metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4⋊F5⋊3S3, D6⋊F5.C2, (C4×S3)⋊1F5, (S3×C20)⋊1C4, (C4×D15)⋊1C4, C60⋊C4⋊3C2, (C2×F5).1D6, D6.5(C2×F5), C4.19(S3×F5), C20.19(C4×S3), C60.19(C2×C4), D30.C2⋊4C4, (Dic3×F5)⋊1C2, (S3×Dic5)⋊4C4, (C4×D5).64D6, D30.5(C2×C4), C12.13(C2×F5), C6.2(C22×F5), C30.2(C22×C4), (C6×F5).1C22, C15⋊1(C42⋊C2), Dic15.7(C2×C4), Dic5.25(C4×S3), Dic3.11(C2×F5), (C6×D5).22C23, D5.2(D4⋊2S3), D5.1(Q8⋊3S3), (D5×C12).50C22, D10.25(C22×S3), (D5×Dic3).12C22, C3⋊2(D10.C23), C5⋊(C4⋊C4⋊7S3), C2.7(C2×S3×F5), (C3×C4⋊F5)⋊3C2, C10.2(S3×C2×C4), (C4×S3×D5).2C2, (S3×C10).5(C2×C4), (C2×C3⋊F5).2C22, (C2×S3×D5).14C22, (C3×D5).2(C4○D4), (C5×Dic3).7(C2×C4), (C3×Dic5).20(C2×C4), SmallGroup(480,983)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4⋊F5⋊3S3
G = < a,b,c,d,e | a4=b5=c4=d3=e2=1, ab=ba, cac-1=a-1, ad=da, ae=ea, cbc-1=b3, bd=db, be=eb, cd=dc, ece=a2c, ede=d-1 >
Subgroups: 820 in 152 conjugacy classes, 50 normal (40 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C6, C2×C4, C23, D5, D5, C10, C10, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C15, C42, C22⋊C4, C4⋊C4, C22×C4, Dic5, Dic5, C20, C20, F5, D10, D10, C2×C10, C4×S3, C4×S3, C2×Dic3, C2×C12, C22×S3, C5×S3, C3×D5, D15, C30, C42⋊C2, C4×D5, C4×D5, C2×Dic5, C2×C20, C2×F5, C2×F5, C22×D5, C4×Dic3, C4⋊Dic3, D6⋊C4, C3×C4⋊C4, S3×C2×C4, C5×Dic3, C3×Dic5, Dic15, C60, C3×F5, C3⋊F5, S3×D5, C6×D5, S3×C10, D30, C4×F5, C4⋊F5, C4⋊F5, C22⋊F5, C2×C4×D5, C4⋊C4⋊7S3, D5×Dic3, S3×Dic5, D30.C2, D5×C12, S3×C20, C4×D15, C6×F5, C2×C3⋊F5, C2×S3×D5, D10.C23, Dic3×F5, D6⋊F5, C3×C4⋊F5, C60⋊C4, C4×S3×D5, C4⋊F5⋊3S3
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, C22×C4, C4○D4, F5, C4×S3, C22×S3, C42⋊C2, C2×F5, S3×C2×C4, D4⋊2S3, Q8⋊3S3, C22×F5, C4⋊C4⋊7S3, S3×F5, D10.C23, C2×S3×F5, C4⋊F5⋊3S3
►On 120 points
Generators in S120
(1 49 19 34)(2 50 20 35)(3 46 16 31)(4 47 17 32)(5 48 18 33)(6 51 21 36)(7 52 22 37)(8 53 23 38)(9 54 24 39)(10 55 25 40)(11 56 26 41)(12 57 27 42)(13 58 28 43)(14 59 29 44)(15 60 30 45)(61 106 76 91)(62 107 77 92)(63 108 78 93)(64 109 79 94)(65 110 80 95)(66 111 81 96)(67 112 82 97)(68 113 83 98)(69 114 84 99)(70 115 85 100)(71 116 86 101)(72 117 87 102)(73 118 88 103)(74 119 89 104)(75 120 90 105)
(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)(111 112 113 114 115)(116 117 118 119 120)
(2 3 5 4)(6 8 7 10)(11 13 12 15)(16 18 17 20)(21 23 22 25)(26 28 27 30)(31 48 32 50)(33 47 35 46)(34 49)(36 53 37 55)(38 52 40 51)(39 54)(41 58 42 60)(43 57 45 56)(44 59)(61 63 62 65)(66 68 67 70)(71 73 72 75)(76 78 77 80)(81 83 82 85)(86 88 87 90)(91 108 92 110)(93 107 95 106)(94 109)(96 113 97 115)(98 112 100 111)(99 114)(101 118 102 120)(103 117 105 116)(104 119)
(1 9 14)(2 10 15)(3 6 11)(4 7 12)(5 8 13)(16 21 26)(17 22 27)(18 23 28)(19 24 29)(20 25 30)(31 36 41)(32 37 42)(33 38 43)(34 39 44)(35 40 45)(46 51 56)(47 52 57)(48 53 58)(49 54 59)(50 55 60)(61 71 66)(62 72 67)(63 73 68)(64 74 69)(65 75 70)(76 86 81)(77 87 82)(78 88 83)(79 89 84)(80 90 85)(91 101 96)(92 102 97)(93 103 98)(94 104 99)(95 105 100)(106 116 111)(107 117 112)(108 118 113)(109 119 114)(110 120 115)
(1 94)(2 95)(3 91)(4 92)(5 93)(6 96)(7 97)(8 98)(9 99)(10 100)(11 101)(12 102)(13 103)(14 104)(15 105)(16 106)(17 107)(18 108)(19 109)(20 110)(21 111)(22 112)(23 113)(24 114)(25 115)(26 116)(27 117)(28 118)(29 119)(30 120)(31 76)(32 77)(33 78)(34 79)(35 80)(36 81)(37 82)(38 83)(39 84)(40 85)(41 86)(42 87)(43 88)(44 89)(45 90)(46 61)(47 62)(48 63)(49 64)(50 65)(51 66)(52 67)(53 68)(54 69)(55 70)(56 71)(57 72)(58 73)(59 74)(60 75)
G:=sub<Sym(120)| (1,49,19,34)(2,50,20,35)(3,46,16,31)(4,47,17,32)(5,48,18,33)(6,51,21,36)(7,52,22,37)(8,53,23,38)(9,54,24,39)(10,55,25,40)(11,56,26,41)(12,57,27,42)(13,58,28,43)(14,59,29,44)(15,60,30,45)(61,106,76,91)(62,107,77,92)(63,108,78,93)(64,109,79,94)(65,110,80,95)(66,111,81,96)(67,112,82,97)(68,113,83,98)(69,114,84,99)(70,115,85,100)(71,116,86,101)(72,117,87,102)(73,118,88,103)(74,119,89,104)(75,120,90,105), (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)(111,112,113,114,115)(116,117,118,119,120), (2,3,5,4)(6,8,7,10)(11,13,12,15)(16,18,17,20)(21,23,22,25)(26,28,27,30)(31,48,32,50)(33,47,35,46)(34,49)(36,53,37,55)(38,52,40,51)(39,54)(41,58,42,60)(43,57,45,56)(44,59)(61,63,62,65)(66,68,67,70)(71,73,72,75)(76,78,77,80)(81,83,82,85)(86,88,87,90)(91,108,92,110)(93,107,95,106)(94,109)(96,113,97,115)(98,112,100,111)(99,114)(101,118,102,120)(103,117,105,116)(104,119), (1,9,14)(2,10,15)(3,6,11)(4,7,12)(5,8,13)(16,21,26)(17,22,27)(18,23,28)(19,24,29)(20,25,30)(31,36,41)(32,37,42)(33,38,43)(34,39,44)(35,40,45)(46,51,56)(47,52,57)(48,53,58)(49,54,59)(50,55,60)(61,71,66)(62,72,67)(63,73,68)(64,74,69)(65,75,70)(76,86,81)(77,87,82)(78,88,83)(79,89,84)(80,90,85)(91,101,96)(92,102,97)(93,103,98)(94,104,99)(95,105,100)(106,116,111)(107,117,112)(108,118,113)(109,119,114)(110,120,115), (1,94)(2,95)(3,91)(4,92)(5,93)(6,96)(7,97)(8,98)(9,99)(10,100)(11,101)(12,102)(13,103)(14,104)(15,105)(16,106)(17,107)(18,108)(19,109)(20,110)(21,111)(22,112)(23,113)(24,114)(25,115)(26,116)(27,117)(28,118)(29,119)(30,120)(31,76)(32,77)(33,78)(34,79)(35,80)(36,81)(37,82)(38,83)(39,84)(40,85)(41,86)(42,87)(43,88)(44,89)(45,90)(46,61)(47,62)(48,63)(49,64)(50,65)(51,66)(52,67)(53,68)(54,69)(55,70)(56,71)(57,72)(58,73)(59,74)(60,75)>;
G:=Group( (1,49,19,34)(2,50,20,35)(3,46,16,31)(4,47,17,32)(5,48,18,33)(6,51,21,36)(7,52,22,37)(8,53,23,38)(9,54,24,39)(10,55,25,40)(11,56,26,41)(12,57,27,42)(13,58,28,43)(14,59,29,44)(15,60,30,45)(61,106,76,91)(62,107,77,92)(63,108,78,93)(64,109,79,94)(65,110,80,95)(66,111,81,96)(67,112,82,97)(68,113,83,98)(69,114,84,99)(70,115,85,100)(71,116,86,101)(72,117,87,102)(73,118,88,103)(74,119,89,104)(75,120,90,105), (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)(111,112,113,114,115)(116,117,118,119,120), (2,3,5,4)(6,8,7,10)(11,13,12,15)(16,18,17,20)(21,23,22,25)(26,28,27,30)(31,48,32,50)(33,47,35,46)(34,49)(36,53,37,55)(38,52,40,51)(39,54)(41,58,42,60)(43,57,45,56)(44,59)(61,63,62,65)(66,68,67,70)(71,73,72,75)(76,78,77,80)(81,83,82,85)(86,88,87,90)(91,108,92,110)(93,107,95,106)(94,109)(96,113,97,115)(98,112,100,111)(99,114)(101,118,102,120)(103,117,105,116)(104,119), (1,9,14)(2,10,15)(3,6,11)(4,7,12)(5,8,13)(16,21,26)(17,22,27)(18,23,28)(19,24,29)(20,25,30)(31,36,41)(32,37,42)(33,38,43)(34,39,44)(35,40,45)(46,51,56)(47,52,57)(48,53,58)(49,54,59)(50,55,60)(61,71,66)(62,72,67)(63,73,68)(64,74,69)(65,75,70)(76,86,81)(77,87,82)(78,88,83)(79,89,84)(80,90,85)(91,101,96)(92,102,97)(93,103,98)(94,104,99)(95,105,100)(106,116,111)(107,117,112)(108,118,113)(109,119,114)(110,120,115), (1,94)(2,95)(3,91)(4,92)(5,93)(6,96)(7,97)(8,98)(9,99)(10,100)(11,101)(12,102)(13,103)(14,104)(15,105)(16,106)(17,107)(18,108)(19,109)(20,110)(21,111)(22,112)(23,113)(24,114)(25,115)(26,116)(27,117)(28,118)(29,119)(30,120)(31,76)(32,77)(33,78)(34,79)(35,80)(36,81)(37,82)(38,83)(39,84)(40,85)(41,86)(42,87)(43,88)(44,89)(45,90)(46,61)(47,62)(48,63)(49,64)(50,65)(51,66)(52,67)(53,68)(54,69)(55,70)(56,71)(57,72)(58,73)(59,74)(60,75) );
G=PermutationGroup([[(1,49,19,34),(2,50,20,35),(3,46,16,31),(4,47,17,32),(5,48,18,33),(6,51,21,36),(7,52,22,37),(8,53,23,38),(9,54,24,39),(10,55,25,40),(11,56,26,41),(12,57,27,42),(13,58,28,43),(14,59,29,44),(15,60,30,45),(61,106,76,91),(62,107,77,92),(63,108,78,93),(64,109,79,94),(65,110,80,95),(66,111,81,96),(67,112,82,97),(68,113,83,98),(69,114,84,99),(70,115,85,100),(71,116,86,101),(72,117,87,102),(73,118,88,103),(74,119,89,104),(75,120,90,105)], [(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),(111,112,113,114,115),(116,117,118,119,120)], [(2,3,5,4),(6,8,7,10),(11,13,12,15),(16,18,17,20),(21,23,22,25),(26,28,27,30),(31,48,32,50),(33,47,35,46),(34,49),(36,53,37,55),(38,52,40,51),(39,54),(41,58,42,60),(43,57,45,56),(44,59),(61,63,62,65),(66,68,67,70),(71,73,72,75),(76,78,77,80),(81,83,82,85),(86,88,87,90),(91,108,92,110),(93,107,95,106),(94,109),(96,113,97,115),(98,112,100,111),(99,114),(101,118,102,120),(103,117,105,116),(104,119)], [(1,9,14),(2,10,15),(3,6,11),(4,7,12),(5,8,13),(16,21,26),(17,22,27),(18,23,28),(19,24,29),(20,25,30),(31,36,41),(32,37,42),(33,38,43),(34,39,44),(35,40,45),(46,51,56),(47,52,57),(48,53,58),(49,54,59),(50,55,60),(61,71,66),(62,72,67),(63,73,68),(64,74,69),(65,75,70),(76,86,81),(77,87,82),(78,88,83),(79,89,84),(80,90,85),(91,101,96),(92,102,97),(93,103,98),(94,104,99),(95,105,100),(106,116,111),(107,117,112),(108,118,113),(109,119,114),(110,120,115)], [(1,94),(2,95),(3,91),(4,92),(5,93),(6,96),(7,97),(8,98),(9,99),(10,100),(11,101),(12,102),(13,103),(14,104),(15,105),(16,106),(17,107),(18,108),(19,109),(20,110),(21,111),(22,112),(23,113),(24,114),(25,115),(26,116),(27,117),(28,118),(29,119),(30,120),(31,76),(32,77),(33,78),(34,79),(35,80),(36,81),(37,82),(38,83),(39,84),(40,85),(41,86),(42,87),(43,88),(44,89),(45,90),(46,61),(47,62),(48,63),(49,64),(50,65),(51,66),(52,67),(53,68),(54,69),(55,70),(56,71),(57,72),(58,73),(59,74),(60,75)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | ··· | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 5 | 6A | 6B | 6C | 10A | 10B | 10C | 12A | 12B | ··· | 12F | 15 | 20A | 20B | 20C | 20D | 30 | 60A | 60B |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 6 | 6 | 6 | 10 | 10 | 10 | 12 | 12 | ··· | 12 | 15 | 20 | 20 | 20 | 20 | 30 | 60 | 60 |
| size | 1 | 1 | 5 | 5 | 6 | 30 | 2 | 2 | 3 | 3 | 10 | ··· | 10 | 15 | 15 | 30 | 30 | 30 | 30 | 4 | 2 | 10 | 10 | 4 | 12 | 12 | 4 | 20 | ··· | 20 | 8 | 4 | 4 | 12 | 12 | 8 | 8 | 8 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | S3 | D6 | D6 | C4○D4 | C4×S3 | C4×S3 | F5 | C2×F5 | C2×F5 | C2×F5 | D4⋊2S3 | Q8⋊3S3 | D10.C23 | S3×F5 | C2×S3×F5 | C4⋊F5⋊3S3 |
| kernel | C4⋊F5⋊3S3 | Dic3×F5 | D6⋊F5 | C3×C4⋊F5 | C60⋊C4 | C4×S3×D5 | S3×Dic5 | D30.C2 | S3×C20 | C4×D15 | C4⋊F5 | C4×D5 | C2×F5 | C3×D5 | Dic5 | C20 | C4×S3 | Dic3 | C12 | D6 | D5 | D5 | C3 | C4 | C2 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 4 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 2 |
Matrix representation of C4⋊F5⋊3S3 ►in GL8(𝔽61)
| 1 | 59 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 60 | 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 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 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 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 60 | 0 | 0 | 0 |
| 50 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 50 | 11 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 50 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 50 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 60 | 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 |
| 50 | 22 | 0 | 0 | 0 | 0 | 0 | 0 |
| 50 | 11 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 19 | 0 | 0 | 0 | 0 |
| 0 | 0 | 31 | 49 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 60 |
G:=sub<GL(8,GF(61))| [1,1,0,0,0,0,0,0,59,60,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,60,60,60,60,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0],[50,50,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,50,0,0,0,0,0,0,0,0,50,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,60,60,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[50,50,0,0,0,0,0,0,22,11,0,0,0,0,0,0,0,0,12,31,0,0,0,0,0,0,19,49,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60] >;
C4⋊F5⋊3S3 in GAP, Magma, Sage, TeX
C_4\rtimes F_5\rtimes_3S_3
% in TeX
G:=Group("C4:F5:3S3"); // GroupNames label
G:=SmallGroup(480,983);
// by ID
G=gap.SmallGroup(480,983);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,120,422,100,1356,9414,2379]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^5=c^4=d^3=e^2=1,a*b=b*a,c*a*c^-1=a^-1,a*d=d*a,a*e=e*a,c*b*c^-1=b^3,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e=a^2*c,e*d*e=d^-1>;
// generators/relations