p-group, metabelian, nilpotent (class 3), monomial
Aliases: Q8.2M4(2), C42.644C23, Q8⋊C8.8C2, C8⋊2C8.6C2, C8⋊C8.2C2, C2.8(C8○D8), (C2×C8).309D4, (C2×Q16).7C4, (C8×Q8).15C2, C2.D8.12C4, C4.36(C8○D4), Q8⋊C4.8C4, (C4×Q16).12C2, C8⋊4Q8.12C2, C2.14(C8⋊9D4), C4⋊C8.278C22, (C4×C8).316C22, C22.135(C4×D4), C4.30(C2×M4(2)), (C4×Q8).12C22, C2.4(Q16⋊C4), C4.138(C8.C22), C4⋊C4.138(C2×C4), (C2×C8).105(C2×C4), (C2×C4).1480(C2×D4), (C2×Q8).137(C2×C4), (C2×C4).505(C4○D4), (C2×C4).336(C22×C4), SmallGroup(128,319)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for Q8.M4(2)
G = < a,b,c,d | a4=c8=1, b2=d2=a2, bab-1=cac-1=dad-1=a-1, cbc-1=dbd-1=a-1b, dcd-1=a2c5 >
Subgroups: 120 in 76 conjugacy classes, 42 normal (40 characteristic)
C1, C2, C4, C4, C22, C8, C2×C4, C2×C4, Q8, Q8, C42, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, Q16, C2×Q8, C4×C8, C4×C8, C8⋊C4, Q8⋊C4, C4⋊C8, C4⋊C8, C2.D8, C4×Q8, C2×Q16, C8⋊C8, Q8⋊C8, C8⋊2C8, C4×Q16, C8×Q8, C8⋊4Q8, Q8.M4(2)
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, M4(2), C22×C4, C2×D4, C4○D4, C4×D4, C2×M4(2), C8○D4, C8.C22, C8⋊9D4, Q16⋊C4, C8○D8, Q8.M4(2)
►Regular action on 128 points
Generators in S128
(1 44 97 73)(2 74 98 45)(3 46 99 75)(4 76 100 47)(5 48 101 77)(6 78 102 41)(7 42 103 79)(8 80 104 43)(9 33 71 126)(10 127 72 34)(11 35 65 128)(12 121 66 36)(13 37 67 122)(14 123 68 38)(15 39 69 124)(16 125 70 40)(17 86 110 49)(18 50 111 87)(19 88 112 51)(20 52 105 81)(21 82 106 53)(22 54 107 83)(23 84 108 55)(24 56 109 85)(25 57 118 94)(26 95 119 58)(27 59 120 96)(28 89 113 60)(29 61 114 90)(30 91 115 62)(31 63 116 92)(32 93 117 64)
(1 117 97 32)(2 57 98 94)(3 119 99 26)(4 59 100 96)(5 113 101 28)(6 61 102 90)(7 115 103 30)(8 63 104 92)(9 21 71 106)(10 83 72 54)(11 23 65 108)(12 85 66 56)(13 17 67 110)(14 87 68 50)(15 19 69 112)(16 81 70 52)(18 123 111 38)(20 125 105 40)(22 127 107 34)(24 121 109 36)(25 45 118 74)(27 47 120 76)(29 41 114 78)(31 43 116 80)(33 53 126 82)(35 55 128 84)(37 49 122 86)(39 51 124 88)(42 91 79 62)(44 93 73 64)(46 95 75 58)(48 89 77 60)
(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 97 106)(2 111 98 18)(3 23 99 108)(4 105 100 20)(5 17 101 110)(6 107 102 22)(7 19 103 112)(8 109 104 24)(9 64 71 93)(10 90 72 61)(11 58 65 95)(12 92 66 63)(13 60 67 89)(14 94 68 57)(15 62 69 91)(16 96 70 59)(25 123 118 38)(26 35 119 128)(27 125 120 40)(28 37 113 122)(29 127 114 34)(30 39 115 124)(31 121 116 36)(32 33 117 126)(41 83 78 54)(42 51 79 88)(43 85 80 56)(44 53 73 82)(45 87 74 50)(46 55 75 84)(47 81 76 52)(48 49 77 86)
G:=sub<Sym(128)| (1,44,97,73)(2,74,98,45)(3,46,99,75)(4,76,100,47)(5,48,101,77)(6,78,102,41)(7,42,103,79)(8,80,104,43)(9,33,71,126)(10,127,72,34)(11,35,65,128)(12,121,66,36)(13,37,67,122)(14,123,68,38)(15,39,69,124)(16,125,70,40)(17,86,110,49)(18,50,111,87)(19,88,112,51)(20,52,105,81)(21,82,106,53)(22,54,107,83)(23,84,108,55)(24,56,109,85)(25,57,118,94)(26,95,119,58)(27,59,120,96)(28,89,113,60)(29,61,114,90)(30,91,115,62)(31,63,116,92)(32,93,117,64), (1,117,97,32)(2,57,98,94)(3,119,99,26)(4,59,100,96)(5,113,101,28)(6,61,102,90)(7,115,103,30)(8,63,104,92)(9,21,71,106)(10,83,72,54)(11,23,65,108)(12,85,66,56)(13,17,67,110)(14,87,68,50)(15,19,69,112)(16,81,70,52)(18,123,111,38)(20,125,105,40)(22,127,107,34)(24,121,109,36)(25,45,118,74)(27,47,120,76)(29,41,114,78)(31,43,116,80)(33,53,126,82)(35,55,128,84)(37,49,122,86)(39,51,124,88)(42,91,79,62)(44,93,73,64)(46,95,75,58)(48,89,77,60), (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), (1,21,97,106)(2,111,98,18)(3,23,99,108)(4,105,100,20)(5,17,101,110)(6,107,102,22)(7,19,103,112)(8,109,104,24)(9,64,71,93)(10,90,72,61)(11,58,65,95)(12,92,66,63)(13,60,67,89)(14,94,68,57)(15,62,69,91)(16,96,70,59)(25,123,118,38)(26,35,119,128)(27,125,120,40)(28,37,113,122)(29,127,114,34)(30,39,115,124)(31,121,116,36)(32,33,117,126)(41,83,78,54)(42,51,79,88)(43,85,80,56)(44,53,73,82)(45,87,74,50)(46,55,75,84)(47,81,76,52)(48,49,77,86)>;
G:=Group( (1,44,97,73)(2,74,98,45)(3,46,99,75)(4,76,100,47)(5,48,101,77)(6,78,102,41)(7,42,103,79)(8,80,104,43)(9,33,71,126)(10,127,72,34)(11,35,65,128)(12,121,66,36)(13,37,67,122)(14,123,68,38)(15,39,69,124)(16,125,70,40)(17,86,110,49)(18,50,111,87)(19,88,112,51)(20,52,105,81)(21,82,106,53)(22,54,107,83)(23,84,108,55)(24,56,109,85)(25,57,118,94)(26,95,119,58)(27,59,120,96)(28,89,113,60)(29,61,114,90)(30,91,115,62)(31,63,116,92)(32,93,117,64), (1,117,97,32)(2,57,98,94)(3,119,99,26)(4,59,100,96)(5,113,101,28)(6,61,102,90)(7,115,103,30)(8,63,104,92)(9,21,71,106)(10,83,72,54)(11,23,65,108)(12,85,66,56)(13,17,67,110)(14,87,68,50)(15,19,69,112)(16,81,70,52)(18,123,111,38)(20,125,105,40)(22,127,107,34)(24,121,109,36)(25,45,118,74)(27,47,120,76)(29,41,114,78)(31,43,116,80)(33,53,126,82)(35,55,128,84)(37,49,122,86)(39,51,124,88)(42,91,79,62)(44,93,73,64)(46,95,75,58)(48,89,77,60), (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), (1,21,97,106)(2,111,98,18)(3,23,99,108)(4,105,100,20)(5,17,101,110)(6,107,102,22)(7,19,103,112)(8,109,104,24)(9,64,71,93)(10,90,72,61)(11,58,65,95)(12,92,66,63)(13,60,67,89)(14,94,68,57)(15,62,69,91)(16,96,70,59)(25,123,118,38)(26,35,119,128)(27,125,120,40)(28,37,113,122)(29,127,114,34)(30,39,115,124)(31,121,116,36)(32,33,117,126)(41,83,78,54)(42,51,79,88)(43,85,80,56)(44,53,73,82)(45,87,74,50)(46,55,75,84)(47,81,76,52)(48,49,77,86) );
G=PermutationGroup([[(1,44,97,73),(2,74,98,45),(3,46,99,75),(4,76,100,47),(5,48,101,77),(6,78,102,41),(7,42,103,79),(8,80,104,43),(9,33,71,126),(10,127,72,34),(11,35,65,128),(12,121,66,36),(13,37,67,122),(14,123,68,38),(15,39,69,124),(16,125,70,40),(17,86,110,49),(18,50,111,87),(19,88,112,51),(20,52,105,81),(21,82,106,53),(22,54,107,83),(23,84,108,55),(24,56,109,85),(25,57,118,94),(26,95,119,58),(27,59,120,96),(28,89,113,60),(29,61,114,90),(30,91,115,62),(31,63,116,92),(32,93,117,64)], [(1,117,97,32),(2,57,98,94),(3,119,99,26),(4,59,100,96),(5,113,101,28),(6,61,102,90),(7,115,103,30),(8,63,104,92),(9,21,71,106),(10,83,72,54),(11,23,65,108),(12,85,66,56),(13,17,67,110),(14,87,68,50),(15,19,69,112),(16,81,70,52),(18,123,111,38),(20,125,105,40),(22,127,107,34),(24,121,109,36),(25,45,118,74),(27,47,120,76),(29,41,114,78),(31,43,116,80),(33,53,126,82),(35,55,128,84),(37,49,122,86),(39,51,124,88),(42,91,79,62),(44,93,73,64),(46,95,75,58),(48,89,77,60)], [(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)], [(1,21,97,106),(2,111,98,18),(3,23,99,108),(4,105,100,20),(5,17,101,110),(6,107,102,22),(7,19,103,112),(8,109,104,24),(9,64,71,93),(10,90,72,61),(11,58,65,95),(12,92,66,63),(13,60,67,89),(14,94,68,57),(15,62,69,91),(16,96,70,59),(25,123,118,38),(26,35,119,128),(27,125,120,40),(28,37,113,122),(29,127,114,34),(30,39,115,124),(31,121,116,36),(32,33,117,126),(41,83,78,54),(42,51,79,88),(43,85,80,56),(44,53,73,82),(45,87,74,50),(46,55,75,84),(47,81,76,52),(48,49,77,86)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 8A | 8B | 8C | 8D | 8E | ··· | 8R | 8S | 8T |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 8 | 8 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | - | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | C4○D4 | M4(2) | C8○D4 | C8○D8 | C8.C22 |
| kernel | Q8.M4(2) | C8⋊C8 | Q8⋊C8 | C8⋊2C8 | C4×Q16 | C8×Q8 | C8⋊4Q8 | Q8⋊C4 | C2.D8 | C2×Q16 | C2×C8 | C2×C4 | Q8 | C4 | C2 | C4 |
| # reps | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 2 |
Matrix representation of Q8.M4(2) ►in GL4(𝔽17) generated by
| 1 | 15 | 0 | 0 |
| 1 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 0 | 7 | 0 | 0 |
| 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 9 |
| 0 | 0 | 2 | 0 |
| 9 | 16 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 9 |
| 13 | 8 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 2 |
| 0 | 0 | 9 | 0 |
G:=sub<GL(4,GF(17))| [1,1,0,0,15,16,0,0,0,0,16,0,0,0,0,16],[0,12,0,0,7,0,0,0,0,0,0,2,0,0,9,0],[9,0,0,0,16,8,0,0,0,0,8,0,0,0,0,9],[13,0,0,0,8,4,0,0,0,0,0,9,0,0,2,0] >;
Q8.M4(2) in GAP, Magma, Sage, TeX
Q_8.M_4(2)
% in TeX
G:=Group("Q8.M4(2)"); // GroupNames label
G:=SmallGroup(128,319);
// by ID
G=gap.SmallGroup(128,319);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,232,2102,100,1123,570,136,172]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^8=1,b^2=d^2=a^2,b*a*b^-1=c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=d*b*d^-1=a^-1*b,d*c*d^-1=a^2*c^5>;
// generators/relations