direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D8⋊D7, D8⋊9D14, C56⋊5C23, C28.2C24, D28.1C23, Dic14⋊1C23, (C2×C8)⋊8D14, C7⋊C8⋊1C23, (C2×D8)⋊11D7, C4.40(D4×D7), C8⋊3(C22×D7), (C2×D4)⋊28D14, (C14×D8)⋊11C2, D4⋊D7⋊9C22, (C4×D7).14D4, C28.77(C2×D4), (C7×D4)⋊2C23, (D4×D7)⋊5C22, D4⋊2(C22×D7), C4.2(C23×D7), C14⋊2(C8⋊C22), (C2×C56)⋊16C22, D14.49(C2×D4), (C7×D8)⋊14C22, D4.D7⋊7C22, (C4×D7).1C23, (D4×C14)⋊19C22, D4⋊2D7⋊5C22, C8⋊D7⋊12C22, C56⋊C2⋊13C22, Dic7.54(C2×D4), (C22×D7).97D4, C22.136(D4×D7), (C2×C28).519C23, (C2×Dic7).191D4, C14.103(C22×D4), (C2×Dic14)⋊36C22, (C2×D28).176C22, (C2×D4×D7)⋊22C2, C7⋊2(C2×C8⋊C22), C2.76(C2×D4×D7), (C2×D4⋊D7)⋊26C2, (C2×C8⋊D7)⋊8C2, (C2×C7⋊C8)⋊14C22, (C2×C56⋊C2)⋊24C2, (C2×D4.D7)⋊25C2, (C2×D4⋊2D7)⋊23C2, (C2×C14).392(C2×D4), (C2×C4×D7).155C22, (C2×C4).609(C22×D7), SmallGroup(448,1208)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1700 in 298 conjugacy classes, 103 normal (33 characteristic)
C1, C2, C2 [×2], C2 [×8], C4 [×2], C4 [×4], C22, C22 [×24], C7, C8 [×2], C8 [×2], C2×C4, C2×C4 [×10], D4 [×4], D4 [×13], Q8 [×3], C23 [×12], D7 [×4], C14, C14 [×2], C14 [×4], C2×C8, C2×C8, M4(2) [×4], D8 [×4], D8 [×4], SD16 [×8], C22×C4 [×2], C2×D4 [×2], C2×D4 [×9], C2×Q8, C4○D4 [×6], C24, Dic7 [×2], Dic7 [×2], C28 [×2], D14 [×2], D14 [×14], C2×C14, C2×C14 [×8], C2×M4(2), C2×D8, C2×D8, C2×SD16 [×2], C8⋊C22 [×8], C22×D4, C2×C4○D4, C7⋊C8 [×2], C56 [×2], Dic14 [×2], Dic14, C4×D7 [×4], D28 [×2], D28, C2×Dic7, C2×Dic7 [×5], C7⋊D4 [×8], C2×C28, C7×D4 [×4], C7×D4 [×2], C22×D7, C22×D7 [×9], C22×C14 [×2], C2×C8⋊C22, C8⋊D7 [×4], C56⋊C2 [×4], C2×C7⋊C8, D4⋊D7 [×4], D4.D7 [×4], C2×C56, C7×D8 [×4], C2×Dic14, C2×C4×D7, C2×D28, D4×D7 [×4], D4×D7 [×2], D4⋊2D7 [×4], D4⋊2D7 [×2], C22×Dic7, C2×C7⋊D4 [×2], D4×C14 [×2], C23×D7, C2×C8⋊D7, C2×C56⋊C2, D8⋊D7 [×8], C2×D4⋊D7, C2×D4.D7, C14×D8, C2×D4×D7, C2×D4⋊2D7, C2×D8⋊D7
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D7, C2×D4 [×6], C24, D14 [×7], C8⋊C22 [×2], C22×D4, C22×D7 [×7], C2×C8⋊C22, D4×D7 [×2], C23×D7, D8⋊D7 [×2], C2×D4×D7, C2×D8⋊D7
Generators and relations
G = < a,b,c,d,e | a2=b8=c2=d7=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, ebe=b5, cd=dc, ce=ec, ede=d-1 >
►On 112 points
Generators in S112
(1 108)(2 109)(3 110)(4 111)(5 112)(6 105)(7 106)(8 107)(9 100)(10 101)(11 102)(12 103)(13 104)(14 97)(15 98)(16 99)(17 94)(18 95)(19 96)(20 89)(21 90)(22 91)(23 92)(24 93)(25 66)(26 67)(27 68)(28 69)(29 70)(30 71)(31 72)(32 65)(33 60)(34 61)(35 62)(36 63)(37 64)(38 57)(39 58)(40 59)(41 75)(42 76)(43 77)(44 78)(45 79)(46 80)(47 73)(48 74)(49 83)(50 84)(51 85)(52 86)(53 87)(54 88)(55 81)(56 82)
(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)
(1 106)(2 105)(3 112)(4 111)(5 110)(6 109)(7 108)(8 107)(9 98)(10 97)(11 104)(12 103)(13 102)(14 101)(15 100)(16 99)(17 92)(18 91)(19 90)(20 89)(21 96)(22 95)(23 94)(24 93)(25 68)(26 67)(27 66)(28 65)(29 72)(30 71)(31 70)(32 69)(33 58)(34 57)(35 64)(36 63)(37 62)(38 61)(39 60)(40 59)(41 75)(42 74)(43 73)(44 80)(45 79)(46 78)(47 77)(48 76)(49 85)(50 84)(51 83)(52 82)(53 81)(54 88)(55 87)(56 86)
(1 72 104 64 81 90 80)(2 65 97 57 82 91 73)(3 66 98 58 83 92 74)(4 67 99 59 84 93 75)(5 68 100 60 85 94 76)(6 69 101 61 86 95 77)(7 70 102 62 87 96 78)(8 71 103 63 88 89 79)(9 33 51 17 42 112 27)(10 34 52 18 43 105 28)(11 35 53 19 44 106 29)(12 36 54 20 45 107 30)(13 37 55 21 46 108 31)(14 38 56 22 47 109 32)(15 39 49 23 48 110 25)(16 40 50 24 41 111 26)
(1 46)(2 43)(3 48)(4 45)(5 42)(6 47)(7 44)(8 41)(9 85)(10 82)(11 87)(12 84)(13 81)(14 86)(15 83)(16 88)(17 68)(18 65)(19 70)(20 67)(21 72)(22 69)(23 66)(24 71)(25 92)(26 89)(27 94)(28 91)(29 96)(30 93)(31 90)(32 95)(33 60)(34 57)(35 62)(36 59)(37 64)(38 61)(39 58)(40 63)(49 98)(50 103)(51 100)(52 97)(53 102)(54 99)(55 104)(56 101)(73 105)(74 110)(75 107)(76 112)(77 109)(78 106)(79 111)(80 108)
G:=sub<Sym(112)| (1,108)(2,109)(3,110)(4,111)(5,112)(6,105)(7,106)(8,107)(9,100)(10,101)(11,102)(12,103)(13,104)(14,97)(15,98)(16,99)(17,94)(18,95)(19,96)(20,89)(21,90)(22,91)(23,92)(24,93)(25,66)(26,67)(27,68)(28,69)(29,70)(30,71)(31,72)(32,65)(33,60)(34,61)(35,62)(36,63)(37,64)(38,57)(39,58)(40,59)(41,75)(42,76)(43,77)(44,78)(45,79)(46,80)(47,73)(48,74)(49,83)(50,84)(51,85)(52,86)(53,87)(54,88)(55,81)(56,82), (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), (1,106)(2,105)(3,112)(4,111)(5,110)(6,109)(7,108)(8,107)(9,98)(10,97)(11,104)(12,103)(13,102)(14,101)(15,100)(16,99)(17,92)(18,91)(19,90)(20,89)(21,96)(22,95)(23,94)(24,93)(25,68)(26,67)(27,66)(28,65)(29,72)(30,71)(31,70)(32,69)(33,58)(34,57)(35,64)(36,63)(37,62)(38,61)(39,60)(40,59)(41,75)(42,74)(43,73)(44,80)(45,79)(46,78)(47,77)(48,76)(49,85)(50,84)(51,83)(52,82)(53,81)(54,88)(55,87)(56,86), (1,72,104,64,81,90,80)(2,65,97,57,82,91,73)(3,66,98,58,83,92,74)(4,67,99,59,84,93,75)(5,68,100,60,85,94,76)(6,69,101,61,86,95,77)(7,70,102,62,87,96,78)(8,71,103,63,88,89,79)(9,33,51,17,42,112,27)(10,34,52,18,43,105,28)(11,35,53,19,44,106,29)(12,36,54,20,45,107,30)(13,37,55,21,46,108,31)(14,38,56,22,47,109,32)(15,39,49,23,48,110,25)(16,40,50,24,41,111,26), (1,46)(2,43)(3,48)(4,45)(5,42)(6,47)(7,44)(8,41)(9,85)(10,82)(11,87)(12,84)(13,81)(14,86)(15,83)(16,88)(17,68)(18,65)(19,70)(20,67)(21,72)(22,69)(23,66)(24,71)(25,92)(26,89)(27,94)(28,91)(29,96)(30,93)(31,90)(32,95)(33,60)(34,57)(35,62)(36,59)(37,64)(38,61)(39,58)(40,63)(49,98)(50,103)(51,100)(52,97)(53,102)(54,99)(55,104)(56,101)(73,105)(74,110)(75,107)(76,112)(77,109)(78,106)(79,111)(80,108)>;
G:=Group( (1,108)(2,109)(3,110)(4,111)(5,112)(6,105)(7,106)(8,107)(9,100)(10,101)(11,102)(12,103)(13,104)(14,97)(15,98)(16,99)(17,94)(18,95)(19,96)(20,89)(21,90)(22,91)(23,92)(24,93)(25,66)(26,67)(27,68)(28,69)(29,70)(30,71)(31,72)(32,65)(33,60)(34,61)(35,62)(36,63)(37,64)(38,57)(39,58)(40,59)(41,75)(42,76)(43,77)(44,78)(45,79)(46,80)(47,73)(48,74)(49,83)(50,84)(51,85)(52,86)(53,87)(54,88)(55,81)(56,82), (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), (1,106)(2,105)(3,112)(4,111)(5,110)(6,109)(7,108)(8,107)(9,98)(10,97)(11,104)(12,103)(13,102)(14,101)(15,100)(16,99)(17,92)(18,91)(19,90)(20,89)(21,96)(22,95)(23,94)(24,93)(25,68)(26,67)(27,66)(28,65)(29,72)(30,71)(31,70)(32,69)(33,58)(34,57)(35,64)(36,63)(37,62)(38,61)(39,60)(40,59)(41,75)(42,74)(43,73)(44,80)(45,79)(46,78)(47,77)(48,76)(49,85)(50,84)(51,83)(52,82)(53,81)(54,88)(55,87)(56,86), (1,72,104,64,81,90,80)(2,65,97,57,82,91,73)(3,66,98,58,83,92,74)(4,67,99,59,84,93,75)(5,68,100,60,85,94,76)(6,69,101,61,86,95,77)(7,70,102,62,87,96,78)(8,71,103,63,88,89,79)(9,33,51,17,42,112,27)(10,34,52,18,43,105,28)(11,35,53,19,44,106,29)(12,36,54,20,45,107,30)(13,37,55,21,46,108,31)(14,38,56,22,47,109,32)(15,39,49,23,48,110,25)(16,40,50,24,41,111,26), (1,46)(2,43)(3,48)(4,45)(5,42)(6,47)(7,44)(8,41)(9,85)(10,82)(11,87)(12,84)(13,81)(14,86)(15,83)(16,88)(17,68)(18,65)(19,70)(20,67)(21,72)(22,69)(23,66)(24,71)(25,92)(26,89)(27,94)(28,91)(29,96)(30,93)(31,90)(32,95)(33,60)(34,57)(35,62)(36,59)(37,64)(38,61)(39,58)(40,63)(49,98)(50,103)(51,100)(52,97)(53,102)(54,99)(55,104)(56,101)(73,105)(74,110)(75,107)(76,112)(77,109)(78,106)(79,111)(80,108) );
G=PermutationGroup([(1,108),(2,109),(3,110),(4,111),(5,112),(6,105),(7,106),(8,107),(9,100),(10,101),(11,102),(12,103),(13,104),(14,97),(15,98),(16,99),(17,94),(18,95),(19,96),(20,89),(21,90),(22,91),(23,92),(24,93),(25,66),(26,67),(27,68),(28,69),(29,70),(30,71),(31,72),(32,65),(33,60),(34,61),(35,62),(36,63),(37,64),(38,57),(39,58),(40,59),(41,75),(42,76),(43,77),(44,78),(45,79),(46,80),(47,73),(48,74),(49,83),(50,84),(51,85),(52,86),(53,87),(54,88),(55,81),(56,82)], [(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)], [(1,106),(2,105),(3,112),(4,111),(5,110),(6,109),(7,108),(8,107),(9,98),(10,97),(11,104),(12,103),(13,102),(14,101),(15,100),(16,99),(17,92),(18,91),(19,90),(20,89),(21,96),(22,95),(23,94),(24,93),(25,68),(26,67),(27,66),(28,65),(29,72),(30,71),(31,70),(32,69),(33,58),(34,57),(35,64),(36,63),(37,62),(38,61),(39,60),(40,59),(41,75),(42,74),(43,73),(44,80),(45,79),(46,78),(47,77),(48,76),(49,85),(50,84),(51,83),(52,82),(53,81),(54,88),(55,87),(56,86)], [(1,72,104,64,81,90,80),(2,65,97,57,82,91,73),(3,66,98,58,83,92,74),(4,67,99,59,84,93,75),(5,68,100,60,85,94,76),(6,69,101,61,86,95,77),(7,70,102,62,87,96,78),(8,71,103,63,88,89,79),(9,33,51,17,42,112,27),(10,34,52,18,43,105,28),(11,35,53,19,44,106,29),(12,36,54,20,45,107,30),(13,37,55,21,46,108,31),(14,38,56,22,47,109,32),(15,39,49,23,48,110,25),(16,40,50,24,41,111,26)], [(1,46),(2,43),(3,48),(4,45),(5,42),(6,47),(7,44),(8,41),(9,85),(10,82),(11,87),(12,84),(13,81),(14,86),(15,83),(16,88),(17,68),(18,65),(19,70),(20,67),(21,72),(22,69),(23,66),(24,71),(25,92),(26,89),(27,94),(28,91),(29,96),(30,93),(31,90),(32,95),(33,60),(34,57),(35,62),(36,59),(37,64),(38,61),(39,58),(40,63),(49,98),(50,103),(51,100),(52,97),(53,102),(54,99),(55,104),(56,101),(73,105),(74,110),(75,107),(76,112),(77,109),(78,106),(79,111),(80,108)])
Matrix representation ►G ⊆ GL8(𝔽113)
| 112 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 112 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 91 | 28 |
| 0 | 0 | 0 | 0 | 0 | 0 | 58 | 69 |
| 0 | 0 | 0 | 0 | 2 | 104 | 105 | 84 |
| 0 | 0 | 0 | 0 | 54 | 1 | 45 | 8 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 112 | 0 | 80 | 71 |
| 0 | 0 | 0 | 0 | 0 | 112 | 42 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 79 | 112 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 79 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 112 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 88 | 112 |
| 0 | 0 | 0 | 0 | 0 | 0 | 60 | 34 |
| 34 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 88 | 79 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 88 | 79 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 112 | 0 | 0 |
| 0 | 0 | 0 | 0 | 80 | 104 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 25 | 33 |
| 0 | 0 | 0 | 0 | 0 | 0 | 53 | 88 |
G:=sub<GL(8,GF(113))| [112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,2,54,0,0,0,0,0,0,104,1,0,0,0,0,91,58,105,45,0,0,0,0,28,69,84,8],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,80,42,1,0,0,0,0,0,71,0,0,1],[79,1,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,0,79,1,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,0,9,1,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,0,88,60,0,0,0,0,0,0,112,34],[34,88,0,0,0,0,0,0,1,79,0,0,0,0,0,0,0,0,34,88,0,0,0,0,0,0,1,79,0,0,0,0,0,0,0,0,9,80,0,0,0,0,0,0,112,104,0,0,0,0,0,0,0,0,25,53,0,0,0,0,0,0,33,88] >;
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | 4F | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 14J | ··· | 14U | 28A | ··· | 28F | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 14 | 14 | 28 | 28 | 2 | 2 | 14 | 14 | 28 | 28 | 2 | 2 | 2 | 4 | 4 | 28 | 28 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | ··· | 4 | 4 | ··· | 4 |
64 irreducible representations
| dim | 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 | D4 | D4 | D4 | D7 | D14 | D14 | D14 | C8⋊C22 | D4×D7 | D4×D7 | D8⋊D7 |
| kernel | C2×D8⋊D7 | C2×C8⋊D7 | C2×C56⋊C2 | D8⋊D7 | C2×D4⋊D7 | C2×D4.D7 | C14×D8 | C2×D4×D7 | C2×D4⋊2D7 | C4×D7 | C2×Dic7 | C22×D7 | C2×D8 | C2×C8 | D8 | C2×D4 | C14 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 8 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 3 | 3 | 12 | 6 | 2 | 3 | 3 | 12 |
In GAP, Magma, Sage, TeX
C_2\times D_8\rtimes D_7
% in TeX
G:=Group("C2xD8:D7"); // GroupNames label
G:=SmallGroup(448,1208);
// by ID
G=gap.SmallGroup(448,1208);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,1123,185,438,235,102,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^8=c^2=d^7=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,e*b*e=b^5,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀