p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.251C23, C4⋊C4.73D4, (C2×C8).189D4, (C2×Q8).62D4, C8⋊4Q8.8C2, C8⋊1C8.12C2, C4⋊C8.35C22, C4⋊2Q16.9C2, C4⋊Q8.72C22, C4.106(C4○D8), C2.15(C8⋊D4), C4.46(C8⋊C22), (C4×C8).217C22, Q8⋊Q8.10C2, C4.SD16.9C2, (C4×Q8).50C22, C4.10D8.13C2, C4.95(C8.C22), C2.17(D4.5D4), C2.13(Q8.D4), C22.212(C4⋊D4), (C2×C4).36(C4○D4), (C2×C4).1286(C2×D4), SmallGroup(128,432)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.251C23
G = < a,b,c,d,e | a4=b4=1, c2=a2, d2=ab2, e2=b2, ab=ba, cac-1=a-1, ad=da, ae=ea, cbc-1=ebe-1=b-1, bd=db, dcd-1=a-1c, ece-1=bc, ede-1=a2d >
Subgroups: 152 in 73 conjugacy classes, 32 normal (all characteristic)
C1, C2, C4, C4, C22, C8, C2×C4, C2×C4, Q8, C42, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, Q16, C2×Q8, C2×Q8, C4×C8, C8⋊C4, Q8⋊C4, C4⋊C8, C4⋊C8, C4.Q8, C4×Q8, C4⋊Q8, C2×Q16, C4.10D8, C8⋊1C8, C8⋊4Q8, C4⋊2Q16, Q8⋊Q8, C4.SD16, C42.251C23
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C4⋊D4, C4○D8, C8⋊C22, C8.C22, Q8.D4, C8⋊D4, D4.5D4, C42.251C23
Character table of C42.251C23
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 8 | 8 | 16 | 16 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | 1 | -1 | -1 | -1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | linear of order 2 |
| ρ4 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | linear of order 2 |
| ρ6 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | -1 | -1 | -1 | 1 | linear of order 2 |
| ρ7 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | linear of order 2 |
| ρ8 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | linear of order 2 |
| ρ9 | 2 | 2 | 2 | 2 | -2 | 2 | -2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | 2 | -2 | 2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | 2 | 2 | 2 | -2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ12 | 2 | 2 | 2 | 2 | 2 | -2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ13 | 2 | 2 | 2 | 2 | -2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | 0 | complex lifted from C4○D4 |
| ρ14 | 2 | 2 | 2 | 2 | -2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | 0 | complex lifted from C4○D4 |
| ρ15 | 2 | -2 | -2 | 2 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | 0 | √2 | -√-2 | -√2 | 0 | 0 | √-2 | complex lifted from C4○D8 |
| ρ16 | 2 | -2 | -2 | 2 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | 0 | -√2 | -√-2 | √2 | 0 | 0 | √-2 | complex lifted from C4○D8 |
| ρ17 | 2 | -2 | -2 | 2 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | 0 | -√2 | √-2 | √2 | 0 | 0 | -√-2 | complex lifted from C4○D8 |
| ρ18 | 2 | -2 | -2 | 2 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | 0 | √2 | √-2 | -√2 | 0 | 0 | -√-2 | complex lifted from C4○D8 |
| ρ19 | 4 | -4 | 4 | -4 | 0 | -4 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C8⋊C22 |
| ρ20 | 4 | -4 | -4 | 4 | 4 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C8.C22, Schur index 2 |
| ρ21 | 4 | -4 | 4 | -4 | 0 | 4 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C8.C22, Schur index 2 |
| ρ22 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2√2 | 0 | 0 | -2√2 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from D4.5D4, Schur index 2 |
| ρ23 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2√2 | 0 | 0 | 2√2 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from D4.5D4, Schur index 2 |
►Regular action on 128 points
Generators in S128
(1 56 5 52)(2 49 6 53)(3 50 7 54)(4 51 8 55)(9 121 13 125)(10 122 14 126)(11 123 15 127)(12 124 16 128)(17 27 21 31)(18 28 22 32)(19 29 23 25)(20 30 24 26)(33 42 37 46)(34 43 38 47)(35 44 39 48)(36 45 40 41)(57 98 61 102)(58 99 62 103)(59 100 63 104)(60 101 64 97)(65 80 69 76)(66 73 70 77)(67 74 71 78)(68 75 72 79)(81 107 85 111)(82 108 86 112)(83 109 87 105)(84 110 88 106)(89 120 93 116)(90 113 94 117)(91 114 95 118)(92 115 96 119)
(1 61 54 100)(2 62 55 101)(3 63 56 102)(4 64 49 103)(5 57 50 104)(6 58 51 97)(7 59 52 98)(8 60 53 99)(9 105 127 81)(10 106 128 82)(11 107 121 83)(12 108 122 84)(13 109 123 85)(14 110 124 86)(15 111 125 87)(16 112 126 88)(17 80 25 67)(18 73 26 68)(19 74 27 69)(20 75 28 70)(21 76 29 71)(22 77 30 72)(23 78 31 65)(24 79 32 66)(33 114 48 93)(34 115 41 94)(35 116 42 95)(36 117 43 96)(37 118 44 89)(38 119 45 90)(39 120 46 91)(40 113 47 92)
(1 109 5 105)(2 88 6 84)(3 107 7 111)(4 86 8 82)(9 61 13 57)(10 103 14 99)(11 59 15 63)(12 101 16 97)(17 94 21 90)(18 118 22 114)(19 92 23 96)(20 116 24 120)(25 115 29 119)(26 89 30 93)(27 113 31 117)(28 95 32 91)(33 73 37 77)(34 71 38 67)(35 79 39 75)(36 69 40 65)(41 76 45 80)(42 66 46 70)(43 74 47 78)(44 72 48 68)(49 110 53 106)(50 81 54 85)(51 108 55 112)(52 87 56 83)(58 122 62 126)(60 128 64 124)(98 125 102 121)(100 123 104 127)
(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)(121 122 123 124 125 126 127 128)
(1 21 54 29)(2 18 55 26)(3 23 56 31)(4 20 49 28)(5 17 50 25)(6 22 51 30)(7 19 52 27)(8 24 53 32)(9 94 127 115)(10 91 128 120)(11 96 121 117)(12 93 122 114)(13 90 123 119)(14 95 124 116)(15 92 125 113)(16 89 126 118)(33 108 48 84)(34 105 41 81)(35 110 42 86)(36 107 43 83)(37 112 44 88)(38 109 45 85)(39 106 46 82)(40 111 47 87)(57 67 104 80)(58 72 97 77)(59 69 98 74)(60 66 99 79)(61 71 100 76)(62 68 101 73)(63 65 102 78)(64 70 103 75)
G:=sub<Sym(128)| (1,56,5,52)(2,49,6,53)(3,50,7,54)(4,51,8,55)(9,121,13,125)(10,122,14,126)(11,123,15,127)(12,124,16,128)(17,27,21,31)(18,28,22,32)(19,29,23,25)(20,30,24,26)(33,42,37,46)(34,43,38,47)(35,44,39,48)(36,45,40,41)(57,98,61,102)(58,99,62,103)(59,100,63,104)(60,101,64,97)(65,80,69,76)(66,73,70,77)(67,74,71,78)(68,75,72,79)(81,107,85,111)(82,108,86,112)(83,109,87,105)(84,110,88,106)(89,120,93,116)(90,113,94,117)(91,114,95,118)(92,115,96,119), (1,61,54,100)(2,62,55,101)(3,63,56,102)(4,64,49,103)(5,57,50,104)(6,58,51,97)(7,59,52,98)(8,60,53,99)(9,105,127,81)(10,106,128,82)(11,107,121,83)(12,108,122,84)(13,109,123,85)(14,110,124,86)(15,111,125,87)(16,112,126,88)(17,80,25,67)(18,73,26,68)(19,74,27,69)(20,75,28,70)(21,76,29,71)(22,77,30,72)(23,78,31,65)(24,79,32,66)(33,114,48,93)(34,115,41,94)(35,116,42,95)(36,117,43,96)(37,118,44,89)(38,119,45,90)(39,120,46,91)(40,113,47,92), (1,109,5,105)(2,88,6,84)(3,107,7,111)(4,86,8,82)(9,61,13,57)(10,103,14,99)(11,59,15,63)(12,101,16,97)(17,94,21,90)(18,118,22,114)(19,92,23,96)(20,116,24,120)(25,115,29,119)(26,89,30,93)(27,113,31,117)(28,95,32,91)(33,73,37,77)(34,71,38,67)(35,79,39,75)(36,69,40,65)(41,76,45,80)(42,66,46,70)(43,74,47,78)(44,72,48,68)(49,110,53,106)(50,81,54,85)(51,108,55,112)(52,87,56,83)(58,122,62,126)(60,128,64,124)(98,125,102,121)(100,123,104,127), (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)(121,122,123,124,125,126,127,128), (1,21,54,29)(2,18,55,26)(3,23,56,31)(4,20,49,28)(5,17,50,25)(6,22,51,30)(7,19,52,27)(8,24,53,32)(9,94,127,115)(10,91,128,120)(11,96,121,117)(12,93,122,114)(13,90,123,119)(14,95,124,116)(15,92,125,113)(16,89,126,118)(33,108,48,84)(34,105,41,81)(35,110,42,86)(36,107,43,83)(37,112,44,88)(38,109,45,85)(39,106,46,82)(40,111,47,87)(57,67,104,80)(58,72,97,77)(59,69,98,74)(60,66,99,79)(61,71,100,76)(62,68,101,73)(63,65,102,78)(64,70,103,75)>;
G:=Group( (1,56,5,52)(2,49,6,53)(3,50,7,54)(4,51,8,55)(9,121,13,125)(10,122,14,126)(11,123,15,127)(12,124,16,128)(17,27,21,31)(18,28,22,32)(19,29,23,25)(20,30,24,26)(33,42,37,46)(34,43,38,47)(35,44,39,48)(36,45,40,41)(57,98,61,102)(58,99,62,103)(59,100,63,104)(60,101,64,97)(65,80,69,76)(66,73,70,77)(67,74,71,78)(68,75,72,79)(81,107,85,111)(82,108,86,112)(83,109,87,105)(84,110,88,106)(89,120,93,116)(90,113,94,117)(91,114,95,118)(92,115,96,119), (1,61,54,100)(2,62,55,101)(3,63,56,102)(4,64,49,103)(5,57,50,104)(6,58,51,97)(7,59,52,98)(8,60,53,99)(9,105,127,81)(10,106,128,82)(11,107,121,83)(12,108,122,84)(13,109,123,85)(14,110,124,86)(15,111,125,87)(16,112,126,88)(17,80,25,67)(18,73,26,68)(19,74,27,69)(20,75,28,70)(21,76,29,71)(22,77,30,72)(23,78,31,65)(24,79,32,66)(33,114,48,93)(34,115,41,94)(35,116,42,95)(36,117,43,96)(37,118,44,89)(38,119,45,90)(39,120,46,91)(40,113,47,92), (1,109,5,105)(2,88,6,84)(3,107,7,111)(4,86,8,82)(9,61,13,57)(10,103,14,99)(11,59,15,63)(12,101,16,97)(17,94,21,90)(18,118,22,114)(19,92,23,96)(20,116,24,120)(25,115,29,119)(26,89,30,93)(27,113,31,117)(28,95,32,91)(33,73,37,77)(34,71,38,67)(35,79,39,75)(36,69,40,65)(41,76,45,80)(42,66,46,70)(43,74,47,78)(44,72,48,68)(49,110,53,106)(50,81,54,85)(51,108,55,112)(52,87,56,83)(58,122,62,126)(60,128,64,124)(98,125,102,121)(100,123,104,127), (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)(121,122,123,124,125,126,127,128), (1,21,54,29)(2,18,55,26)(3,23,56,31)(4,20,49,28)(5,17,50,25)(6,22,51,30)(7,19,52,27)(8,24,53,32)(9,94,127,115)(10,91,128,120)(11,96,121,117)(12,93,122,114)(13,90,123,119)(14,95,124,116)(15,92,125,113)(16,89,126,118)(33,108,48,84)(34,105,41,81)(35,110,42,86)(36,107,43,83)(37,112,44,88)(38,109,45,85)(39,106,46,82)(40,111,47,87)(57,67,104,80)(58,72,97,77)(59,69,98,74)(60,66,99,79)(61,71,100,76)(62,68,101,73)(63,65,102,78)(64,70,103,75) );
G=PermutationGroup([[(1,56,5,52),(2,49,6,53),(3,50,7,54),(4,51,8,55),(9,121,13,125),(10,122,14,126),(11,123,15,127),(12,124,16,128),(17,27,21,31),(18,28,22,32),(19,29,23,25),(20,30,24,26),(33,42,37,46),(34,43,38,47),(35,44,39,48),(36,45,40,41),(57,98,61,102),(58,99,62,103),(59,100,63,104),(60,101,64,97),(65,80,69,76),(66,73,70,77),(67,74,71,78),(68,75,72,79),(81,107,85,111),(82,108,86,112),(83,109,87,105),(84,110,88,106),(89,120,93,116),(90,113,94,117),(91,114,95,118),(92,115,96,119)], [(1,61,54,100),(2,62,55,101),(3,63,56,102),(4,64,49,103),(5,57,50,104),(6,58,51,97),(7,59,52,98),(8,60,53,99),(9,105,127,81),(10,106,128,82),(11,107,121,83),(12,108,122,84),(13,109,123,85),(14,110,124,86),(15,111,125,87),(16,112,126,88),(17,80,25,67),(18,73,26,68),(19,74,27,69),(20,75,28,70),(21,76,29,71),(22,77,30,72),(23,78,31,65),(24,79,32,66),(33,114,48,93),(34,115,41,94),(35,116,42,95),(36,117,43,96),(37,118,44,89),(38,119,45,90),(39,120,46,91),(40,113,47,92)], [(1,109,5,105),(2,88,6,84),(3,107,7,111),(4,86,8,82),(9,61,13,57),(10,103,14,99),(11,59,15,63),(12,101,16,97),(17,94,21,90),(18,118,22,114),(19,92,23,96),(20,116,24,120),(25,115,29,119),(26,89,30,93),(27,113,31,117),(28,95,32,91),(33,73,37,77),(34,71,38,67),(35,79,39,75),(36,69,40,65),(41,76,45,80),(42,66,46,70),(43,74,47,78),(44,72,48,68),(49,110,53,106),(50,81,54,85),(51,108,55,112),(52,87,56,83),(58,122,62,126),(60,128,64,124),(98,125,102,121),(100,123,104,127)], [(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),(121,122,123,124,125,126,127,128)], [(1,21,54,29),(2,18,55,26),(3,23,56,31),(4,20,49,28),(5,17,50,25),(6,22,51,30),(7,19,52,27),(8,24,53,32),(9,94,127,115),(10,91,128,120),(11,96,121,117),(12,93,122,114),(13,90,123,119),(14,95,124,116),(15,92,125,113),(16,89,126,118),(33,108,48,84),(34,105,41,81),(35,110,42,86),(36,107,43,83),(37,112,44,88),(38,109,45,85),(39,106,46,82),(40,111,47,87),(57,67,104,80),(58,72,97,77),(59,69,98,74),(60,66,99,79),(61,71,100,76),(62,68,101,73),(63,65,102,78),(64,70,103,75)]])
Matrix representation of C42.251C23 ►in GL6(𝔽17)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 12 | 5 | 13 | 0 |
| 0 | 0 | 5 | 5 | 0 | 13 |
| 1 | 15 | 0 | 0 | 0 | 0 |
| 1 | 16 | 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 |
| 7 | 8 | 0 | 0 | 0 | 0 |
| 11 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 14 | 15 | 0 |
| 0 | 0 | 14 | 14 | 0 | 15 |
| 0 | 0 | 1 | 0 | 14 | 3 |
| 0 | 0 | 0 | 1 | 3 | 3 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 4 |
| 0 | 0 | 4 | 14 | 16 | 0 |
| 9 | 2 | 0 | 0 | 0 | 0 |
| 10 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 3 | 1 | 0 |
| 0 | 0 | 14 | 0 | 0 | 16 |
G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,12,5,0,0,0,4,5,5,0,0,0,0,13,0,0,0,0,0,0,13],[1,1,0,0,0,0,15,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[7,11,0,0,0,0,8,10,0,0,0,0,0,0,3,14,1,0,0,0,14,14,0,1,0,0,15,0,14,3,0,0,0,15,3,3],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,4,0,0,1,0,16,14,0,0,0,0,0,16,0,0,0,0,4,0],[9,10,0,0,0,0,2,8,0,0,0,0,0,0,1,0,0,14,0,0,0,16,3,0,0,0,0,0,1,0,0,0,0,0,0,16] >;
C42.251C23 in GAP, Magma, Sage, TeX
C_4^2._{251}C_2^3 % in TeX
G:=Group("C4^2.251C2^3"); // GroupNames label
G:=SmallGroup(128,432);
// by ID
G=gap.SmallGroup(128,432);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,224,141,512,422,387,352,1123,136,2804,718,172]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=1,c^2=a^2,d^2=a*b^2,e^2=b^2,a*b=b*a,c*a*c^-1=a^-1,a*d=d*a,a*e=e*a,c*b*c^-1=e*b*e^-1=b^-1,b*d=d*b,d*c*d^-1=a^-1*c,e*c*e^-1=b*c,e*d*e^-1=a^2*d>;
// generators/relations
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