p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.515C23, C4.362- 1+4, C4⋊C4.180D4, C8⋊4Q8.5C2, Q8.Q8.2C2, Q8⋊Q8.1C2, (C4×Q16).20C2, Q8⋊3Q8.5C2, (C2×Q8).246D4, C2.64(Q8○D8), C4⋊C4.440C23, C4⋊C8.139C22, (C2×C8).120C23, (C4×C8).233C22, (C2×C4).566C24, Q8.38(C4○D4), Q16⋊C4.1C2, C4⋊2Q16.12C2, C4⋊Q8.195C22, C8⋊C4.65C22, C4.Q8.75C22, C2.74(Q8⋊5D4), C4.80(C8.C22), C4.SD16.10C2, (C2×Q8).260C23, (C4×Q8).197C22, (C2×Q16).92C22, C2.D8.207C22, Q8⋊C4.91C22, C22.826(C22×D4), C42.C2.68C22, C42.30C22.1C2, C4.267(C2×C4○D4), (C2×C4).642(C2×D4), C2.88(C2×C8.C22), SmallGroup(128,2106)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.515C23
G = < a,b,c,d,e | a4=b4=1, c2=e2=b2, d2=a2, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc-1=ebe-1=b-1, bd=db, dcd-1=a2c, ece-1=bc, ede-1=b2d >
Subgroups: 256 in 163 conjugacy classes, 88 normal (38 characteristic)
C1, C2, C4, C4, C4, C22, C8, C2×C4, C2×C4, C2×C4, Q8, Q8, C42, C42, C42, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, Q16, C2×Q8, C2×Q8, C2×Q8, C4×C8, C8⋊C4, Q8⋊C4, Q8⋊C4, C4⋊C8, C4⋊C8, C4.Q8, C2.D8, C4×Q8, C4×Q8, C4×Q8, C42.C2, C42.C2, C4⋊Q8, C4⋊Q8, C4⋊Q8, C2×Q16, C2×Q16, C4×Q16, Q16⋊C4, C8⋊4Q8, C4⋊2Q16, C4⋊2Q16, Q8⋊Q8, Q8.Q8, C4.SD16, C42.30C22, Q8⋊3Q8, C42.515C23
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C8.C22, C22×D4, C2×C4○D4, 2- 1+4, Q8⋊5D4, C2×C8.C22, Q8○D8, C42.515C23
Character table of C42.515C23
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 4Q | 4R | 4S | 8A | 8B | 8C | 8D | 8E | 8F | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 | 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 | 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 | -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 | -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 | 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 | 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 | -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 | -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 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ9 | 1 | 1 | 1 | 1 | 1 | 1 | 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 |
| ρ10 | 1 | 1 | 1 | 1 | -1 | 1 | -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 |
| ρ11 | 1 | 1 | 1 | 1 | -1 | 1 | -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 |
| ρ12 | 1 | 1 | 1 | 1 | 1 | 1 | 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 |
| ρ13 | 1 | 1 | 1 | 1 | 1 | 1 | 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 |
| ρ14 | 1 | 1 | 1 | 1 | -1 | 1 | -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 |
| ρ15 | 1 | 1 | 1 | 1 | -1 | 1 | -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 |
| ρ16 | 1 | 1 | 1 | 1 | 1 | 1 | 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 |
| ρ17 | 2 | 2 | 2 | 2 | 2 | -2 | 2 | -2 | 2 | 2 | -2 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | 2 | 2 | -2 | -2 | -2 | -2 | 2 | -2 | 2 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | 2 | 2 | 2 | -2 | 2 | -2 | -2 | -2 | -2 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | 2 | 2 | 2 | -2 | -2 | -2 | -2 | -2 | 2 | 2 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 2 | -2 | 2 | -2 | 0 | 2 | 0 | -2 | 0 | 0 | 0 | -2 | 0 | 0 | -2i | 2 | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ22 | 2 | -2 | 2 | -2 | 0 | 2 | 0 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | 2i | -2 | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ23 | 2 | -2 | 2 | -2 | 0 | 2 | 0 | -2 | 0 | 0 | 0 | -2 | 0 | 0 | 2i | 2 | -2i | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ24 | 2 | -2 | 2 | -2 | 0 | 2 | 0 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | -2i | -2 | 2i | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ25 | 4 | -4 | -4 | 4 | 4 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C8.C22, Schur index 2 |
| ρ26 | 4 | -4 | 4 | -4 | 0 | -4 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from 2- 1+4, Schur index 2 |
| ρ27 | 4 | -4 | -4 | 4 | -4 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C8.C22, Schur index 2 |
| ρ28 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2√2 | -2√2 | 0 | 0 | symplectic lifted from Q8○D8, Schur index 2 |
| ρ29 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2√2 | 2√2 | 0 | 0 | symplectic lifted from Q8○D8, Schur index 2 |
►Regular action on 128 points
Generators in S128
(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 23 25 17)(2 24 26 18)(3 21 27 19)(4 22 28 20)(5 128 9 15)(6 125 10 16)(7 126 11 13)(8 127 12 14)(29 37 41 35)(30 38 42 36)(31 39 43 33)(32 40 44 34)(45 53 51 57)(46 54 52 58)(47 55 49 59)(48 56 50 60)(61 73 71 65)(62 74 72 66)(63 75 69 67)(64 76 70 68)(77 81 89 87)(78 82 90 88)(79 83 91 85)(80 84 92 86)(93 99 105 101)(94 100 106 102)(95 97 107 103)(96 98 108 104)(109 121 115 117)(110 122 116 118)(111 123 113 119)(112 124 114 120)
(1 89 25 77)(2 90 26 78)(3 91 27 79)(4 92 28 80)(5 76 9 68)(6 73 10 65)(7 74 11 66)(8 75 12 67)(13 72 126 62)(14 69 127 63)(15 70 128 64)(16 71 125 61)(17 87 23 81)(18 88 24 82)(19 85 21 83)(20 86 22 84)(29 105 41 93)(30 106 42 94)(31 107 43 95)(32 108 44 96)(33 103 39 97)(34 104 40 98)(35 101 37 99)(36 102 38 100)(45 115 51 109)(46 116 52 110)(47 113 49 111)(48 114 50 112)(53 121 57 117)(54 122 58 118)(55 123 59 119)(56 124 60 120)
(1 53 3 55)(2 56 4 54)(5 106 7 108)(6 105 8 107)(9 94 11 96)(10 93 12 95)(13 98 15 100)(14 97 16 99)(17 45 19 47)(18 48 20 46)(21 49 23 51)(22 52 24 50)(25 57 27 59)(26 60 28 58)(29 65 31 67)(30 68 32 66)(33 69 35 71)(34 72 36 70)(37 61 39 63)(38 64 40 62)(41 73 43 75)(42 76 44 74)(77 119 79 117)(78 118 80 120)(81 111 83 109)(82 110 84 112)(85 115 87 113)(86 114 88 116)(89 123 91 121)(90 122 92 124)(101 127 103 125)(102 126 104 128)
(1 41 25 29)(2 42 26 30)(3 43 27 31)(4 44 28 32)(5 114 9 112)(6 115 10 109)(7 116 11 110)(8 113 12 111)(13 118 126 122)(14 119 127 123)(15 120 128 124)(16 117 125 121)(17 35 23 37)(18 36 24 38)(19 33 21 39)(20 34 22 40)(45 61 51 71)(46 62 52 72)(47 63 49 69)(48 64 50 70)(53 65 57 73)(54 66 58 74)(55 67 59 75)(56 68 60 76)(77 99 89 101)(78 100 90 102)(79 97 91 103)(80 98 92 104)(81 93 87 105)(82 94 88 106)(83 95 85 107)(84 96 86 108)
G:=sub<Sym(128)| (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), (1,23,25,17)(2,24,26,18)(3,21,27,19)(4,22,28,20)(5,128,9,15)(6,125,10,16)(7,126,11,13)(8,127,12,14)(29,37,41,35)(30,38,42,36)(31,39,43,33)(32,40,44,34)(45,53,51,57)(46,54,52,58)(47,55,49,59)(48,56,50,60)(61,73,71,65)(62,74,72,66)(63,75,69,67)(64,76,70,68)(77,81,89,87)(78,82,90,88)(79,83,91,85)(80,84,92,86)(93,99,105,101)(94,100,106,102)(95,97,107,103)(96,98,108,104)(109,121,115,117)(110,122,116,118)(111,123,113,119)(112,124,114,120), (1,89,25,77)(2,90,26,78)(3,91,27,79)(4,92,28,80)(5,76,9,68)(6,73,10,65)(7,74,11,66)(8,75,12,67)(13,72,126,62)(14,69,127,63)(15,70,128,64)(16,71,125,61)(17,87,23,81)(18,88,24,82)(19,85,21,83)(20,86,22,84)(29,105,41,93)(30,106,42,94)(31,107,43,95)(32,108,44,96)(33,103,39,97)(34,104,40,98)(35,101,37,99)(36,102,38,100)(45,115,51,109)(46,116,52,110)(47,113,49,111)(48,114,50,112)(53,121,57,117)(54,122,58,118)(55,123,59,119)(56,124,60,120), (1,53,3,55)(2,56,4,54)(5,106,7,108)(6,105,8,107)(9,94,11,96)(10,93,12,95)(13,98,15,100)(14,97,16,99)(17,45,19,47)(18,48,20,46)(21,49,23,51)(22,52,24,50)(25,57,27,59)(26,60,28,58)(29,65,31,67)(30,68,32,66)(33,69,35,71)(34,72,36,70)(37,61,39,63)(38,64,40,62)(41,73,43,75)(42,76,44,74)(77,119,79,117)(78,118,80,120)(81,111,83,109)(82,110,84,112)(85,115,87,113)(86,114,88,116)(89,123,91,121)(90,122,92,124)(101,127,103,125)(102,126,104,128), (1,41,25,29)(2,42,26,30)(3,43,27,31)(4,44,28,32)(5,114,9,112)(6,115,10,109)(7,116,11,110)(8,113,12,111)(13,118,126,122)(14,119,127,123)(15,120,128,124)(16,117,125,121)(17,35,23,37)(18,36,24,38)(19,33,21,39)(20,34,22,40)(45,61,51,71)(46,62,52,72)(47,63,49,69)(48,64,50,70)(53,65,57,73)(54,66,58,74)(55,67,59,75)(56,68,60,76)(77,99,89,101)(78,100,90,102)(79,97,91,103)(80,98,92,104)(81,93,87,105)(82,94,88,106)(83,95,85,107)(84,96,86,108)>;
G:=Group( (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), (1,23,25,17)(2,24,26,18)(3,21,27,19)(4,22,28,20)(5,128,9,15)(6,125,10,16)(7,126,11,13)(8,127,12,14)(29,37,41,35)(30,38,42,36)(31,39,43,33)(32,40,44,34)(45,53,51,57)(46,54,52,58)(47,55,49,59)(48,56,50,60)(61,73,71,65)(62,74,72,66)(63,75,69,67)(64,76,70,68)(77,81,89,87)(78,82,90,88)(79,83,91,85)(80,84,92,86)(93,99,105,101)(94,100,106,102)(95,97,107,103)(96,98,108,104)(109,121,115,117)(110,122,116,118)(111,123,113,119)(112,124,114,120), (1,89,25,77)(2,90,26,78)(3,91,27,79)(4,92,28,80)(5,76,9,68)(6,73,10,65)(7,74,11,66)(8,75,12,67)(13,72,126,62)(14,69,127,63)(15,70,128,64)(16,71,125,61)(17,87,23,81)(18,88,24,82)(19,85,21,83)(20,86,22,84)(29,105,41,93)(30,106,42,94)(31,107,43,95)(32,108,44,96)(33,103,39,97)(34,104,40,98)(35,101,37,99)(36,102,38,100)(45,115,51,109)(46,116,52,110)(47,113,49,111)(48,114,50,112)(53,121,57,117)(54,122,58,118)(55,123,59,119)(56,124,60,120), (1,53,3,55)(2,56,4,54)(5,106,7,108)(6,105,8,107)(9,94,11,96)(10,93,12,95)(13,98,15,100)(14,97,16,99)(17,45,19,47)(18,48,20,46)(21,49,23,51)(22,52,24,50)(25,57,27,59)(26,60,28,58)(29,65,31,67)(30,68,32,66)(33,69,35,71)(34,72,36,70)(37,61,39,63)(38,64,40,62)(41,73,43,75)(42,76,44,74)(77,119,79,117)(78,118,80,120)(81,111,83,109)(82,110,84,112)(85,115,87,113)(86,114,88,116)(89,123,91,121)(90,122,92,124)(101,127,103,125)(102,126,104,128), (1,41,25,29)(2,42,26,30)(3,43,27,31)(4,44,28,32)(5,114,9,112)(6,115,10,109)(7,116,11,110)(8,113,12,111)(13,118,126,122)(14,119,127,123)(15,120,128,124)(16,117,125,121)(17,35,23,37)(18,36,24,38)(19,33,21,39)(20,34,22,40)(45,61,51,71)(46,62,52,72)(47,63,49,69)(48,64,50,70)(53,65,57,73)(54,66,58,74)(55,67,59,75)(56,68,60,76)(77,99,89,101)(78,100,90,102)(79,97,91,103)(80,98,92,104)(81,93,87,105)(82,94,88,106)(83,95,85,107)(84,96,86,108) );
G=PermutationGroup([[(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)], [(1,23,25,17),(2,24,26,18),(3,21,27,19),(4,22,28,20),(5,128,9,15),(6,125,10,16),(7,126,11,13),(8,127,12,14),(29,37,41,35),(30,38,42,36),(31,39,43,33),(32,40,44,34),(45,53,51,57),(46,54,52,58),(47,55,49,59),(48,56,50,60),(61,73,71,65),(62,74,72,66),(63,75,69,67),(64,76,70,68),(77,81,89,87),(78,82,90,88),(79,83,91,85),(80,84,92,86),(93,99,105,101),(94,100,106,102),(95,97,107,103),(96,98,108,104),(109,121,115,117),(110,122,116,118),(111,123,113,119),(112,124,114,120)], [(1,89,25,77),(2,90,26,78),(3,91,27,79),(4,92,28,80),(5,76,9,68),(6,73,10,65),(7,74,11,66),(8,75,12,67),(13,72,126,62),(14,69,127,63),(15,70,128,64),(16,71,125,61),(17,87,23,81),(18,88,24,82),(19,85,21,83),(20,86,22,84),(29,105,41,93),(30,106,42,94),(31,107,43,95),(32,108,44,96),(33,103,39,97),(34,104,40,98),(35,101,37,99),(36,102,38,100),(45,115,51,109),(46,116,52,110),(47,113,49,111),(48,114,50,112),(53,121,57,117),(54,122,58,118),(55,123,59,119),(56,124,60,120)], [(1,53,3,55),(2,56,4,54),(5,106,7,108),(6,105,8,107),(9,94,11,96),(10,93,12,95),(13,98,15,100),(14,97,16,99),(17,45,19,47),(18,48,20,46),(21,49,23,51),(22,52,24,50),(25,57,27,59),(26,60,28,58),(29,65,31,67),(30,68,32,66),(33,69,35,71),(34,72,36,70),(37,61,39,63),(38,64,40,62),(41,73,43,75),(42,76,44,74),(77,119,79,117),(78,118,80,120),(81,111,83,109),(82,110,84,112),(85,115,87,113),(86,114,88,116),(89,123,91,121),(90,122,92,124),(101,127,103,125),(102,126,104,128)], [(1,41,25,29),(2,42,26,30),(3,43,27,31),(4,44,28,32),(5,114,9,112),(6,115,10,109),(7,116,11,110),(8,113,12,111),(13,118,126,122),(14,119,127,123),(15,120,128,124),(16,117,125,121),(17,35,23,37),(18,36,24,38),(19,33,21,39),(20,34,22,40),(45,61,51,71),(46,62,52,72),(47,63,49,69),(48,64,50,70),(53,65,57,73),(54,66,58,74),(55,67,59,75),(56,68,60,76),(77,99,89,101),(78,100,90,102),(79,97,91,103),(80,98,92,104),(81,93,87,105),(82,94,88,106),(83,95,85,107),(84,96,86,108)]])
Matrix representation of C42.515C23 ►in GL6(𝔽17)
| 16 | 16 | 0 | 0 | 0 | 0 |
| 2 | 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 | 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 13 | 13 | 0 | 0 | 0 | 0 |
| 8 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 5 | 9 | 8 |
| 0 | 0 | 5 | 4 | 9 | 9 |
| 0 | 0 | 9 | 9 | 12 | 13 |
| 0 | 0 | 8 | 9 | 13 | 5 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 7 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 12 | 16 |
| 0 | 0 | 0 | 3 | 16 | 5 |
| 0 | 0 | 12 | 16 | 14 | 0 |
| 0 | 0 | 16 | 5 | 0 | 14 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
G:=sub<GL(6,GF(17))| [16,2,0,0,0,0,16,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0],[13,8,0,0,0,0,13,4,0,0,0,0,0,0,13,5,9,8,0,0,5,4,9,9,0,0,9,9,12,13,0,0,8,9,13,5],[0,7,0,0,0,0,12,0,0,0,0,0,0,0,3,0,12,16,0,0,0,3,16,5,0,0,12,16,14,0,0,0,16,5,0,14],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,1,0,0,0,0,0,0,1,0,0] >;
C42.515C23 in GAP, Magma, Sage, TeX
C_4^2._{515}C_2^3 % in TeX
G:=Group("C4^2.515C2^3"); // GroupNames label
G:=SmallGroup(128,2106);
// by ID
G=gap.SmallGroup(128,2106);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,120,758,723,352,346,80,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=1,c^2=e^2=b^2,d^2=a^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,a*e=e*a,c*b*c^-1=e*b*e^-1=b^-1,b*d=d*b,d*c*d^-1=a^2*c,e*c*e^-1=b*c,e*d*e^-1=b^2*d>;
// generators/relations
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