metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24⋊7D14, C14.892+ 1+4, (C2×D4)⋊40D14, (C22×D4)⋊10D7, (C22×C4)⋊28D14, (C22×C14)⋊13D4, C7⋊5(C23⋊3D4), C23⋊4(C7⋊D4), C23⋊D14⋊30C2, D14⋊C4⋊36C22, (D4×C14)⋊58C22, C24⋊D7⋊12C2, Dic7⋊D4⋊41C2, (C2×C14).299C24, (C2×C28).644C23, Dic7⋊C4⋊38C22, (C23×C14)⋊14C22, (C22×C28)⋊44C22, C14.146(C22×D4), (C23×D7)⋊15C22, C2.92(D4⋊6D14), C23.D7⋊64C22, C22.312(C23×D7), C23.206(C22×D7), C23.18D14⋊29C2, C23.23D14⋊28C2, (C22×C14).233C23, (C2×Dic7).154C23, (C22×Dic7)⋊34C22, (C22×D7).130C23, (D4×C2×C14)⋊17C2, (C2×C14).582(C2×D4), (C2×C7⋊D4)⋊48C22, (C22×C7⋊D4)⋊17C2, (C2×C23.D7)⋊30C2, C22.20(C2×C7⋊D4), C2.19(C22×C7⋊D4), (C2×C4).238(C22×D7), SmallGroup(448,1257)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24⋊7D14
G = < a,b,c,d,e,f | a2=b2=c2=d2=e14=f2=1, ab=ba, ac=ca, faf=ad=da, ae=ea, bc=cb, ebe-1=bd=db, fbf=bcd, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=e-1 >
Subgroups: 1620 in 346 conjugacy classes, 111 normal (25 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C7, C2×C4, C2×C4, D4, C23, C23, C23, D7, C14, C14, C14, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, C24, Dic7, C28, D14, C2×C14, C2×C14, C2×C14, C2×C22⋊C4, C22≀C2, C4⋊D4, C22.D4, C22×D4, C22×D4, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×D4, C22×D7, C22×D7, C22×C14, C22×C14, C22×C14, C23⋊3D4, Dic7⋊C4, D14⋊C4, C23.D7, C22×Dic7, C22×Dic7, C2×C7⋊D4, C2×C7⋊D4, C22×C28, D4×C14, D4×C14, C23×D7, C23×C14, C23.23D14, C23.18D14, C23⋊D14, Dic7⋊D4, C2×C23.D7, C24⋊D7, C22×C7⋊D4, D4×C2×C14, C24⋊7D14
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C24, D14, C22×D4, 2+ 1+4, C7⋊D4, C22×D7, C23⋊3D4, C2×C7⋊D4, C23×D7, D4⋊6D14, C22×C7⋊D4, C24⋊7D14
►On 112 points
Generators in S112
(1 19)(2 20)(3 21)(4 15)(5 16)(6 17)(7 18)(8 22)(9 23)(10 24)(11 25)(12 26)(13 27)(14 28)(29 47)(30 48)(31 49)(32 43)(33 44)(34 45)(35 46)(36 56)(37 50)(38 51)(39 52)(40 53)(41 54)(42 55)(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)(85 109)(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 95)(2 89)(3 97)(4 91)(5 85)(6 93)(7 87)(8 98)(9 92)(10 86)(11 94)(12 88)(13 96)(14 90)(15 101)(16 109)(17 103)(18 111)(19 105)(20 99)(21 107)(22 108)(23 102)(24 110)(25 104)(26 112)(27 106)(28 100)(29 73)(30 81)(31 75)(32 83)(33 77)(34 71)(35 79)(36 72)(37 80)(38 74)(39 82)(40 76)(41 84)(42 78)(43 65)(44 59)(45 67)(46 61)(47 69)(48 63)(49 57)(50 62)(51 70)(52 64)(53 58)(54 66)(55 60)(56 68)
(1 19)(2 20)(3 21)(4 15)(5 16)(6 17)(7 18)(8 22)(9 23)(10 24)(11 25)(12 26)(13 27)(14 28)(29 50)(30 51)(31 52)(32 53)(33 54)(34 55)(35 56)(36 46)(37 47)(38 48)(39 49)(40 43)(41 44)(42 45)(57 82)(58 83)(59 84)(60 71)(61 72)(62 73)(63 74)(64 75)(65 76)(66 77)(67 78)(68 79)(69 80)(70 81)(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 12)(2 13)(3 14)(4 8)(5 9)(6 10)(7 11)(15 22)(16 23)(17 24)(18 25)(19 26)(20 27)(21 28)(29 37)(30 38)(31 39)(32 40)(33 41)(34 42)(35 36)(43 53)(44 54)(45 55)(46 56)(47 50)(48 51)(49 52)(57 64)(58 65)(59 66)(60 67)(61 68)(62 69)(63 70)(71 78)(72 79)(73 80)(74 81)(75 82)(76 83)(77 84)(85 92)(86 93)(87 94)(88 95)(89 96)(90 97)(91 98)(99 106)(100 107)(101 108)(102 109)(103 110)(104 111)(105 112)
(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 50)(2 56)(3 55)(4 54)(5 53)(6 52)(7 51)(8 44)(9 43)(10 49)(11 48)(12 47)(13 46)(14 45)(15 33)(16 32)(17 31)(18 30)(19 29)(20 35)(21 34)(22 41)(23 40)(24 39)(25 38)(26 37)(27 36)(28 42)(57 103)(58 102)(59 101)(60 100)(61 99)(62 112)(63 111)(64 110)(65 109)(66 108)(67 107)(68 106)(69 105)(70 104)(71 90)(72 89)(73 88)(74 87)(75 86)(76 85)(77 98)(78 97)(79 96)(80 95)(81 94)(82 93)(83 92)(84 91)
G:=sub<Sym(112)| (1,19)(2,20)(3,21)(4,15)(5,16)(6,17)(7,18)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(29,47)(30,48)(31,49)(32,43)(33,44)(34,45)(35,46)(36,56)(37,50)(38,51)(39,52)(40,53)(41,54)(42,55)(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)(85,109)(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), (1,95)(2,89)(3,97)(4,91)(5,85)(6,93)(7,87)(8,98)(9,92)(10,86)(11,94)(12,88)(13,96)(14,90)(15,101)(16,109)(17,103)(18,111)(19,105)(20,99)(21,107)(22,108)(23,102)(24,110)(25,104)(26,112)(27,106)(28,100)(29,73)(30,81)(31,75)(32,83)(33,77)(34,71)(35,79)(36,72)(37,80)(38,74)(39,82)(40,76)(41,84)(42,78)(43,65)(44,59)(45,67)(46,61)(47,69)(48,63)(49,57)(50,62)(51,70)(52,64)(53,58)(54,66)(55,60)(56,68), (1,19)(2,20)(3,21)(4,15)(5,16)(6,17)(7,18)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(29,50)(30,51)(31,52)(32,53)(33,54)(34,55)(35,56)(36,46)(37,47)(38,48)(39,49)(40,43)(41,44)(42,45)(57,82)(58,83)(59,84)(60,71)(61,72)(62,73)(63,74)(64,75)(65,76)(66,77)(67,78)(68,79)(69,80)(70,81)(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), (1,12)(2,13)(3,14)(4,8)(5,9)(6,10)(7,11)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28)(29,37)(30,38)(31,39)(32,40)(33,41)(34,42)(35,36)(43,53)(44,54)(45,55)(46,56)(47,50)(48,51)(49,52)(57,64)(58,65)(59,66)(60,67)(61,68)(62,69)(63,70)(71,78)(72,79)(73,80)(74,81)(75,82)(76,83)(77,84)(85,92)(86,93)(87,94)(88,95)(89,96)(90,97)(91,98)(99,106)(100,107)(101,108)(102,109)(103,110)(104,111)(105,112), (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), (1,50)(2,56)(3,55)(4,54)(5,53)(6,52)(7,51)(8,44)(9,43)(10,49)(11,48)(12,47)(13,46)(14,45)(15,33)(16,32)(17,31)(18,30)(19,29)(20,35)(21,34)(22,41)(23,40)(24,39)(25,38)(26,37)(27,36)(28,42)(57,103)(58,102)(59,101)(60,100)(61,99)(62,112)(63,111)(64,110)(65,109)(66,108)(67,107)(68,106)(69,105)(70,104)(71,90)(72,89)(73,88)(74,87)(75,86)(76,85)(77,98)(78,97)(79,96)(80,95)(81,94)(82,93)(83,92)(84,91)>;
G:=Group( (1,19)(2,20)(3,21)(4,15)(5,16)(6,17)(7,18)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(29,47)(30,48)(31,49)(32,43)(33,44)(34,45)(35,46)(36,56)(37,50)(38,51)(39,52)(40,53)(41,54)(42,55)(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)(85,109)(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), (1,95)(2,89)(3,97)(4,91)(5,85)(6,93)(7,87)(8,98)(9,92)(10,86)(11,94)(12,88)(13,96)(14,90)(15,101)(16,109)(17,103)(18,111)(19,105)(20,99)(21,107)(22,108)(23,102)(24,110)(25,104)(26,112)(27,106)(28,100)(29,73)(30,81)(31,75)(32,83)(33,77)(34,71)(35,79)(36,72)(37,80)(38,74)(39,82)(40,76)(41,84)(42,78)(43,65)(44,59)(45,67)(46,61)(47,69)(48,63)(49,57)(50,62)(51,70)(52,64)(53,58)(54,66)(55,60)(56,68), (1,19)(2,20)(3,21)(4,15)(5,16)(6,17)(7,18)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(29,50)(30,51)(31,52)(32,53)(33,54)(34,55)(35,56)(36,46)(37,47)(38,48)(39,49)(40,43)(41,44)(42,45)(57,82)(58,83)(59,84)(60,71)(61,72)(62,73)(63,74)(64,75)(65,76)(66,77)(67,78)(68,79)(69,80)(70,81)(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), (1,12)(2,13)(3,14)(4,8)(5,9)(6,10)(7,11)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28)(29,37)(30,38)(31,39)(32,40)(33,41)(34,42)(35,36)(43,53)(44,54)(45,55)(46,56)(47,50)(48,51)(49,52)(57,64)(58,65)(59,66)(60,67)(61,68)(62,69)(63,70)(71,78)(72,79)(73,80)(74,81)(75,82)(76,83)(77,84)(85,92)(86,93)(87,94)(88,95)(89,96)(90,97)(91,98)(99,106)(100,107)(101,108)(102,109)(103,110)(104,111)(105,112), (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), (1,50)(2,56)(3,55)(4,54)(5,53)(6,52)(7,51)(8,44)(9,43)(10,49)(11,48)(12,47)(13,46)(14,45)(15,33)(16,32)(17,31)(18,30)(19,29)(20,35)(21,34)(22,41)(23,40)(24,39)(25,38)(26,37)(27,36)(28,42)(57,103)(58,102)(59,101)(60,100)(61,99)(62,112)(63,111)(64,110)(65,109)(66,108)(67,107)(68,106)(69,105)(70,104)(71,90)(72,89)(73,88)(74,87)(75,86)(76,85)(77,98)(78,97)(79,96)(80,95)(81,94)(82,93)(83,92)(84,91) );
G=PermutationGroup([[(1,19),(2,20),(3,21),(4,15),(5,16),(6,17),(7,18),(8,22),(9,23),(10,24),(11,25),(12,26),(13,27),(14,28),(29,47),(30,48),(31,49),(32,43),(33,44),(34,45),(35,46),(36,56),(37,50),(38,51),(39,52),(40,53),(41,54),(42,55),(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),(85,109),(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)], [(1,95),(2,89),(3,97),(4,91),(5,85),(6,93),(7,87),(8,98),(9,92),(10,86),(11,94),(12,88),(13,96),(14,90),(15,101),(16,109),(17,103),(18,111),(19,105),(20,99),(21,107),(22,108),(23,102),(24,110),(25,104),(26,112),(27,106),(28,100),(29,73),(30,81),(31,75),(32,83),(33,77),(34,71),(35,79),(36,72),(37,80),(38,74),(39,82),(40,76),(41,84),(42,78),(43,65),(44,59),(45,67),(46,61),(47,69),(48,63),(49,57),(50,62),(51,70),(52,64),(53,58),(54,66),(55,60),(56,68)], [(1,19),(2,20),(3,21),(4,15),(5,16),(6,17),(7,18),(8,22),(9,23),(10,24),(11,25),(12,26),(13,27),(14,28),(29,50),(30,51),(31,52),(32,53),(33,54),(34,55),(35,56),(36,46),(37,47),(38,48),(39,49),(40,43),(41,44),(42,45),(57,82),(58,83),(59,84),(60,71),(61,72),(62,73),(63,74),(64,75),(65,76),(66,77),(67,78),(68,79),(69,80),(70,81),(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)], [(1,12),(2,13),(3,14),(4,8),(5,9),(6,10),(7,11),(15,22),(16,23),(17,24),(18,25),(19,26),(20,27),(21,28),(29,37),(30,38),(31,39),(32,40),(33,41),(34,42),(35,36),(43,53),(44,54),(45,55),(46,56),(47,50),(48,51),(49,52),(57,64),(58,65),(59,66),(60,67),(61,68),(62,69),(63,70),(71,78),(72,79),(73,80),(74,81),(75,82),(76,83),(77,84),(85,92),(86,93),(87,94),(88,95),(89,96),(90,97),(91,98),(99,106),(100,107),(101,108),(102,109),(103,110),(104,111),(105,112)], [(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)], [(1,50),(2,56),(3,55),(4,54),(5,53),(6,52),(7,51),(8,44),(9,43),(10,49),(11,48),(12,47),(13,46),(14,45),(15,33),(16,32),(17,31),(18,30),(19,29),(20,35),(21,34),(22,41),(23,40),(24,39),(25,38),(26,37),(27,36),(28,42),(57,103),(58,102),(59,101),(60,100),(61,99),(62,112),(63,111),(64,110),(65,109),(66,108),(67,107),(68,106),(69,105),(70,104),(71,90),(72,89),(73,88),(74,87),(75,86),(76,85),(77,98),(78,97),(79,96),(80,95),(81,94),(82,93),(83,92),(84,91)]])
82 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 2J | 2K | 2L | 2M | 4A | 4B | 4C | ··· | 4H | 7A | 7B | 7C | 14A | ··· | 14U | 14V | ··· | 14AS | 28A | ··· | 28L |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 28 | 28 | 4 | 4 | 28 | ··· | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
82 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D7 | D14 | D14 | D14 | C7⋊D4 | 2+ 1+4 | D4⋊6D14 |
| kernel | C24⋊7D14 | C23.23D14 | C23.18D14 | C23⋊D14 | Dic7⋊D4 | C2×C23.D7 | C24⋊D7 | C22×C7⋊D4 | D4×C2×C14 | C22×C14 | C22×D4 | C22×C4 | C2×D4 | C24 | C23 | C14 | C2 |
| # reps | 1 | 2 | 2 | 2 | 4 | 1 | 2 | 1 | 1 | 4 | 3 | 3 | 12 | 6 | 24 | 2 | 12 |
Matrix representation of C24⋊7D14 ►in GL6(𝔽29)
| 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 | 13 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 7 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 16 | 27 | 28 | 6 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 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 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 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 |
| 6 | 0 | 0 | 0 | 0 | 0 |
| 14 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 8 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 7 | 0 | 10 | 28 |
| 24 | 19 | 0 | 0 | 0 | 0 |
| 14 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 19 | 0 | 23 | 0 |
| 0 | 0 | 7 | 0 | 10 | 28 |
| 0 | 0 | 2 | 0 | 10 | 0 |
| 0 | 0 | 8 | 28 | 0 | 0 |
G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,13,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[28,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,16,0,0,0,7,28,27,0,0,0,0,0,28,0,0,0,0,0,6,1],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[6,14,0,0,0,0,0,5,0,0,0,0,0,0,1,8,0,7,0,0,0,28,0,0,0,0,0,0,1,10,0,0,0,0,0,28],[24,14,0,0,0,0,19,5,0,0,0,0,0,0,19,7,2,8,0,0,0,0,0,28,0,0,23,10,10,0,0,0,0,28,0,0] >;
C24⋊7D14 in GAP, Magma, Sage, TeX
C_2^4\rtimes_7D_{14} % in TeX
G:=Group("C2^4:7D14"); // GroupNames label
G:=SmallGroup(448,1257);
// by ID
G=gap.SmallGroup(448,1257);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,758,675,570,18822]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^14=f^2=1,a*b=b*a,a*c=c*a,f*a*f=a*d=d*a,a*e=e*a,b*c=c*b,e*b*e^-1=b*d=d*b,f*b*f=b*c*d,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f=e^-1>;
// generators/relations