p-group, metabelian, nilpotent (class 2), monomial, rational
Aliases: C42⋊12Q8, C23.743C24, C22.5162+ 1+4, C22.3952- 1+4, C42⋊5C4.19C2, C42⋊9C4.40C2, (C22×C4).254C23, (C2×C42).746C22, C22.175(C22×Q8), (C22×Q8).245C22, C2.59(C22.54C24), C23.83C23.50C2, C23.81C23.53C2, C2.C42.444C22, C23.78C23.31C2, C2.75(C22.57C24), C2.50(C23.41C23), (C2×C4).138(C2×Q8), (C2×C4⋊C4).550C22, SmallGroup(128,1575)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42⋊12Q8
G = < a,b,c,d | a4=b4=c4=1, d2=c2, ab=ba, cac-1=a-1, dad-1=ab2, cbc-1=b-1, dbd-1=a2b-1, dcd-1=c-1 >
Subgroups: 324 in 174 conjugacy classes, 92 normal (9 characteristic)
C1, C2, C2, C4, C22, C22, C2×C4, C2×C4, Q8, C23, C42, C4⋊C4, C22×C4, C22×C4, C2×Q8, C2.C42, C2×C42, C2×C4⋊C4, C22×Q8, C42⋊5C4, C42⋊9C4, C23.78C23, C23.81C23, C23.83C23, C42⋊12Q8
Quotients: C1, C2, C22, Q8, C23, C2×Q8, C24, C22×Q8, 2+ 1+4, 2- 1+4, C23.41C23, C22.54C24, C22.57C24, C42⋊12Q8
Character table of C42⋊12Q8
| 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 75 38 10)(2 76 39 11)(3 73 40 12)(4 74 37 9)(5 41 70 102)(6 42 71 103)(7 43 72 104)(8 44 69 101)(13 110 78 49)(14 111 79 50)(15 112 80 51)(16 109 77 52)(17 45 82 106)(18 46 83 107)(19 47 84 108)(20 48 81 105)(21 120 86 59)(22 117 87 60)(23 118 88 57)(24 119 85 58)(25 55 90 116)(26 56 91 113)(27 53 92 114)(28 54 89 115)(29 68 94 128)(30 65 95 125)(31 66 96 126)(32 67 93 127)(33 124 98 63)(34 121 99 64)(35 122 100 61)(36 123 97 62)
(1 14 6 107)(2 13 7 106)(3 16 8 105)(4 15 5 108)(9 112 102 19)(10 111 103 18)(11 110 104 17)(12 109 101 20)(21 31 114 124)(22 30 115 123)(23 29 116 122)(24 32 113 121)(25 35 118 128)(26 34 119 127)(27 33 120 126)(28 36 117 125)(37 80 70 47)(38 79 71 46)(39 78 72 45)(40 77 69 48)(41 84 74 51)(42 83 75 50)(43 82 76 49)(44 81 73 52)(53 63 86 96)(54 62 87 95)(55 61 88 94)(56 64 85 93)(57 68 90 100)(58 67 91 99)(59 66 92 98)(60 65 89 97)
(1 22 6 115)(2 88 7 55)(3 24 8 113)(4 86 5 53)(9 57 102 90)(10 119 103 26)(11 59 104 92)(12 117 101 28)(13 61 106 94)(14 123 107 30)(15 63 108 96)(16 121 105 32)(17 66 110 98)(18 127 111 34)(19 68 112 100)(20 125 109 36)(21 70 114 37)(23 72 116 39)(25 74 118 41)(27 76 120 43)(29 78 122 45)(31 80 124 47)(33 82 126 49)(35 84 128 51)(38 87 71 54)(40 85 69 56)(42 91 75 58)(44 89 73 60)(46 95 79 62)(48 93 77 64)(50 99 83 67)(52 97 81 65)
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), (1,75,38,10)(2,76,39,11)(3,73,40,12)(4,74,37,9)(5,41,70,102)(6,42,71,103)(7,43,72,104)(8,44,69,101)(13,110,78,49)(14,111,79,50)(15,112,80,51)(16,109,77,52)(17,45,82,106)(18,46,83,107)(19,47,84,108)(20,48,81,105)(21,120,86,59)(22,117,87,60)(23,118,88,57)(24,119,85,58)(25,55,90,116)(26,56,91,113)(27,53,92,114)(28,54,89,115)(29,68,94,128)(30,65,95,125)(31,66,96,126)(32,67,93,127)(33,124,98,63)(34,121,99,64)(35,122,100,61)(36,123,97,62), (1,14,6,107)(2,13,7,106)(3,16,8,105)(4,15,5,108)(9,112,102,19)(10,111,103,18)(11,110,104,17)(12,109,101,20)(21,31,114,124)(22,30,115,123)(23,29,116,122)(24,32,113,121)(25,35,118,128)(26,34,119,127)(27,33,120,126)(28,36,117,125)(37,80,70,47)(38,79,71,46)(39,78,72,45)(40,77,69,48)(41,84,74,51)(42,83,75,50)(43,82,76,49)(44,81,73,52)(53,63,86,96)(54,62,87,95)(55,61,88,94)(56,64,85,93)(57,68,90,100)(58,67,91,99)(59,66,92,98)(60,65,89,97), (1,22,6,115)(2,88,7,55)(3,24,8,113)(4,86,5,53)(9,57,102,90)(10,119,103,26)(11,59,104,92)(12,117,101,28)(13,61,106,94)(14,123,107,30)(15,63,108,96)(16,121,105,32)(17,66,110,98)(18,127,111,34)(19,68,112,100)(20,125,109,36)(21,70,114,37)(23,72,116,39)(25,74,118,41)(27,76,120,43)(29,78,122,45)(31,80,124,47)(33,82,126,49)(35,84,128,51)(38,87,71,54)(40,85,69,56)(42,91,75,58)(44,89,73,60)(46,95,79,62)(48,93,77,64)(50,99,83,67)(52,97,81,65)>;
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), (1,75,38,10)(2,76,39,11)(3,73,40,12)(4,74,37,9)(5,41,70,102)(6,42,71,103)(7,43,72,104)(8,44,69,101)(13,110,78,49)(14,111,79,50)(15,112,80,51)(16,109,77,52)(17,45,82,106)(18,46,83,107)(19,47,84,108)(20,48,81,105)(21,120,86,59)(22,117,87,60)(23,118,88,57)(24,119,85,58)(25,55,90,116)(26,56,91,113)(27,53,92,114)(28,54,89,115)(29,68,94,128)(30,65,95,125)(31,66,96,126)(32,67,93,127)(33,124,98,63)(34,121,99,64)(35,122,100,61)(36,123,97,62), (1,14,6,107)(2,13,7,106)(3,16,8,105)(4,15,5,108)(9,112,102,19)(10,111,103,18)(11,110,104,17)(12,109,101,20)(21,31,114,124)(22,30,115,123)(23,29,116,122)(24,32,113,121)(25,35,118,128)(26,34,119,127)(27,33,120,126)(28,36,117,125)(37,80,70,47)(38,79,71,46)(39,78,72,45)(40,77,69,48)(41,84,74,51)(42,83,75,50)(43,82,76,49)(44,81,73,52)(53,63,86,96)(54,62,87,95)(55,61,88,94)(56,64,85,93)(57,68,90,100)(58,67,91,99)(59,66,92,98)(60,65,89,97), (1,22,6,115)(2,88,7,55)(3,24,8,113)(4,86,5,53)(9,57,102,90)(10,119,103,26)(11,59,104,92)(12,117,101,28)(13,61,106,94)(14,123,107,30)(15,63,108,96)(16,121,105,32)(17,66,110,98)(18,127,111,34)(19,68,112,100)(20,125,109,36)(21,70,114,37)(23,72,116,39)(25,74,118,41)(27,76,120,43)(29,78,122,45)(31,80,124,47)(33,82,126,49)(35,84,128,51)(38,87,71,54)(40,85,69,56)(42,91,75,58)(44,89,73,60)(46,95,79,62)(48,93,77,64)(50,99,83,67)(52,97,81,65) );
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)], [(1,75,38,10),(2,76,39,11),(3,73,40,12),(4,74,37,9),(5,41,70,102),(6,42,71,103),(7,43,72,104),(8,44,69,101),(13,110,78,49),(14,111,79,50),(15,112,80,51),(16,109,77,52),(17,45,82,106),(18,46,83,107),(19,47,84,108),(20,48,81,105),(21,120,86,59),(22,117,87,60),(23,118,88,57),(24,119,85,58),(25,55,90,116),(26,56,91,113),(27,53,92,114),(28,54,89,115),(29,68,94,128),(30,65,95,125),(31,66,96,126),(32,67,93,127),(33,124,98,63),(34,121,99,64),(35,122,100,61),(36,123,97,62)], [(1,14,6,107),(2,13,7,106),(3,16,8,105),(4,15,5,108),(9,112,102,19),(10,111,103,18),(11,110,104,17),(12,109,101,20),(21,31,114,124),(22,30,115,123),(23,29,116,122),(24,32,113,121),(25,35,118,128),(26,34,119,127),(27,33,120,126),(28,36,117,125),(37,80,70,47),(38,79,71,46),(39,78,72,45),(40,77,69,48),(41,84,74,51),(42,83,75,50),(43,82,76,49),(44,81,73,52),(53,63,86,96),(54,62,87,95),(55,61,88,94),(56,64,85,93),(57,68,90,100),(58,67,91,99),(59,66,92,98),(60,65,89,97)], [(1,22,6,115),(2,88,7,55),(3,24,8,113),(4,86,5,53),(9,57,102,90),(10,119,103,26),(11,59,104,92),(12,117,101,28),(13,61,106,94),(14,123,107,30),(15,63,108,96),(16,121,105,32),(17,66,110,98),(18,127,111,34),(19,68,112,100),(20,125,109,36),(21,70,114,37),(23,72,116,39),(25,74,118,41),(27,76,120,43),(29,78,122,45),(31,80,124,47),(33,82,126,49),(35,84,128,51),(38,87,71,54),(40,85,69,56),(42,91,75,58),(44,89,73,60),(46,95,79,62),(48,93,77,64),(50,99,83,67),(52,97,81,65)]])
Matrix representation of C42⋊12Q8 ►in GL10(𝔽5)
| 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 2 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 3 | 2 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 2 | 3 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | 2 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 3 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 3 | 3 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 3 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 3 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 4 | 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 2 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 1 | 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 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 |
| 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 3 | 2 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 1 | 0 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 2 | 3 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 4 | 4 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 4 | 4 | 1 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 4 | 1 | 4 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |
G:=sub<GL(10,GF(5))| [4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,1,4,0,0,0,0,0,0,0,0,3,2,4,0,0,0,0,0,0,2,2,3,4,0,0,0,0,0,0,4,4,2,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,2,0,2,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,2,3,3,0],[4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,1,4,0,0,0,0,0,0,0,3,1,4,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,2,4,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,2,4],[0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,3,1,4,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0],[2,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,2,2,0,2,0,0,0,0,0,0,3,1,0,2,0,0,0,0,0,0,2,0,1,3,0,0,0,0,0,0,3,2,0,1,0,0,0,0,0,0,0,0,0,0,1,4,4,1,0,0,0,0,0,0,4,4,1,1,0,0,0,0,0,0,4,1,4,1,0,0,0,0,0,0,1,1,1,1] >;
C42⋊12Q8 in GAP, Magma, Sage, TeX
C_4^2\rtimes_{12}Q_8 % in TeX
G:=Group("C4^2:12Q8"); // GroupNames label
G:=SmallGroup(128,1575);
// by ID
G=gap.SmallGroup(128,1575);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,456,758,723,352,794,185,80]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=c^2,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a*b^2,c*b*c^-1=b^-1,d*b*d^-1=a^2*b^-1,d*c*d^-1=c^-1>;
// generators/relations
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