p-group, metabelian, nilpotent (class 2), monomial, rational
Aliases: C42.40Q8, C23.745C24, C22.5182+ 1+4, C22.3972- 1+4, C42⋊8C4.52C2, (C2×C42).748C22, (C22×C4).256C23, C22.177(C22×Q8), C2.C42.446C22, C23.81C23.54C2, C2.8(C22.58C24), C2.64(C22.56C24), C2.52(C23.41C23), (C2×C4).140(C2×Q8), (C2×C4⋊C4).552C22, SmallGroup(128,1577)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.40Q8
G = < a,b,c,d | a4=b4=c4=1, d2=b2c2, ab=ba, cac-1=a-1b2, dad-1=ab2, cbc-1=b-1, dbd-1=a2b, dcd-1=b2c-1 >
Subgroups: 308 in 170 conjugacy classes, 92 normal (6 characteristic)
C1, C2, C2, C4, C22, C22, C2×C4, C2×C4, C23, C42, C4⋊C4, C22×C4, C2.C42, C2×C42, C2×C4⋊C4, C42⋊8C4, C23.81C23, C42.40Q8
Quotients: C1, C2, C22, Q8, C23, C2×Q8, C24, C22×Q8, 2+ 1+4, 2- 1+4, C23.41C23, C22.56C24, C22.58C24, C42.40Q8
Character table of C42.40Q8
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 4Q | 4R | |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | linear of order 2 |
| ρ17 | 2 | -2 | 2 | -2 | 2 | -2 | 2 | -2 | -2 | 2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ18 | 2 | -2 | 2 | -2 | 2 | -2 | 2 | -2 | 2 | -2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ19 | 2 | -2 | 2 | -2 | 2 | -2 | 2 | -2 | -2 | 2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ20 | 2 | -2 | 2 | -2 | 2 | -2 | 2 | -2 | 2 | -2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ21 | 4 | -4 | -4 | 4 | -4 | 4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from 2+ 1+4 |
| ρ22 | 4 | -4 | 4 | 4 | -4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from 2+ 1+4 |
| ρ23 | 4 | -4 | -4 | -4 | 4 | 4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from 2+ 1+4 |
| ρ24 | 4 | 4 | -4 | 4 | 4 | -4 | -4 | -4 | 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 |
| ρ25 | 4 | 4 | 4 | -4 | -4 | 4 | -4 | -4 | 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 |
| ρ26 | 4 | 4 | -4 | -4 | -4 | -4 | 4 | 4 | 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 |
►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 76 103 13)(2 73 104 14)(3 74 101 15)(4 75 102 16)(5 36 70 95)(6 33 71 96)(7 34 72 93)(8 35 69 94)(9 107 40 48)(10 108 37 45)(11 105 38 46)(12 106 39 47)(17 56 80 115)(18 53 77 116)(19 54 78 113)(20 55 79 114)(21 111 84 52)(22 112 81 49)(23 109 82 50)(24 110 83 51)(25 121 88 62)(26 122 85 63)(27 123 86 64)(28 124 87 61)(29 58 92 117)(30 59 89 118)(31 60 90 119)(32 57 91 120)(41 126 97 68)(42 127 98 65)(43 128 99 66)(44 125 100 67)
(1 78 38 52)(2 18 39 110)(3 80 40 50)(4 20 37 112)(5 30 43 122)(6 92 44 62)(7 32 41 124)(8 90 42 64)(9 109 101 17)(10 49 102 79)(11 111 103 19)(12 51 104 77)(13 113 105 21)(14 53 106 83)(15 115 107 23)(16 55 108 81)(22 75 114 45)(24 73 116 47)(25 96 117 67)(26 36 118 128)(27 94 119 65)(28 34 120 126)(29 100 121 71)(31 98 123 69)(33 58 125 88)(35 60 127 86)(46 84 76 54)(48 82 74 56)(57 68 87 93)(59 66 85 95)(61 72 91 97)(63 70 89 99)
(1 27 11 60)(2 87 12 120)(3 25 9 58)(4 85 10 118)(5 53 99 24)(6 113 100 84)(7 55 97 22)(8 115 98 82)(13 62 46 29)(14 122 47 89)(15 64 48 31)(16 124 45 91)(17 67 50 33)(18 126 51 93)(19 65 52 35)(20 128 49 95)(21 71 54 44)(23 69 56 42)(26 37 59 102)(28 39 57 104)(30 73 63 106)(32 75 61 108)(34 77 68 110)(36 79 66 112)(38 119 103 86)(40 117 101 88)(41 81 72 114)(43 83 70 116)(74 123 107 90)(76 121 105 92)(78 127 111 94)(80 125 109 96)
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), (1,76,103,13)(2,73,104,14)(3,74,101,15)(4,75,102,16)(5,36,70,95)(6,33,71,96)(7,34,72,93)(8,35,69,94)(9,107,40,48)(10,108,37,45)(11,105,38,46)(12,106,39,47)(17,56,80,115)(18,53,77,116)(19,54,78,113)(20,55,79,114)(21,111,84,52)(22,112,81,49)(23,109,82,50)(24,110,83,51)(25,121,88,62)(26,122,85,63)(27,123,86,64)(28,124,87,61)(29,58,92,117)(30,59,89,118)(31,60,90,119)(32,57,91,120)(41,126,97,68)(42,127,98,65)(43,128,99,66)(44,125,100,67), (1,78,38,52)(2,18,39,110)(3,80,40,50)(4,20,37,112)(5,30,43,122)(6,92,44,62)(7,32,41,124)(8,90,42,64)(9,109,101,17)(10,49,102,79)(11,111,103,19)(12,51,104,77)(13,113,105,21)(14,53,106,83)(15,115,107,23)(16,55,108,81)(22,75,114,45)(24,73,116,47)(25,96,117,67)(26,36,118,128)(27,94,119,65)(28,34,120,126)(29,100,121,71)(31,98,123,69)(33,58,125,88)(35,60,127,86)(46,84,76,54)(48,82,74,56)(57,68,87,93)(59,66,85,95)(61,72,91,97)(63,70,89,99), (1,27,11,60)(2,87,12,120)(3,25,9,58)(4,85,10,118)(5,53,99,24)(6,113,100,84)(7,55,97,22)(8,115,98,82)(13,62,46,29)(14,122,47,89)(15,64,48,31)(16,124,45,91)(17,67,50,33)(18,126,51,93)(19,65,52,35)(20,128,49,95)(21,71,54,44)(23,69,56,42)(26,37,59,102)(28,39,57,104)(30,73,63,106)(32,75,61,108)(34,77,68,110)(36,79,66,112)(38,119,103,86)(40,117,101,88)(41,81,72,114)(43,83,70,116)(74,123,107,90)(76,121,105,92)(78,127,111,94)(80,125,109,96)>;
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), (1,76,103,13)(2,73,104,14)(3,74,101,15)(4,75,102,16)(5,36,70,95)(6,33,71,96)(7,34,72,93)(8,35,69,94)(9,107,40,48)(10,108,37,45)(11,105,38,46)(12,106,39,47)(17,56,80,115)(18,53,77,116)(19,54,78,113)(20,55,79,114)(21,111,84,52)(22,112,81,49)(23,109,82,50)(24,110,83,51)(25,121,88,62)(26,122,85,63)(27,123,86,64)(28,124,87,61)(29,58,92,117)(30,59,89,118)(31,60,90,119)(32,57,91,120)(41,126,97,68)(42,127,98,65)(43,128,99,66)(44,125,100,67), (1,78,38,52)(2,18,39,110)(3,80,40,50)(4,20,37,112)(5,30,43,122)(6,92,44,62)(7,32,41,124)(8,90,42,64)(9,109,101,17)(10,49,102,79)(11,111,103,19)(12,51,104,77)(13,113,105,21)(14,53,106,83)(15,115,107,23)(16,55,108,81)(22,75,114,45)(24,73,116,47)(25,96,117,67)(26,36,118,128)(27,94,119,65)(28,34,120,126)(29,100,121,71)(31,98,123,69)(33,58,125,88)(35,60,127,86)(46,84,76,54)(48,82,74,56)(57,68,87,93)(59,66,85,95)(61,72,91,97)(63,70,89,99), (1,27,11,60)(2,87,12,120)(3,25,9,58)(4,85,10,118)(5,53,99,24)(6,113,100,84)(7,55,97,22)(8,115,98,82)(13,62,46,29)(14,122,47,89)(15,64,48,31)(16,124,45,91)(17,67,50,33)(18,126,51,93)(19,65,52,35)(20,128,49,95)(21,71,54,44)(23,69,56,42)(26,37,59,102)(28,39,57,104)(30,73,63,106)(32,75,61,108)(34,77,68,110)(36,79,66,112)(38,119,103,86)(40,117,101,88)(41,81,72,114)(43,83,70,116)(74,123,107,90)(76,121,105,92)(78,127,111,94)(80,125,109,96) );
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)], [(1,76,103,13),(2,73,104,14),(3,74,101,15),(4,75,102,16),(5,36,70,95),(6,33,71,96),(7,34,72,93),(8,35,69,94),(9,107,40,48),(10,108,37,45),(11,105,38,46),(12,106,39,47),(17,56,80,115),(18,53,77,116),(19,54,78,113),(20,55,79,114),(21,111,84,52),(22,112,81,49),(23,109,82,50),(24,110,83,51),(25,121,88,62),(26,122,85,63),(27,123,86,64),(28,124,87,61),(29,58,92,117),(30,59,89,118),(31,60,90,119),(32,57,91,120),(41,126,97,68),(42,127,98,65),(43,128,99,66),(44,125,100,67)], [(1,78,38,52),(2,18,39,110),(3,80,40,50),(4,20,37,112),(5,30,43,122),(6,92,44,62),(7,32,41,124),(8,90,42,64),(9,109,101,17),(10,49,102,79),(11,111,103,19),(12,51,104,77),(13,113,105,21),(14,53,106,83),(15,115,107,23),(16,55,108,81),(22,75,114,45),(24,73,116,47),(25,96,117,67),(26,36,118,128),(27,94,119,65),(28,34,120,126),(29,100,121,71),(31,98,123,69),(33,58,125,88),(35,60,127,86),(46,84,76,54),(48,82,74,56),(57,68,87,93),(59,66,85,95),(61,72,91,97),(63,70,89,99)], [(1,27,11,60),(2,87,12,120),(3,25,9,58),(4,85,10,118),(5,53,99,24),(6,113,100,84),(7,55,97,22),(8,115,98,82),(13,62,46,29),(14,122,47,89),(15,64,48,31),(16,124,45,91),(17,67,50,33),(18,126,51,93),(19,65,52,35),(20,128,49,95),(21,71,54,44),(23,69,56,42),(26,37,59,102),(28,39,57,104),(30,73,63,106),(32,75,61,108),(34,77,68,110),(36,79,66,112),(38,119,103,86),(40,117,101,88),(41,81,72,114),(43,83,70,116),(74,123,107,90),(76,121,105,92),(78,127,111,94),(80,125,109,96)]])
Matrix representation of C42.40Q8 ►in GL12(𝔽5)
| 0 | 1 | 1 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 4 | 0 | 3 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 4 | 2 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 2 | 1 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 | 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 3 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 2 | 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 4 | 2 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 4 | 4 | 2 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 3 | 3 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 2 | 3 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 2 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 3 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 2 | 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | 4 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | 4 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 3 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 2 | 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 3 | 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | 4 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 4 | 1 |
| 3 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 2 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 3 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 1 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 0 | 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 | 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 4 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 4 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
G:=sub<GL(12,GF(5))| [0,4,4,2,0,0,0,0,0,0,0,0,1,0,2,1,0,0,0,0,0,0,0,0,1,3,0,4,0,0,0,0,0,0,0,0,3,4,1,0,0,0,0,0,0,0,0,0,0,0,0,0,4,1,2,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,4,3,1,4,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,1,0,0,0,0,0,0,0,0,1,4,1,1,0,0,0,0,0,0,0,0,4,4,3,2,0,0,0,0,0,0,0,0,2,2,3,3],[0,4,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,1,4,3,0,0,0,0,0,0,0,0,0,2,4,3,2,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,4,0,0,0,0,0,0,0,0,0,1,0,4,4,0,0,0,0,0,0,0,0,0,0,4,4,0,0,0,0,0,0,0,0,0,0,0,1],[0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,2,3,0,0,0,0,0,0,0,0,3,1,2,3,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,1,1,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,1,4,0,4,0,0,0,0,0,0,0,0,0,0,0,1],[3,1,0,1,0,0,0,0,0,0,0,0,1,2,4,0,0,0,0,0,0,0,0,0,0,4,3,1,0,0,0,0,0,0,0,0,1,0,1,2,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,3,1,2,3,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,4,0,0,0,0,0,0,0,0,0,1,4,1,0] >;
C42.40Q8 in GAP, Magma, Sage, TeX
C_4^2._{40}Q_8 % in TeX
G:=Group("C4^2.40Q8"); // GroupNames label
G:=SmallGroup(128,1577);
// by ID
G=gap.SmallGroup(128,1577);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,336,253,456,758,723,436,794,185,80]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=b^2*c^2,a*b=b*a,c*a*c^-1=a^-1*b^2,d*a*d^-1=a*b^2,c*b*c^-1=b^-1,d*b*d^-1=a^2*b,d*c*d^-1=b^2*c^-1>;
// generators/relations
Export