metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24⋊2D14, C14.252+ (1+4), (C2×D4)⋊5D14, C22≀C2⋊2D7, C22⋊C4⋊5D14, D14.2(C2×D4), (C22×D7)⋊6D4, C23⋊D14⋊3C2, C28⋊2D4⋊12C2, (D4×C14)⋊6C22, C24⋊D7⋊6C2, C7⋊2(C23⋊3D4), D14⋊D4⋊12C2, C22.40(D4×D7), D14⋊C4⋊10C22, (C2×D28)⋊18C22, (C2×C28).27C23, Dic7⋊C4⋊8C22, C4⋊Dic7⋊25C22, (C23×C14)⋊9C22, C14.55(C22×D4), (C23×D7)⋊6C22, D14.D4⋊12C2, (C2×C14).133C24, C22.D28⋊9C2, (C22×C14).8C23, C2.27(D4⋊6D14), C23.D7⋊14C22, C23.18D14⋊4C2, (C2×Dic7).60C23, C22.154(C23×D7), C23.107(C22×D7), (C22×Dic7)⋊12C22, (C22×D7).182C23, (C2×D4×D7)⋊6C2, C2.28(C2×D4×D7), (C2×C4×D7)⋊6C22, (D7×C22⋊C4)⋊2C2, (C7×C22≀C2)⋊4C2, (C2×C14).53(C2×D4), (C22×C7⋊D4)⋊7C2, (C2×C7⋊D4)⋊38C22, (C7×C22⋊C4)⋊4C22, (C2×C4).27(C22×D7), SmallGroup(448,1042)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1932 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 [×2], C2×C4 [×11], D4 [×20], C23 [×2], C23 [×2], C23 [×17], D7 [×5], C14, C14 [×2], C14 [×5], C22⋊C4, C22⋊C4 [×2], C22⋊C4 [×9], C4⋊C4 [×4], C22×C4 [×4], C2×D4, C2×D4 [×2], C2×D4 [×17], C24, C24 [×2], Dic7 [×5], C28 [×3], D14 [×4], D14 [×15], C2×C14, C2×C14 [×2], C2×C14 [×15], C2×C22⋊C4, C22≀C2, C22≀C2 [×3], C4⋊D4 [×4], C22.D4 [×4], C22×D4 [×2], C4×D7 [×4], D28 [×2], C2×Dic7, C2×Dic7 [×4], C2×Dic7 [×2], C7⋊D4 [×14], C2×C28, C2×C28 [×2], C7×D4 [×4], C22×D7 [×3], C22×D7 [×4], C22×D7 [×6], C22×C14 [×2], C22×C14 [×2], C22×C14 [×4], C23⋊3D4, Dic7⋊C4 [×2], C4⋊Dic7 [×2], D14⋊C4 [×4], C23.D7, C23.D7 [×4], C7×C22⋊C4, C7×C22⋊C4 [×2], C2×C4×D7, C2×C4×D7 [×2], C2×D28, D4×D7 [×4], C22×Dic7, C2×C7⋊D4 [×2], C2×C7⋊D4 [×6], C2×C7⋊D4 [×4], D4×C14, D4×C14 [×2], C23×D7 [×2], C23×C14, D7×C22⋊C4, D14.D4 [×2], D14⋊D4 [×2], C22.D28, C23.18D14, C23⋊D14 [×2], C28⋊2D4 [×2], C24⋊D7, C7×C22≀C2, C2×D4×D7, C22×C7⋊D4, C24⋊2D14
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⋊6D14 [×2], C24⋊2D14
Generators and relations
G = < a,b,c,d,e,f | a2=b2=c2=d2=e14=f2=1, ab=ba, eae-1=faf=ac=ca, ad=da, bc=cb, ebe-1=bd=db, fbf=bcd, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=e-1 >
►On 112 points
Generators in S112
(2 28)(4 16)(6 18)(8 20)(10 22)(12 24)(14 26)(29 94)(31 96)(33 98)(35 86)(37 88)(39 90)(41 92)(44 69)(46 57)(48 59)(50 61)(52 63)(54 65)(56 67)(72 107)(74 109)(76 111)(78 99)(80 101)(82 103)(84 105)
(1 81)(2 41)(3 83)(4 29)(5 71)(6 31)(7 73)(8 33)(9 75)(10 35)(11 77)(12 37)(13 79)(14 39)(15 104)(16 94)(17 106)(18 96)(19 108)(20 98)(21 110)(22 86)(23 112)(24 88)(25 100)(26 90)(27 102)(28 92)(30 58)(32 60)(34 62)(36 64)(38 66)(40 68)(42 70)(43 91)(44 103)(45 93)(46 105)(47 95)(48 107)(49 97)(50 109)(51 85)(52 111)(53 87)(54 99)(55 89)(56 101)(57 84)(59 72)(61 74)(63 76)(65 78)(67 80)(69 82)
(1 27)(2 28)(3 15)(4 16)(5 17)(6 18)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)(13 25)(14 26)(29 94)(30 95)(31 96)(32 97)(33 98)(34 85)(35 86)(36 87)(37 88)(38 89)(39 90)(40 91)(41 92)(42 93)(43 68)(44 69)(45 70)(46 57)(47 58)(48 59)(49 60)(50 61)(51 62)(52 63)(53 64)(54 65)(55 66)(56 67)(71 106)(72 107)(73 108)(74 109)(75 110)(76 111)(77 112)(78 99)(79 100)(80 101)(81 102)(82 103)(83 104)(84 105)
(1 68)(2 69)(3 70)(4 57)(5 58)(6 59)(7 60)(8 61)(9 62)(10 63)(11 64)(12 65)(13 66)(14 67)(15 45)(16 46)(17 47)(18 48)(19 49)(20 50)(21 51)(22 52)(23 53)(24 54)(25 55)(26 56)(27 43)(28 44)(29 84)(30 71)(31 72)(32 73)(33 74)(34 75)(35 76)(36 77)(37 78)(38 79)(39 80)(40 81)(41 82)(42 83)(85 110)(86 111)(87 112)(88 99)(89 100)(90 101)(91 102)(92 103)(93 104)(94 105)(95 106)(96 107)(97 108)(98 109)
(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 67)(2 66)(3 65)(4 64)(5 63)(6 62)(7 61)(8 60)(9 59)(10 58)(11 57)(12 70)(13 69)(14 68)(15 54)(16 53)(17 52)(18 51)(19 50)(20 49)(21 48)(22 47)(23 46)(24 45)(25 44)(26 43)(27 56)(28 55)(29 112)(30 111)(31 110)(32 109)(33 108)(34 107)(35 106)(36 105)(37 104)(38 103)(39 102)(40 101)(41 100)(42 99)(71 86)(72 85)(73 98)(74 97)(75 96)(76 95)(77 94)(78 93)(79 92)(80 91)(81 90)(82 89)(83 88)(84 87)
G:=sub<Sym(112)| (2,28)(4,16)(6,18)(8,20)(10,22)(12,24)(14,26)(29,94)(31,96)(33,98)(35,86)(37,88)(39,90)(41,92)(44,69)(46,57)(48,59)(50,61)(52,63)(54,65)(56,67)(72,107)(74,109)(76,111)(78,99)(80,101)(82,103)(84,105), (1,81)(2,41)(3,83)(4,29)(5,71)(6,31)(7,73)(8,33)(9,75)(10,35)(11,77)(12,37)(13,79)(14,39)(15,104)(16,94)(17,106)(18,96)(19,108)(20,98)(21,110)(22,86)(23,112)(24,88)(25,100)(26,90)(27,102)(28,92)(30,58)(32,60)(34,62)(36,64)(38,66)(40,68)(42,70)(43,91)(44,103)(45,93)(46,105)(47,95)(48,107)(49,97)(50,109)(51,85)(52,111)(53,87)(54,99)(55,89)(56,101)(57,84)(59,72)(61,74)(63,76)(65,78)(67,80)(69,82), (1,27)(2,28)(3,15)(4,16)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)(13,25)(14,26)(29,94)(30,95)(31,96)(32,97)(33,98)(34,85)(35,86)(36,87)(37,88)(38,89)(39,90)(40,91)(41,92)(42,93)(43,68)(44,69)(45,70)(46,57)(47,58)(48,59)(49,60)(50,61)(51,62)(52,63)(53,64)(54,65)(55,66)(56,67)(71,106)(72,107)(73,108)(74,109)(75,110)(76,111)(77,112)(78,99)(79,100)(80,101)(81,102)(82,103)(83,104)(84,105), (1,68)(2,69)(3,70)(4,57)(5,58)(6,59)(7,60)(8,61)(9,62)(10,63)(11,64)(12,65)(13,66)(14,67)(15,45)(16,46)(17,47)(18,48)(19,49)(20,50)(21,51)(22,52)(23,53)(24,54)(25,55)(26,56)(27,43)(28,44)(29,84)(30,71)(31,72)(32,73)(33,74)(34,75)(35,76)(36,77)(37,78)(38,79)(39,80)(40,81)(41,82)(42,83)(85,110)(86,111)(87,112)(88,99)(89,100)(90,101)(91,102)(92,103)(93,104)(94,105)(95,106)(96,107)(97,108)(98,109), (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), (1,67)(2,66)(3,65)(4,64)(5,63)(6,62)(7,61)(8,60)(9,59)(10,58)(11,57)(12,70)(13,69)(14,68)(15,54)(16,53)(17,52)(18,51)(19,50)(20,49)(21,48)(22,47)(23,46)(24,45)(25,44)(26,43)(27,56)(28,55)(29,112)(30,111)(31,110)(32,109)(33,108)(34,107)(35,106)(36,105)(37,104)(38,103)(39,102)(40,101)(41,100)(42,99)(71,86)(72,85)(73,98)(74,97)(75,96)(76,95)(77,94)(78,93)(79,92)(80,91)(81,90)(82,89)(83,88)(84,87)>;
G:=Group( (2,28)(4,16)(6,18)(8,20)(10,22)(12,24)(14,26)(29,94)(31,96)(33,98)(35,86)(37,88)(39,90)(41,92)(44,69)(46,57)(48,59)(50,61)(52,63)(54,65)(56,67)(72,107)(74,109)(76,111)(78,99)(80,101)(82,103)(84,105), (1,81)(2,41)(3,83)(4,29)(5,71)(6,31)(7,73)(8,33)(9,75)(10,35)(11,77)(12,37)(13,79)(14,39)(15,104)(16,94)(17,106)(18,96)(19,108)(20,98)(21,110)(22,86)(23,112)(24,88)(25,100)(26,90)(27,102)(28,92)(30,58)(32,60)(34,62)(36,64)(38,66)(40,68)(42,70)(43,91)(44,103)(45,93)(46,105)(47,95)(48,107)(49,97)(50,109)(51,85)(52,111)(53,87)(54,99)(55,89)(56,101)(57,84)(59,72)(61,74)(63,76)(65,78)(67,80)(69,82), (1,27)(2,28)(3,15)(4,16)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)(13,25)(14,26)(29,94)(30,95)(31,96)(32,97)(33,98)(34,85)(35,86)(36,87)(37,88)(38,89)(39,90)(40,91)(41,92)(42,93)(43,68)(44,69)(45,70)(46,57)(47,58)(48,59)(49,60)(50,61)(51,62)(52,63)(53,64)(54,65)(55,66)(56,67)(71,106)(72,107)(73,108)(74,109)(75,110)(76,111)(77,112)(78,99)(79,100)(80,101)(81,102)(82,103)(83,104)(84,105), (1,68)(2,69)(3,70)(4,57)(5,58)(6,59)(7,60)(8,61)(9,62)(10,63)(11,64)(12,65)(13,66)(14,67)(15,45)(16,46)(17,47)(18,48)(19,49)(20,50)(21,51)(22,52)(23,53)(24,54)(25,55)(26,56)(27,43)(28,44)(29,84)(30,71)(31,72)(32,73)(33,74)(34,75)(35,76)(36,77)(37,78)(38,79)(39,80)(40,81)(41,82)(42,83)(85,110)(86,111)(87,112)(88,99)(89,100)(90,101)(91,102)(92,103)(93,104)(94,105)(95,106)(96,107)(97,108)(98,109), (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), (1,67)(2,66)(3,65)(4,64)(5,63)(6,62)(7,61)(8,60)(9,59)(10,58)(11,57)(12,70)(13,69)(14,68)(15,54)(16,53)(17,52)(18,51)(19,50)(20,49)(21,48)(22,47)(23,46)(24,45)(25,44)(26,43)(27,56)(28,55)(29,112)(30,111)(31,110)(32,109)(33,108)(34,107)(35,106)(36,105)(37,104)(38,103)(39,102)(40,101)(41,100)(42,99)(71,86)(72,85)(73,98)(74,97)(75,96)(76,95)(77,94)(78,93)(79,92)(80,91)(81,90)(82,89)(83,88)(84,87) );
G=PermutationGroup([(2,28),(4,16),(6,18),(8,20),(10,22),(12,24),(14,26),(29,94),(31,96),(33,98),(35,86),(37,88),(39,90),(41,92),(44,69),(46,57),(48,59),(50,61),(52,63),(54,65),(56,67),(72,107),(74,109),(76,111),(78,99),(80,101),(82,103),(84,105)], [(1,81),(2,41),(3,83),(4,29),(5,71),(6,31),(7,73),(8,33),(9,75),(10,35),(11,77),(12,37),(13,79),(14,39),(15,104),(16,94),(17,106),(18,96),(19,108),(20,98),(21,110),(22,86),(23,112),(24,88),(25,100),(26,90),(27,102),(28,92),(30,58),(32,60),(34,62),(36,64),(38,66),(40,68),(42,70),(43,91),(44,103),(45,93),(46,105),(47,95),(48,107),(49,97),(50,109),(51,85),(52,111),(53,87),(54,99),(55,89),(56,101),(57,84),(59,72),(61,74),(63,76),(65,78),(67,80),(69,82)], [(1,27),(2,28),(3,15),(4,16),(5,17),(6,18),(7,19),(8,20),(9,21),(10,22),(11,23),(12,24),(13,25),(14,26),(29,94),(30,95),(31,96),(32,97),(33,98),(34,85),(35,86),(36,87),(37,88),(38,89),(39,90),(40,91),(41,92),(42,93),(43,68),(44,69),(45,70),(46,57),(47,58),(48,59),(49,60),(50,61),(51,62),(52,63),(53,64),(54,65),(55,66),(56,67),(71,106),(72,107),(73,108),(74,109),(75,110),(76,111),(77,112),(78,99),(79,100),(80,101),(81,102),(82,103),(83,104),(84,105)], [(1,68),(2,69),(3,70),(4,57),(5,58),(6,59),(7,60),(8,61),(9,62),(10,63),(11,64),(12,65),(13,66),(14,67),(15,45),(16,46),(17,47),(18,48),(19,49),(20,50),(21,51),(22,52),(23,53),(24,54),(25,55),(26,56),(27,43),(28,44),(29,84),(30,71),(31,72),(32,73),(33,74),(34,75),(35,76),(36,77),(37,78),(38,79),(39,80),(40,81),(41,82),(42,83),(85,110),(86,111),(87,112),(88,99),(89,100),(90,101),(91,102),(92,103),(93,104),(94,105),(95,106),(96,107),(97,108),(98,109)], [(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)], [(1,67),(2,66),(3,65),(4,64),(5,63),(6,62),(7,61),(8,60),(9,59),(10,58),(11,57),(12,70),(13,69),(14,68),(15,54),(16,53),(17,52),(18,51),(19,50),(20,49),(21,48),(22,47),(23,46),(24,45),(25,44),(26,43),(27,56),(28,55),(29,112),(30,111),(31,110),(32,109),(33,108),(34,107),(35,106),(36,105),(37,104),(38,103),(39,102),(40,101),(41,100),(42,99),(71,86),(72,85),(73,98),(74,97),(75,96),(76,95),(77,94),(78,93),(79,92),(80,91),(81,90),(82,89),(83,88),(84,87)])
Matrix representation ►G ⊆ GL6(𝔽29)
| 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 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 13 | 16 | 0 | 0 | 0 | 0 |
| 4 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 18 | 2 | 0 | 0 |
| 0 | 0 | 27 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 18 | 2 |
| 0 | 0 | 0 | 0 | 27 | 11 |
| 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 |
| 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 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 2 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 25 |
| 0 | 0 | 0 | 0 | 4 | 25 |
| 0 | 0 | 11 | 25 | 0 | 0 |
| 0 | 0 | 4 | 25 | 0 | 0 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 27 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 25 | 11 |
| 0 | 0 | 0 | 0 | 25 | 4 |
| 0 | 0 | 25 | 11 | 0 | 0 |
| 0 | 0 | 25 | 4 | 0 | 0 |
G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[13,4,0,0,0,0,16,16,0,0,0,0,0,0,18,27,0,0,0,0,2,11,0,0,0,0,0,0,18,27,0,0,0,0,2,11],[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],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,2,0,0,0,0,0,28,0,0,0,0,0,0,0,0,11,4,0,0,0,0,25,25,0,0,11,4,0,0,0,0,25,25,0,0],[28,27,0,0,0,0,0,1,0,0,0,0,0,0,0,0,25,25,0,0,0,0,11,4,0,0,25,25,0,0,0,0,11,4,0,0] >;
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 4A | 4B | 4C | 4D | ··· | 4H | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14AA | 14AB | 14AC | 14AD | 28A | ··· | 28I |
| 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 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 14 | 14 | 14 | 14 | 28 | 4 | 4 | 4 | 28 | ··· | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | ··· | 8 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D7 | D14 | D14 | D14 | 2+ (1+4) | D4×D7 | D4⋊6D14 |
| kernel | C24⋊2D14 | D7×C22⋊C4 | D14.D4 | D14⋊D4 | C22.D28 | C23.18D14 | C23⋊D14 | C28⋊2D4 | C24⋊D7 | C7×C22≀C2 | C2×D4×D7 | C22×C7⋊D4 | C22×D7 | C22≀C2 | C22⋊C4 | C2×D4 | C24 | C14 | C22 | C2 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 4 | 3 | 9 | 9 | 3 | 2 | 6 | 12 |
In GAP, Magma, Sage, TeX
C_2^4\rtimes_2D_{14} % in TeX
G:=Group("C2^4:2D14"); // GroupNames label
G:=SmallGroup(448,1042);
// by ID
G=gap.SmallGroup(448,1042);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,219,675,297,18822]);
// 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,e*a*e^-1=f*a*f=a*c=c*a,a*d=d*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
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