?
G = C14.402+ (1+4) order 448 = 26·7
40th non-split extension by C14 of 2+ (1+4) acting via 2+ (1+4)/C2×D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C14.402+ (1+4), C4⋊C4⋊6D14, C7⋊D4⋊2D4, (C2×D4)⋊24D14, C4⋊D4⋊13D7, C7⋊5(D4⋊5D4), D14⋊8(C4○D4), C22.6(D4×D7), C22⋊C4⋊27D14, D14.17(C2×D4), (C22×C4)⋊18D14, C23⋊D14⋊23C2, D14⋊D4⋊21C2, D14⋊C4⋊67C22, D14⋊Q8⋊14C2, Dic7⋊4D4⋊9C2, (D4×C14)⋊30C22, (C2×C28).41C23, Dic7.22(C2×D4), C14.69(C22×D4), Dic7⋊D4⋊30C2, D14.5D4⋊12C2, (C2×C14).154C24, Dic7⋊C4⋊16C22, (C22×C28)⋊40C22, (C4×Dic7)⋊54C22, C23.D7⋊52C22, C2.42(D4⋊6D14), Dic7.D4⋊19C2, (C2×Dic14)⋊25C22, (C2×D28).143C22, C23.18D14⋊9C2, (C22×C14).21C23, (C2×Dic7).74C23, (C23×D7).47C22, C22.175(C23×D7), C23.182(C22×D7), (C22×Dic7)⋊20C22, (C22×D7).188C23, (C2×D4×D7)⋊12C2, C2.42(C2×D4×D7), (C4×C7⋊D4)⋊54C2, (D7×C22⋊C4)⋊5C2, C2.39(D7×C4○D4), (C2×C14).6(C2×D4), (C2×C4×D7)⋊50C22, (C2×D14⋊C4)⋊36C2, (C7×C4⋊D4)⋊16C2, (C7×C4⋊C4)⋊12C22, (C2×D4⋊2D7)⋊14C2, C14.152(C2×C4○D4), (C2×C7⋊D4)⋊16C22, (C7×C22⋊C4)⋊14C22, (C2×C4).177(C22×D7), SmallGroup(448,1063)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1836 in 334 conjugacy classes, 105 normal (91 characteristic)
C1, C2 [×3], C2 [×9], C4 [×10], C22, C22 [×2], C22 [×27], C7, C2×C4 [×4], C2×C4 [×15], D4 [×18], Q8 [×2], C23 [×3], C23 [×13], D7 [×5], C14 [×3], C14 [×4], C42, C22⋊C4 [×2], C22⋊C4 [×10], C4⋊C4, C4⋊C4 [×3], C22×C4, C22×C4 [×5], C2×D4 [×3], C2×D4 [×10], C2×Q8, C4○D4 [×4], C24 [×2], Dic7 [×2], Dic7 [×4], C28 [×4], D14 [×4], D14 [×15], C2×C14, C2×C14 [×2], C2×C14 [×8], C2×C22⋊C4 [×2], C4×D4 [×2], C22≀C2 [×2], C4⋊D4, C4⋊D4 [×2], C22⋊Q8, C22.D4 [×2], C4.4D4, C22×D4, C2×C4○D4, Dic14 [×2], C4×D7 [×5], D28 [×2], C2×Dic7 [×5], C2×Dic7 [×4], C7⋊D4 [×4], C7⋊D4 [×7], C2×C28 [×4], C2×C28, C7×D4 [×5], C22×D7 [×3], C22×D7 [×10], C22×C14 [×3], D4⋊5D4, C4×Dic7, Dic7⋊C4 [×3], D14⋊C4 [×7], C23.D7 [×3], C7×C22⋊C4 [×2], C7×C4⋊C4, C2×Dic14, C2×C4×D7 [×3], C2×D28, D4×D7 [×4], D4⋊2D7 [×4], C22×Dic7 [×2], C2×C7⋊D4 [×5], C22×C28, D4×C14 [×3], C23×D7 [×2], D7×C22⋊C4, Dic7⋊4D4, D14⋊D4, Dic7.D4, D14.5D4, D14⋊Q8, C2×D14⋊C4, C4×C7⋊D4, C23.18D14, C23⋊D14 [×2], Dic7⋊D4, C7×C4⋊D4, C2×D4×D7, C2×D4⋊2D7, C14.402+ (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), C22×D7 [×7], D4⋊5D4, D4×D7 [×2], C23×D7, C2×D4×D7, D4⋊6D14, D7×C4○D4, C14.402+ (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=b-1, dbd-1=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 73 15 59)(2 74 16 60)(3 75 17 61)(4 76 18 62)(5 77 19 63)(6 78 20 64)(7 79 21 65)(8 80 22 66)(9 81 23 67)(10 82 24 68)(11 83 25 69)(12 84 26 70)(13 71 27 57)(14 72 28 58)(29 99 43 90)(30 100 44 91)(31 101 45 92)(32 102 46 93)(33 103 47 94)(34 104 48 95)(35 105 49 96)(36 106 50 97)(37 107 51 98)(38 108 52 85)(39 109 53 86)(40 110 54 87)(41 111 55 88)(42 112 56 89)
(1 66)(2 67)(3 68)(4 69)(5 70)(6 57)(7 58)(8 59)(9 60)(10 61)(11 62)(12 63)(13 64)(14 65)(15 80)(16 81)(17 82)(18 83)(19 84)(20 71)(21 72)(22 73)(23 74)(24 75)(25 76)(26 77)(27 78)(28 79)(29 90)(30 91)(31 92)(32 93)(33 94)(34 95)(35 96)(36 97)(37 98)(38 85)(39 86)(40 87)(41 88)(42 89)(43 99)(44 100)(45 101)(46 102)(47 103)(48 104)(49 105)(50 106)(51 107)(52 108)(53 109)(54 110)(55 111)(56 112)
(1 54 15 40)(2 53 16 39)(3 52 17 38)(4 51 18 37)(5 50 19 36)(6 49 20 35)(7 48 21 34)(8 47 22 33)(9 46 23 32)(10 45 24 31)(11 44 25 30)(12 43 26 29)(13 56 27 42)(14 55 28 41)(57 105 71 96)(58 104 72 95)(59 103 73 94)(60 102 74 93)(61 101 75 92)(62 100 76 91)(63 99 77 90)(64 112 78 89)(65 111 79 88)(66 110 80 87)(67 109 81 86)(68 108 82 85)(69 107 83 98)(70 106 84 97)
(1 40 8 33)(2 41 9 34)(3 42 10 35)(4 29 11 36)(5 30 12 37)(6 31 13 38)(7 32 14 39)(15 54 22 47)(16 55 23 48)(17 56 24 49)(18 43 25 50)(19 44 26 51)(20 45 27 52)(21 46 28 53)(57 92 64 85)(58 93 65 86)(59 94 66 87)(60 95 67 88)(61 96 68 89)(62 97 69 90)(63 98 70 91)(71 101 78 108)(72 102 79 109)(73 103 80 110)(74 104 81 111)(75 105 82 112)(76 106 83 99)(77 107 84 100)
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), (1,73,15,59)(2,74,16,60)(3,75,17,61)(4,76,18,62)(5,77,19,63)(6,78,20,64)(7,79,21,65)(8,80,22,66)(9,81,23,67)(10,82,24,68)(11,83,25,69)(12,84,26,70)(13,71,27,57)(14,72,28,58)(29,99,43,90)(30,100,44,91)(31,101,45,92)(32,102,46,93)(33,103,47,94)(34,104,48,95)(35,105,49,96)(36,106,50,97)(37,107,51,98)(38,108,52,85)(39,109,53,86)(40,110,54,87)(41,111,55,88)(42,112,56,89), (1,66)(2,67)(3,68)(4,69)(5,70)(6,57)(7,58)(8,59)(9,60)(10,61)(11,62)(12,63)(13,64)(14,65)(15,80)(16,81)(17,82)(18,83)(19,84)(20,71)(21,72)(22,73)(23,74)(24,75)(25,76)(26,77)(27,78)(28,79)(29,90)(30,91)(31,92)(32,93)(33,94)(34,95)(35,96)(36,97)(37,98)(38,85)(39,86)(40,87)(41,88)(42,89)(43,99)(44,100)(45,101)(46,102)(47,103)(48,104)(49,105)(50,106)(51,107)(52,108)(53,109)(54,110)(55,111)(56,112), (1,54,15,40)(2,53,16,39)(3,52,17,38)(4,51,18,37)(5,50,19,36)(6,49,20,35)(7,48,21,34)(8,47,22,33)(9,46,23,32)(10,45,24,31)(11,44,25,30)(12,43,26,29)(13,56,27,42)(14,55,28,41)(57,105,71,96)(58,104,72,95)(59,103,73,94)(60,102,74,93)(61,101,75,92)(62,100,76,91)(63,99,77,90)(64,112,78,89)(65,111,79,88)(66,110,80,87)(67,109,81,86)(68,108,82,85)(69,107,83,98)(70,106,84,97), (1,40,8,33)(2,41,9,34)(3,42,10,35)(4,29,11,36)(5,30,12,37)(6,31,13,38)(7,32,14,39)(15,54,22,47)(16,55,23,48)(17,56,24,49)(18,43,25,50)(19,44,26,51)(20,45,27,52)(21,46,28,53)(57,92,64,85)(58,93,65,86)(59,94,66,87)(60,95,67,88)(61,96,68,89)(62,97,69,90)(63,98,70,91)(71,101,78,108)(72,102,79,109)(73,103,80,110)(74,104,81,111)(75,105,82,112)(76,106,83,99)(77,107,84,100)>;
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), (1,73,15,59)(2,74,16,60)(3,75,17,61)(4,76,18,62)(5,77,19,63)(6,78,20,64)(7,79,21,65)(8,80,22,66)(9,81,23,67)(10,82,24,68)(11,83,25,69)(12,84,26,70)(13,71,27,57)(14,72,28,58)(29,99,43,90)(30,100,44,91)(31,101,45,92)(32,102,46,93)(33,103,47,94)(34,104,48,95)(35,105,49,96)(36,106,50,97)(37,107,51,98)(38,108,52,85)(39,109,53,86)(40,110,54,87)(41,111,55,88)(42,112,56,89), (1,66)(2,67)(3,68)(4,69)(5,70)(6,57)(7,58)(8,59)(9,60)(10,61)(11,62)(12,63)(13,64)(14,65)(15,80)(16,81)(17,82)(18,83)(19,84)(20,71)(21,72)(22,73)(23,74)(24,75)(25,76)(26,77)(27,78)(28,79)(29,90)(30,91)(31,92)(32,93)(33,94)(34,95)(35,96)(36,97)(37,98)(38,85)(39,86)(40,87)(41,88)(42,89)(43,99)(44,100)(45,101)(46,102)(47,103)(48,104)(49,105)(50,106)(51,107)(52,108)(53,109)(54,110)(55,111)(56,112), (1,54,15,40)(2,53,16,39)(3,52,17,38)(4,51,18,37)(5,50,19,36)(6,49,20,35)(7,48,21,34)(8,47,22,33)(9,46,23,32)(10,45,24,31)(11,44,25,30)(12,43,26,29)(13,56,27,42)(14,55,28,41)(57,105,71,96)(58,104,72,95)(59,103,73,94)(60,102,74,93)(61,101,75,92)(62,100,76,91)(63,99,77,90)(64,112,78,89)(65,111,79,88)(66,110,80,87)(67,109,81,86)(68,108,82,85)(69,107,83,98)(70,106,84,97), (1,40,8,33)(2,41,9,34)(3,42,10,35)(4,29,11,36)(5,30,12,37)(6,31,13,38)(7,32,14,39)(15,54,22,47)(16,55,23,48)(17,56,24,49)(18,43,25,50)(19,44,26,51)(20,45,27,52)(21,46,28,53)(57,92,64,85)(58,93,65,86)(59,94,66,87)(60,95,67,88)(61,96,68,89)(62,97,69,90)(63,98,70,91)(71,101,78,108)(72,102,79,109)(73,103,80,110)(74,104,81,111)(75,105,82,112)(76,106,83,99)(77,107,84,100) );
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)], [(1,73,15,59),(2,74,16,60),(3,75,17,61),(4,76,18,62),(5,77,19,63),(6,78,20,64),(7,79,21,65),(8,80,22,66),(9,81,23,67),(10,82,24,68),(11,83,25,69),(12,84,26,70),(13,71,27,57),(14,72,28,58),(29,99,43,90),(30,100,44,91),(31,101,45,92),(32,102,46,93),(33,103,47,94),(34,104,48,95),(35,105,49,96),(36,106,50,97),(37,107,51,98),(38,108,52,85),(39,109,53,86),(40,110,54,87),(41,111,55,88),(42,112,56,89)], [(1,66),(2,67),(3,68),(4,69),(5,70),(6,57),(7,58),(8,59),(9,60),(10,61),(11,62),(12,63),(13,64),(14,65),(15,80),(16,81),(17,82),(18,83),(19,84),(20,71),(21,72),(22,73),(23,74),(24,75),(25,76),(26,77),(27,78),(28,79),(29,90),(30,91),(31,92),(32,93),(33,94),(34,95),(35,96),(36,97),(37,98),(38,85),(39,86),(40,87),(41,88),(42,89),(43,99),(44,100),(45,101),(46,102),(47,103),(48,104),(49,105),(50,106),(51,107),(52,108),(53,109),(54,110),(55,111),(56,112)], [(1,54,15,40),(2,53,16,39),(3,52,17,38),(4,51,18,37),(5,50,19,36),(6,49,20,35),(7,48,21,34),(8,47,22,33),(9,46,23,32),(10,45,24,31),(11,44,25,30),(12,43,26,29),(13,56,27,42),(14,55,28,41),(57,105,71,96),(58,104,72,95),(59,103,73,94),(60,102,74,93),(61,101,75,92),(62,100,76,91),(63,99,77,90),(64,112,78,89),(65,111,79,88),(66,110,80,87),(67,109,81,86),(68,108,82,85),(69,107,83,98),(70,106,84,97)], [(1,40,8,33),(2,41,9,34),(3,42,10,35),(4,29,11,36),(5,30,12,37),(6,31,13,38),(7,32,14,39),(15,54,22,47),(16,55,23,48),(17,56,24,49),(18,43,25,50),(19,44,26,51),(20,45,27,52),(21,46,28,53),(57,92,64,85),(58,93,65,86),(59,94,66,87),(60,95,67,88),(61,96,68,89),(62,97,69,90),(63,98,70,91),(71,101,78,108),(72,102,79,109),(73,103,80,110),(74,104,81,111),(75,105,82,112),(76,106,83,99),(77,107,84,100)])
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 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 18 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 17 |
| 0 | 0 | 0 | 0 | 25 | 26 |
| 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 | 26 | 12 |
| 0 | 0 | 0 | 0 | 9 | 3 |
| 11 | 2 | 0 | 0 | 0 | 0 |
| 27 | 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 22 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 1 |
| 0 | 0 | 0 | 0 | 8 | 22 |
| 18 | 27 | 0 | 0 | 0 | 0 |
| 3 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 22 | 28 |
| 0 | 0 | 0 | 0 | 21 | 7 |
G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,10,19,0,0,0,0,10,22,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[1,18,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,3,25,0,0,0,0,17,26],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,26,9,0,0,0,0,12,3],[11,27,0,0,0,0,2,18,0,0,0,0,0,0,28,22,0,0,0,0,0,1,0,0,0,0,0,0,7,8,0,0,0,0,1,22],[18,3,0,0,0,0,27,11,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,22,21,0,0,0,0,28,7] >;
67 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 | ··· | 14O | 14P | ··· | 14U | 28A | ··· | 28L | 28M | ··· | 28R |
| 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 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 14 | 14 | 14 | 14 | 28 | 2 | 2 | 4 | 4 | 4 | 14 | 14 | 14 | 14 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 | 8 | ··· | 8 |
67 irreducible representations
| dim | 1 | 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 | C2 | D4 | D7 | C4○D4 | D14 | D14 | D14 | D14 | 2+ (1+4) | D4×D7 | D4⋊6D14 | D7×C4○D4 |
| kernel | C14.402+ (1+4) | D7×C22⋊C4 | Dic7⋊4D4 | D14⋊D4 | Dic7.D4 | D14.5D4 | D14⋊Q8 | C2×D14⋊C4 | C4×C7⋊D4 | C23.18D14 | C23⋊D14 | Dic7⋊D4 | C7×C4⋊D4 | C2×D4×D7 | C2×D4⋊2D7 | C7⋊D4 | C4⋊D4 | D14 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C14 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 4 | 3 | 4 | 6 | 3 | 3 | 9 | 1 | 6 | 6 | 6 |
In GAP, Magma, Sage, TeX
C_{14}._{40}2_+^{(1+4)} % in TeX
G:=Group("C14.40ES+(2,2)"); // GroupNames label
G:=SmallGroup(448,1063);
// by ID
G=gap.SmallGroup(448,1063);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,219,1571,297,136,18822]);
// 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=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=a^7*b^2*d>;
// generators/relations