direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C7×C23.37D4, (C2×D4)⋊8C28, (D4×C14)⋊20C4, D4.6(C2×C28), C4.55(D4×C14), (C2×C56)⋊38C22, (C2×C28).315D4, C28.462(C2×D4), C4.5(C22×C28), D4⋊C4⋊15C14, C23.37(C7×D4), C42⋊C2⋊3C14, (C22×D4).7C14, C22.45(D4×C14), C28.82(C22⋊C4), (C14×M4(2))⋊29C2, (C2×M4(2))⋊11C14, C28.150(C22×C4), (C2×C28).894C23, (C22×C14).159D4, C14.127(C8⋊C22), (D4×C14).289C22, (C22×C28).411C22, C4⋊C4⋊8(C2×C14), (C2×C8)⋊8(C2×C14), (D4×C2×C14).19C2, (C2×C4).23(C7×D4), C2.2(C7×C8⋊C22), (C7×C4⋊C4)⋊64C22, (C2×C4).21(C2×C28), (C7×D4).28(C2×C4), C4.14(C7×C22⋊C4), (C7×D4⋊C4)⋊38C2, (C2×C28).194(C2×C4), (C2×D4).47(C2×C14), (C2×C14).621(C2×D4), C2.21(C14×C22⋊C4), (C7×C42⋊C2)⋊24C2, C14.109(C2×C22⋊C4), (C22×C4).30(C2×C14), (C2×C4).69(C22×C14), C22.20(C7×C22⋊C4), (C2×C14).81(C22⋊C4), SmallGroup(448,826)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C23.37D4
G = < a,b,c,d,e,f | a7=b2=c2=d2=1, e4=d, f2=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, ebe-1=fbf-1=bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=cde3 >
Subgroups: 386 in 190 conjugacy classes, 82 normal (26 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, D4, D4, C23, C23, C14, C14, C14, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C2×D4, C2×D4, C24, C28, C28, C28, C2×C14, C2×C14, C2×C14, D4⋊C4, C42⋊C2, C2×M4(2), C22×D4, C56, C2×C28, C2×C28, C2×C28, C7×D4, C7×D4, C22×C14, C22×C14, C23.37D4, C4×C28, C7×C22⋊C4, C7×C4⋊C4, C2×C56, C7×M4(2), C22×C28, D4×C14, D4×C14, C23×C14, C7×D4⋊C4, C7×C42⋊C2, C14×M4(2), D4×C2×C14, C7×C23.37D4
Quotients: C1, C2, C4, C22, C7, C2×C4, D4, C23, C14, C22⋊C4, C22×C4, C2×D4, C28, C2×C14, C2×C22⋊C4, C8⋊C22, C2×C28, C7×D4, C22×C14, C23.37D4, C7×C22⋊C4, C22×C28, D4×C14, C14×C22⋊C4, C7×C8⋊C22, C7×C23.37D4
►On 112 points
Generators in S112
(1 63 111 54 103 46 95)(2 64 112 55 104 47 96)(3 57 105 56 97 48 89)(4 58 106 49 98 41 90)(5 59 107 50 99 42 91)(6 60 108 51 100 43 92)(7 61 109 52 101 44 93)(8 62 110 53 102 45 94)(9 39 86 29 78 21 70)(10 40 87 30 79 22 71)(11 33 88 31 80 23 72)(12 34 81 32 73 24 65)(13 35 82 25 74 17 66)(14 36 83 26 75 18 67)(15 37 84 27 76 19 68)(16 38 85 28 77 20 69)
(1 67)(2 72)(3 69)(4 66)(5 71)(6 68)(7 65)(8 70)(9 62)(10 59)(11 64)(12 61)(13 58)(14 63)(15 60)(16 57)(17 90)(18 95)(19 92)(20 89)(21 94)(22 91)(23 96)(24 93)(25 98)(26 103)(27 100)(28 97)(29 102)(30 99)(31 104)(32 101)(33 112)(34 109)(35 106)(36 111)(37 108)(38 105)(39 110)(40 107)(41 74)(42 79)(43 76)(44 73)(45 78)(46 75)(47 80)(48 77)(49 82)(50 87)(51 84)(52 81)(53 86)(54 83)(55 88)(56 85)
(1 67)(2 68)(3 69)(4 70)(5 71)(6 72)(7 65)(8 66)(9 58)(10 59)(11 60)(12 61)(13 62)(14 63)(15 64)(16 57)(17 94)(18 95)(19 96)(20 89)(21 90)(22 91)(23 92)(24 93)(25 102)(26 103)(27 104)(28 97)(29 98)(30 99)(31 100)(32 101)(33 108)(34 109)(35 110)(36 111)(37 112)(38 105)(39 106)(40 107)(41 78)(42 79)(43 80)(44 73)(45 74)(46 75)(47 76)(48 77)(49 86)(50 87)(51 88)(52 81)(53 82)(54 83)(55 84)(56 85)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)(33 37)(34 38)(35 39)(36 40)(41 45)(42 46)(43 47)(44 48)(49 53)(50 54)(51 55)(52 56)(57 61)(58 62)(59 63)(60 64)(65 69)(66 70)(67 71)(68 72)(73 77)(74 78)(75 79)(76 80)(81 85)(82 86)(83 87)(84 88)(89 93)(90 94)(91 95)(92 96)(97 101)(98 102)(99 103)(100 104)(105 109)(106 110)(107 111)(108 112)
(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 66 67 8)(2 7 68 65)(3 72 69 6)(4 5 70 71)(9 10 58 59)(11 16 60 57)(12 64 61 15)(13 14 62 63)(17 18 94 95)(19 24 96 93)(20 92 89 23)(21 22 90 91)(25 26 102 103)(27 32 104 101)(28 100 97 31)(29 30 98 99)(33 38 108 105)(34 112 109 37)(35 36 110 111)(39 40 106 107)(41 42 78 79)(43 48 80 77)(44 76 73 47)(45 46 74 75)(49 50 86 87)(51 56 88 85)(52 84 81 55)(53 54 82 83)
G:=sub<Sym(112)| (1,63,111,54,103,46,95)(2,64,112,55,104,47,96)(3,57,105,56,97,48,89)(4,58,106,49,98,41,90)(5,59,107,50,99,42,91)(6,60,108,51,100,43,92)(7,61,109,52,101,44,93)(8,62,110,53,102,45,94)(9,39,86,29,78,21,70)(10,40,87,30,79,22,71)(11,33,88,31,80,23,72)(12,34,81,32,73,24,65)(13,35,82,25,74,17,66)(14,36,83,26,75,18,67)(15,37,84,27,76,19,68)(16,38,85,28,77,20,69), (1,67)(2,72)(3,69)(4,66)(5,71)(6,68)(7,65)(8,70)(9,62)(10,59)(11,64)(12,61)(13,58)(14,63)(15,60)(16,57)(17,90)(18,95)(19,92)(20,89)(21,94)(22,91)(23,96)(24,93)(25,98)(26,103)(27,100)(28,97)(29,102)(30,99)(31,104)(32,101)(33,112)(34,109)(35,106)(36,111)(37,108)(38,105)(39,110)(40,107)(41,74)(42,79)(43,76)(44,73)(45,78)(46,75)(47,80)(48,77)(49,82)(50,87)(51,84)(52,81)(53,86)(54,83)(55,88)(56,85), (1,67)(2,68)(3,69)(4,70)(5,71)(6,72)(7,65)(8,66)(9,58)(10,59)(11,60)(12,61)(13,62)(14,63)(15,64)(16,57)(17,94)(18,95)(19,96)(20,89)(21,90)(22,91)(23,92)(24,93)(25,102)(26,103)(27,104)(28,97)(29,98)(30,99)(31,100)(32,101)(33,108)(34,109)(35,110)(36,111)(37,112)(38,105)(39,106)(40,107)(41,78)(42,79)(43,80)(44,73)(45,74)(46,75)(47,76)(48,77)(49,86)(50,87)(51,88)(52,81)(53,82)(54,83)(55,84)(56,85), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(41,45)(42,46)(43,47)(44,48)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64)(65,69)(66,70)(67,71)(68,72)(73,77)(74,78)(75,79)(76,80)(81,85)(82,86)(83,87)(84,88)(89,93)(90,94)(91,95)(92,96)(97,101)(98,102)(99,103)(100,104)(105,109)(106,110)(107,111)(108,112), (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), (1,66,67,8)(2,7,68,65)(3,72,69,6)(4,5,70,71)(9,10,58,59)(11,16,60,57)(12,64,61,15)(13,14,62,63)(17,18,94,95)(19,24,96,93)(20,92,89,23)(21,22,90,91)(25,26,102,103)(27,32,104,101)(28,100,97,31)(29,30,98,99)(33,38,108,105)(34,112,109,37)(35,36,110,111)(39,40,106,107)(41,42,78,79)(43,48,80,77)(44,76,73,47)(45,46,74,75)(49,50,86,87)(51,56,88,85)(52,84,81,55)(53,54,82,83)>;
G:=Group( (1,63,111,54,103,46,95)(2,64,112,55,104,47,96)(3,57,105,56,97,48,89)(4,58,106,49,98,41,90)(5,59,107,50,99,42,91)(6,60,108,51,100,43,92)(7,61,109,52,101,44,93)(8,62,110,53,102,45,94)(9,39,86,29,78,21,70)(10,40,87,30,79,22,71)(11,33,88,31,80,23,72)(12,34,81,32,73,24,65)(13,35,82,25,74,17,66)(14,36,83,26,75,18,67)(15,37,84,27,76,19,68)(16,38,85,28,77,20,69), (1,67)(2,72)(3,69)(4,66)(5,71)(6,68)(7,65)(8,70)(9,62)(10,59)(11,64)(12,61)(13,58)(14,63)(15,60)(16,57)(17,90)(18,95)(19,92)(20,89)(21,94)(22,91)(23,96)(24,93)(25,98)(26,103)(27,100)(28,97)(29,102)(30,99)(31,104)(32,101)(33,112)(34,109)(35,106)(36,111)(37,108)(38,105)(39,110)(40,107)(41,74)(42,79)(43,76)(44,73)(45,78)(46,75)(47,80)(48,77)(49,82)(50,87)(51,84)(52,81)(53,86)(54,83)(55,88)(56,85), (1,67)(2,68)(3,69)(4,70)(5,71)(6,72)(7,65)(8,66)(9,58)(10,59)(11,60)(12,61)(13,62)(14,63)(15,64)(16,57)(17,94)(18,95)(19,96)(20,89)(21,90)(22,91)(23,92)(24,93)(25,102)(26,103)(27,104)(28,97)(29,98)(30,99)(31,100)(32,101)(33,108)(34,109)(35,110)(36,111)(37,112)(38,105)(39,106)(40,107)(41,78)(42,79)(43,80)(44,73)(45,74)(46,75)(47,76)(48,77)(49,86)(50,87)(51,88)(52,81)(53,82)(54,83)(55,84)(56,85), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(41,45)(42,46)(43,47)(44,48)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64)(65,69)(66,70)(67,71)(68,72)(73,77)(74,78)(75,79)(76,80)(81,85)(82,86)(83,87)(84,88)(89,93)(90,94)(91,95)(92,96)(97,101)(98,102)(99,103)(100,104)(105,109)(106,110)(107,111)(108,112), (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), (1,66,67,8)(2,7,68,65)(3,72,69,6)(4,5,70,71)(9,10,58,59)(11,16,60,57)(12,64,61,15)(13,14,62,63)(17,18,94,95)(19,24,96,93)(20,92,89,23)(21,22,90,91)(25,26,102,103)(27,32,104,101)(28,100,97,31)(29,30,98,99)(33,38,108,105)(34,112,109,37)(35,36,110,111)(39,40,106,107)(41,42,78,79)(43,48,80,77)(44,76,73,47)(45,46,74,75)(49,50,86,87)(51,56,88,85)(52,84,81,55)(53,54,82,83) );
G=PermutationGroup([[(1,63,111,54,103,46,95),(2,64,112,55,104,47,96),(3,57,105,56,97,48,89),(4,58,106,49,98,41,90),(5,59,107,50,99,42,91),(6,60,108,51,100,43,92),(7,61,109,52,101,44,93),(8,62,110,53,102,45,94),(9,39,86,29,78,21,70),(10,40,87,30,79,22,71),(11,33,88,31,80,23,72),(12,34,81,32,73,24,65),(13,35,82,25,74,17,66),(14,36,83,26,75,18,67),(15,37,84,27,76,19,68),(16,38,85,28,77,20,69)], [(1,67),(2,72),(3,69),(4,66),(5,71),(6,68),(7,65),(8,70),(9,62),(10,59),(11,64),(12,61),(13,58),(14,63),(15,60),(16,57),(17,90),(18,95),(19,92),(20,89),(21,94),(22,91),(23,96),(24,93),(25,98),(26,103),(27,100),(28,97),(29,102),(30,99),(31,104),(32,101),(33,112),(34,109),(35,106),(36,111),(37,108),(38,105),(39,110),(40,107),(41,74),(42,79),(43,76),(44,73),(45,78),(46,75),(47,80),(48,77),(49,82),(50,87),(51,84),(52,81),(53,86),(54,83),(55,88),(56,85)], [(1,67),(2,68),(3,69),(4,70),(5,71),(6,72),(7,65),(8,66),(9,58),(10,59),(11,60),(12,61),(13,62),(14,63),(15,64),(16,57),(17,94),(18,95),(19,96),(20,89),(21,90),(22,91),(23,92),(24,93),(25,102),(26,103),(27,104),(28,97),(29,98),(30,99),(31,100),(32,101),(33,108),(34,109),(35,110),(36,111),(37,112),(38,105),(39,106),(40,107),(41,78),(42,79),(43,80),(44,73),(45,74),(46,75),(47,76),(48,77),(49,86),(50,87),(51,88),(52,81),(53,82),(54,83),(55,84),(56,85)], [(1,5),(2,6),(3,7),(4,8),(9,13),(10,14),(11,15),(12,16),(17,21),(18,22),(19,23),(20,24),(25,29),(26,30),(27,31),(28,32),(33,37),(34,38),(35,39),(36,40),(41,45),(42,46),(43,47),(44,48),(49,53),(50,54),(51,55),(52,56),(57,61),(58,62),(59,63),(60,64),(65,69),(66,70),(67,71),(68,72),(73,77),(74,78),(75,79),(76,80),(81,85),(82,86),(83,87),(84,88),(89,93),(90,94),(91,95),(92,96),(97,101),(98,102),(99,103),(100,104),(105,109),(106,110),(107,111),(108,112)], [(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)], [(1,66,67,8),(2,7,68,65),(3,72,69,6),(4,5,70,71),(9,10,58,59),(11,16,60,57),(12,64,61,15),(13,14,62,63),(17,18,94,95),(19,24,96,93),(20,92,89,23),(21,22,90,91),(25,26,102,103),(27,32,104,101),(28,100,97,31),(29,30,98,99),(33,38,108,105),(34,112,109,37),(35,36,110,111),(39,40,106,107),(41,42,78,79),(43,48,80,77),(44,76,73,47),(45,46,74,75),(49,50,86,87),(51,56,88,85),(52,84,81,55),(53,54,82,83)]])
154 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 7A | ··· | 7F | 8A | 8B | 8C | 8D | 14A | ··· | 14R | 14S | ··· | 14AD | 14AE | ··· | 14BB | 28A | ··· | 28X | 28Y | ··· | 28AV | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | ··· | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
154 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C7 | C14 | C14 | C14 | C14 | C28 | D4 | D4 | C7×D4 | C7×D4 | C8⋊C22 | C7×C8⋊C22 |
| kernel | C7×C23.37D4 | C7×D4⋊C4 | C7×C42⋊C2 | C14×M4(2) | D4×C2×C14 | D4×C14 | C23.37D4 | D4⋊C4 | C42⋊C2 | C2×M4(2) | C22×D4 | C2×D4 | C2×C28 | C22×C14 | C2×C4 | C23 | C14 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 8 | 6 | 24 | 6 | 6 | 6 | 48 | 3 | 1 | 18 | 6 | 2 | 12 |
Matrix representation of C7×C23.37D4 ►in GL6(𝔽113)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 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 | 112 | 0 |
| 0 | 0 | 1 | 1 | 0 | 112 |
| 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 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 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 | 0 |
| 0 | 0 | 0 | 112 | 0 | 0 |
| 0 | 0 | 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 0 | 0 | 112 |
| 0 | 15 | 0 | 0 | 0 | 0 |
| 98 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 112 | 112 | 1 | 2 |
| 0 | 0 | 0 | 112 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 0 | 98 | 0 | 0 | 0 | 0 |
| 98 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 1 | 112 | 111 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 112 |
G:=sub<GL(6,GF(113))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,1,0,1,0,0,0,0,112,0,0,0,0,0,0,112],[112,0,0,0,0,0,0,112,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,112],[0,98,0,0,0,0,15,0,0,0,0,0,0,0,0,112,0,0,0,0,0,112,112,0,0,0,1,1,0,1,0,0,0,2,0,1],[0,98,0,0,0,0,98,0,0,0,0,0,0,0,0,1,1,0,0,0,0,1,0,0,0,0,1,112,0,0,0,0,0,111,0,112] >;
C7×C23.37D4 in GAP, Magma, Sage, TeX
C_7\times C_2^3._{37}D_4 % in TeX
G:=Group("C7xC2^3.37D4"); // GroupNames label
G:=SmallGroup(448,826);
// by ID
G=gap.SmallGroup(448,826);
# by ID
G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,784,813,2403,1192,9804,4911,172]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^7=b^2=c^2=d^2=1,e^4=d,f^2=c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,e*b*e^-1=f*b*f^-1=b*d=d*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=c*d*e^3>;
// generators/relations