p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.59Q8, C8⋊4(C4⋊C4), (C4×C8)⋊17C4, (C2×C4).74D8, C4⋊2(C2.D8), C4.2(C4⋊Q8), (C2×C8).42Q8, (C2×C8).243D4, (C2×C4).36Q16, C2.1(C8⋊4D4), C2.1(C8⋊2Q8), C22.35(C2×D8), C42.324(C2×C4), C42⋊9C4.7C2, C2.1(C4⋊Q16), C23.754(C2×D4), (C22×C4).578D4, C22.28(C4⋊Q8), C22.28(C2×Q16), C2.6(C42⋊9C4), C22.28(C4⋊1D4), (C22×C8).524C22, (C2×C42).1060C22, (C22×C4).1344C23, (C2×C4×C8).34C2, C4.34(C2×C4⋊C4), C2.8(C2×C2.D8), (C2×C2.D8).2C2, (C2×C8).221(C2×C4), (C2×C4).729(C2×D4), (C2×C4).194(C2×Q8), (C2×C4).132(C4⋊C4), (C2×C4⋊C4).48C22, C22.103(C2×C4⋊C4), (C2×C4).543(C22×C4), SmallGroup(128,577)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.59Q8
G = < a,b,c,d | a4=b4=1, c4=b2, d2=bc2, ab=ba, ac=ca, dad-1=a-1, bc=cb, dbd-1=b-1, dcd-1=b2c3 >
Subgroups: 252 in 140 conjugacy classes, 92 normal (14 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, C23, C42, C4⋊C4, C2×C8, C22×C4, C22×C4, C22×C4, C4×C8, C2.D8, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C22×C8, C42⋊9C4, C2×C4×C8, C2×C2.D8, C42.59Q8
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, D8, Q16, C22×C4, C2×D4, C2×Q8, C2.D8, C2×C4⋊C4, C4⋊1D4, C4⋊Q8, C2×D8, C2×Q16, C42⋊9C4, C2×C2.D8, C8⋊4D4, C4⋊Q16, C8⋊2Q8, C42.59Q8
►Regular action on 128 points
Generators in S128
(1 25 56 24)(2 26 49 17)(3 27 50 18)(4 28 51 19)(5 29 52 20)(6 30 53 21)(7 31 54 22)(8 32 55 23)(9 41 106 89)(10 42 107 90)(11 43 108 91)(12 44 109 92)(13 45 110 93)(14 46 111 94)(15 47 112 95)(16 48 105 96)(33 119 87 127)(34 120 88 128)(35 113 81 121)(36 114 82 122)(37 115 83 123)(38 116 84 124)(39 117 85 125)(40 118 86 126)(57 67 99 80)(58 68 100 73)(59 69 101 74)(60 70 102 75)(61 71 103 76)(62 72 104 77)(63 65 97 78)(64 66 98 79)
(1 63 5 59)(2 64 6 60)(3 57 7 61)(4 58 8 62)(9 128 13 124)(10 121 14 125)(11 122 15 126)(12 123 16 127)(17 79 21 75)(18 80 22 76)(19 73 23 77)(20 74 24 78)(25 65 29 69)(26 66 30 70)(27 67 31 71)(28 68 32 72)(33 44 37 48)(34 45 38 41)(35 46 39 42)(36 47 40 43)(49 98 53 102)(50 99 54 103)(51 100 55 104)(52 101 56 97)(81 94 85 90)(82 95 86 91)(83 96 87 92)(84 89 88 93)(105 119 109 115)(106 120 110 116)(107 113 111 117)(108 114 112 118)
(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)
(1 111 57 119)(2 110 58 118)(3 109 59 117)(4 108 60 116)(5 107 61 115)(6 106 62 114)(7 105 63 113)(8 112 64 120)(9 104 122 53)(10 103 123 52)(11 102 124 51)(12 101 125 50)(13 100 126 49)(14 99 127 56)(15 98 128 55)(16 97 121 54)(17 93 73 86)(18 92 74 85)(19 91 75 84)(20 90 76 83)(21 89 77 82)(22 96 78 81)(23 95 79 88)(24 94 80 87)(25 46 67 33)(26 45 68 40)(27 44 69 39)(28 43 70 38)(29 42 71 37)(30 41 72 36)(31 48 65 35)(32 47 66 34)
G:=sub<Sym(128)| (1,25,56,24)(2,26,49,17)(3,27,50,18)(4,28,51,19)(5,29,52,20)(6,30,53,21)(7,31,54,22)(8,32,55,23)(9,41,106,89)(10,42,107,90)(11,43,108,91)(12,44,109,92)(13,45,110,93)(14,46,111,94)(15,47,112,95)(16,48,105,96)(33,119,87,127)(34,120,88,128)(35,113,81,121)(36,114,82,122)(37,115,83,123)(38,116,84,124)(39,117,85,125)(40,118,86,126)(57,67,99,80)(58,68,100,73)(59,69,101,74)(60,70,102,75)(61,71,103,76)(62,72,104,77)(63,65,97,78)(64,66,98,79), (1,63,5,59)(2,64,6,60)(3,57,7,61)(4,58,8,62)(9,128,13,124)(10,121,14,125)(11,122,15,126)(12,123,16,127)(17,79,21,75)(18,80,22,76)(19,73,23,77)(20,74,24,78)(25,65,29,69)(26,66,30,70)(27,67,31,71)(28,68,32,72)(33,44,37,48)(34,45,38,41)(35,46,39,42)(36,47,40,43)(49,98,53,102)(50,99,54,103)(51,100,55,104)(52,101,56,97)(81,94,85,90)(82,95,86,91)(83,96,87,92)(84,89,88,93)(105,119,109,115)(106,120,110,116)(107,113,111,117)(108,114,112,118), (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), (1,111,57,119)(2,110,58,118)(3,109,59,117)(4,108,60,116)(5,107,61,115)(6,106,62,114)(7,105,63,113)(8,112,64,120)(9,104,122,53)(10,103,123,52)(11,102,124,51)(12,101,125,50)(13,100,126,49)(14,99,127,56)(15,98,128,55)(16,97,121,54)(17,93,73,86)(18,92,74,85)(19,91,75,84)(20,90,76,83)(21,89,77,82)(22,96,78,81)(23,95,79,88)(24,94,80,87)(25,46,67,33)(26,45,68,40)(27,44,69,39)(28,43,70,38)(29,42,71,37)(30,41,72,36)(31,48,65,35)(32,47,66,34)>;
G:=Group( (1,25,56,24)(2,26,49,17)(3,27,50,18)(4,28,51,19)(5,29,52,20)(6,30,53,21)(7,31,54,22)(8,32,55,23)(9,41,106,89)(10,42,107,90)(11,43,108,91)(12,44,109,92)(13,45,110,93)(14,46,111,94)(15,47,112,95)(16,48,105,96)(33,119,87,127)(34,120,88,128)(35,113,81,121)(36,114,82,122)(37,115,83,123)(38,116,84,124)(39,117,85,125)(40,118,86,126)(57,67,99,80)(58,68,100,73)(59,69,101,74)(60,70,102,75)(61,71,103,76)(62,72,104,77)(63,65,97,78)(64,66,98,79), (1,63,5,59)(2,64,6,60)(3,57,7,61)(4,58,8,62)(9,128,13,124)(10,121,14,125)(11,122,15,126)(12,123,16,127)(17,79,21,75)(18,80,22,76)(19,73,23,77)(20,74,24,78)(25,65,29,69)(26,66,30,70)(27,67,31,71)(28,68,32,72)(33,44,37,48)(34,45,38,41)(35,46,39,42)(36,47,40,43)(49,98,53,102)(50,99,54,103)(51,100,55,104)(52,101,56,97)(81,94,85,90)(82,95,86,91)(83,96,87,92)(84,89,88,93)(105,119,109,115)(106,120,110,116)(107,113,111,117)(108,114,112,118), (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), (1,111,57,119)(2,110,58,118)(3,109,59,117)(4,108,60,116)(5,107,61,115)(6,106,62,114)(7,105,63,113)(8,112,64,120)(9,104,122,53)(10,103,123,52)(11,102,124,51)(12,101,125,50)(13,100,126,49)(14,99,127,56)(15,98,128,55)(16,97,121,54)(17,93,73,86)(18,92,74,85)(19,91,75,84)(20,90,76,83)(21,89,77,82)(22,96,78,81)(23,95,79,88)(24,94,80,87)(25,46,67,33)(26,45,68,40)(27,44,69,39)(28,43,70,38)(29,42,71,37)(30,41,72,36)(31,48,65,35)(32,47,66,34) );
G=PermutationGroup([[(1,25,56,24),(2,26,49,17),(3,27,50,18),(4,28,51,19),(5,29,52,20),(6,30,53,21),(7,31,54,22),(8,32,55,23),(9,41,106,89),(10,42,107,90),(11,43,108,91),(12,44,109,92),(13,45,110,93),(14,46,111,94),(15,47,112,95),(16,48,105,96),(33,119,87,127),(34,120,88,128),(35,113,81,121),(36,114,82,122),(37,115,83,123),(38,116,84,124),(39,117,85,125),(40,118,86,126),(57,67,99,80),(58,68,100,73),(59,69,101,74),(60,70,102,75),(61,71,103,76),(62,72,104,77),(63,65,97,78),(64,66,98,79)], [(1,63,5,59),(2,64,6,60),(3,57,7,61),(4,58,8,62),(9,128,13,124),(10,121,14,125),(11,122,15,126),(12,123,16,127),(17,79,21,75),(18,80,22,76),(19,73,23,77),(20,74,24,78),(25,65,29,69),(26,66,30,70),(27,67,31,71),(28,68,32,72),(33,44,37,48),(34,45,38,41),(35,46,39,42),(36,47,40,43),(49,98,53,102),(50,99,54,103),(51,100,55,104),(52,101,56,97),(81,94,85,90),(82,95,86,91),(83,96,87,92),(84,89,88,93),(105,119,109,115),(106,120,110,116),(107,113,111,117),(108,114,112,118)], [(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)], [(1,111,57,119),(2,110,58,118),(3,109,59,117),(4,108,60,116),(5,107,61,115),(6,106,62,114),(7,105,63,113),(8,112,64,120),(9,104,122,53),(10,103,123,52),(11,102,124,51),(12,101,125,50),(13,100,126,49),(14,99,127,56),(15,98,128,55),(16,97,121,54),(17,93,73,86),(18,92,74,85),(19,91,75,84),(20,90,76,83),(21,89,77,82),(22,96,78,81),(23,95,79,88),(24,94,80,87),(25,46,67,33),(26,45,68,40),(27,44,69,39),(28,43,70,38),(29,42,71,37),(30,41,72,36),(31,48,65,35),(32,47,66,34)]])
44 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4L | 4M | ··· | 4T | 8A | ··· | 8P |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 8 | ··· | 8 | 2 | ··· | 2 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | - | + | + | - | |
| image | C1 | C2 | C2 | C2 | C4 | Q8 | D4 | Q8 | D4 | D8 | Q16 |
| kernel | C42.59Q8 | C42⋊9C4 | C2×C4×C8 | C2×C2.D8 | C4×C8 | C42 | C2×C8 | C2×C8 | C22×C4 | C2×C4 | C2×C4 |
| # reps | 1 | 2 | 1 | 4 | 8 | 2 | 4 | 4 | 2 | 8 | 8 |
Matrix representation of C42.59Q8 ►in GL5(𝔽17)
| 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 1 | 2 |
| 0 | 0 | 0 | 16 | 16 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 6 | 6 |
| 0 | 0 | 0 | 14 | 0 |
| 13 | 0 | 0 | 0 | 0 |
| 0 | 6 | 13 | 0 | 0 |
| 0 | 13 | 11 | 0 | 0 |
| 0 | 0 | 0 | 10 | 10 |
| 0 | 0 | 0 | 12 | 7 |
G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1],[16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,1,16,0,0,0,2,16],[1,0,0,0,0,0,0,1,0,0,0,16,0,0,0,0,0,0,6,14,0,0,0,6,0],[13,0,0,0,0,0,6,13,0,0,0,13,11,0,0,0,0,0,10,12,0,0,0,10,7] >;
C42.59Q8 in GAP, Magma, Sage, TeX
C_4^2._{59}Q_8 % in TeX
G:=Group("C4^2.59Q8"); // GroupNames label
G:=SmallGroup(128,577);
// by ID
G=gap.SmallGroup(128,577);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,512,422,100,2019,248]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=b^2,d^2=b*c^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=b^2*c^3>;
// generators/relations