50),(26,49),(27,48),(28,47),(57,111),(58,110),(59,109),(60,108),(61,107),(62,106),(63,105),(64,104),(65,103),(66,102),(67,101),(68,100),(69,99),(70,112),(71,92),(72,91),(73,90),(74,89),(75,88),(76,87),(77,86),(78,85),(79,98),(80,97),(81,96),(82,95),(83,94),(84,93)])
Matrix representation ►G ⊆ GL6(𝔽29)
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 22 | 4 | 0 | 0 |
| 0 | 0 | 7 | 0 | 0 | 0 |
| 0 | 0 | 14 | 25 | 5 | 4 |
| 0 | 0 | 4 | 27 | 21 | 17 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 17 | 21 | 15 | 21 |
| 0 | 0 | 16 | 21 | 3 | 25 |
| 0 | 0 | 11 | 0 | 3 | 16 |
| 0 | 0 | 8 | 20 | 20 | 17 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 13 | 24 | 28 | 0 |
| 0 | 0 | 25 | 14 | 0 | 28 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 21 | 5 | 0 | 0 |
| 0 | 0 | 16 | 8 | 0 | 0 |
| 0 | 0 | 16 | 1 | 22 | 24 |
| 0 | 0 | 19 | 22 | 10 | 7 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 23 | 26 | 0 | 0 |
| 0 | 0 | 2 | 6 | 0 | 0 |
| 0 | 0 | 8 | 19 | 12 | 7 |
| 0 | 0 | 25 | 25 | 21 | 17 |
G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,22,7,14,4,0,0,4,0,25,27,0,0,0,0,5,21,0,0,0,0,4,17],[0,1,0,0,0,0,28,0,0,0,0,0,0,0,17,16,11,8,0,0,21,21,0,20,0,0,15,3,3,20,0,0,21,25,16,17],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,13,25,0,0,0,1,24,14,0,0,0,0,28,0,0,0,0,0,0,28],[0,28,0,0,0,0,28,0,0,0,0,0,0,0,21,16,16,19,0,0,5,8,1,22,0,0,0,0,22,10,0,0,0,0,24,7],[0,28,0,0,0,0,28,0,0,0,0,0,0,0,23,2,8,25,0,0,26,6,19,25,0,0,0,0,12,21,0,0,0,0,7,17] >;
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 4A | ··· | 4E | 4F | 4G | 4H | 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 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | 14 | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 14 | 14 | 14 | 14 | 28 | 28 | 28 | 4 | ··· | 4 | 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 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D7 | D14 | D14 | D14 | D14 | 2+ (1+4) | D4×D7 | D4⋊8D14 |
| kernel | C14.1202+ (1+4) | D7×C22⋊C4 | C22⋊D28 | D14⋊D4 | D14.5D4 | C28⋊1D4 | C23.23D14 | C23⋊D14 | C7×C22.D4 | C22×D28 | C2×D4×D7 | C22×D7 | C22.D4 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C14 | C22 | C2 |
| # reps | 1 | 1 | 3 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 4 | 3 | 9 | 6 | 3 | 3 | 2 | 6 | 12 |
In GAP, Magma, Sage, TeX
C_{14}._{120}2_+^{(1+4)} % in TeX
G:=Group("C14.120ES+(2,2)"); // GroupNames label
G:=SmallGroup(448,1106);
// by ID
G=gap.SmallGroup(448,1106);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,219,675,297,80,18822]);
// 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,a*d=d*a,e*a*e=a^-1,c*b*c=a^7*b^-1,d*b*d^-1=e*b*e=a^7*b,c*d=d*c,c*e=e*c,e*d*e=a^7*b^2*d>;
// generators/relations