p-group, metabelian, nilpotent (class 3), monomial
Aliases: C8.4M4(2), C42.650C23, C8⋊C8.3C2, C8⋊2C8.7C2, (C2×C8).184D4, Q8⋊C8.10C2, C2.D8.20C4, C4.17(C8○D4), (C2×Q16).12C4, (C4×Q16).16C2, C8⋊4Q8.14C2, C2.10(C8⋊6D4), C4⋊C8.228C22, (C4×C8).319C22, Q8⋊C4.12C4, C22.141(C4×D4), C4.11(C2×M4(2)), (C4×Q8).16C22, C2.11(C8.26D4), C2.5(Q16⋊C4), C4.141(C8.C22), C4⋊C4.63(C2×C4), (C2×C8).58(C2×C4), (C2×Q8).56(C2×C4), (C2×C4).1486(C2×D4), (C2×C4).511(C4○D4), (C2×C4).342(C22×C4), SmallGroup(128,325)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C8.M4(2)
G = < a,b,c | a8=b8=1, c2=a4, bab-1=a3, cac-1=a-1, cbc-1=a6b5 >
Subgroups: 120 in 74 conjugacy classes, 42 normal (22 characteristic)
C1, C2, C4, C4, C4, C22, C8, C8, C2×C4, C2×C4, Q8, C42, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C2×C8, Q16, C2×Q8, 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⋊4Q8, C8.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⋊6D4, Q16⋊C4, C8.26D4, C8.M4(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 86 15 24 54 47 124 92)(2 81 16 19 55 42 125 95)(3 84 9 22 56 45 126 90)(4 87 10 17 49 48 127 93)(5 82 11 20 50 43 128 96)(6 85 12 23 51 46 121 91)(7 88 13 18 52 41 122 94)(8 83 14 21 53 44 123 89)(25 119 79 67 97 109 33 60)(26 114 80 70 98 112 34 63)(27 117 73 65 99 107 35 58)(28 120 74 68 100 110 36 61)(29 115 75 71 101 105 37 64)(30 118 76 66 102 108 38 59)(31 113 77 69 103 111 39 62)(32 116 78 72 104 106 40 57)
(1 34 5 38)(2 33 6 37)(3 40 7 36)(4 39 8 35)(9 32 13 28)(10 31 14 27)(11 30 15 26)(12 29 16 25)(17 105 21 109)(18 112 22 108)(19 111 23 107)(20 110 24 106)(41 63 45 59)(42 62 46 58)(43 61 47 57)(44 60 48 64)(49 77 53 73)(50 76 54 80)(51 75 55 79)(52 74 56 78)(65 81 69 85)(66 88 70 84)(67 87 71 83)(68 86 72 82)(89 119 93 115)(90 118 94 114)(91 117 95 113)(92 116 96 120)(97 121 101 125)(98 128 102 124)(99 127 103 123)(100 126 104 122)
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), (1,86,15,24,54,47,124,92)(2,81,16,19,55,42,125,95)(3,84,9,22,56,45,126,90)(4,87,10,17,49,48,127,93)(5,82,11,20,50,43,128,96)(6,85,12,23,51,46,121,91)(7,88,13,18,52,41,122,94)(8,83,14,21,53,44,123,89)(25,119,79,67,97,109,33,60)(26,114,80,70,98,112,34,63)(27,117,73,65,99,107,35,58)(28,120,74,68,100,110,36,61)(29,115,75,71,101,105,37,64)(30,118,76,66,102,108,38,59)(31,113,77,69,103,111,39,62)(32,116,78,72,104,106,40,57), (1,34,5,38)(2,33,6,37)(3,40,7,36)(4,39,8,35)(9,32,13,28)(10,31,14,27)(11,30,15,26)(12,29,16,25)(17,105,21,109)(18,112,22,108)(19,111,23,107)(20,110,24,106)(41,63,45,59)(42,62,46,58)(43,61,47,57)(44,60,48,64)(49,77,53,73)(50,76,54,80)(51,75,55,79)(52,74,56,78)(65,81,69,85)(66,88,70,84)(67,87,71,83)(68,86,72,82)(89,119,93,115)(90,118,94,114)(91,117,95,113)(92,116,96,120)(97,121,101,125)(98,128,102,124)(99,127,103,123)(100,126,104,122)>;
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), (1,86,15,24,54,47,124,92)(2,81,16,19,55,42,125,95)(3,84,9,22,56,45,126,90)(4,87,10,17,49,48,127,93)(5,82,11,20,50,43,128,96)(6,85,12,23,51,46,121,91)(7,88,13,18,52,41,122,94)(8,83,14,21,53,44,123,89)(25,119,79,67,97,109,33,60)(26,114,80,70,98,112,34,63)(27,117,73,65,99,107,35,58)(28,120,74,68,100,110,36,61)(29,115,75,71,101,105,37,64)(30,118,76,66,102,108,38,59)(31,113,77,69,103,111,39,62)(32,116,78,72,104,106,40,57), (1,34,5,38)(2,33,6,37)(3,40,7,36)(4,39,8,35)(9,32,13,28)(10,31,14,27)(11,30,15,26)(12,29,16,25)(17,105,21,109)(18,112,22,108)(19,111,23,107)(20,110,24,106)(41,63,45,59)(42,62,46,58)(43,61,47,57)(44,60,48,64)(49,77,53,73)(50,76,54,80)(51,75,55,79)(52,74,56,78)(65,81,69,85)(66,88,70,84)(67,87,71,83)(68,86,72,82)(89,119,93,115)(90,118,94,114)(91,117,95,113)(92,116,96,120)(97,121,101,125)(98,128,102,124)(99,127,103,123)(100,126,104,122) );
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)], [(1,86,15,24,54,47,124,92),(2,81,16,19,55,42,125,95),(3,84,9,22,56,45,126,90),(4,87,10,17,49,48,127,93),(5,82,11,20,50,43,128,96),(6,85,12,23,51,46,121,91),(7,88,13,18,52,41,122,94),(8,83,14,21,53,44,123,89),(25,119,79,67,97,109,33,60),(26,114,80,70,98,112,34,63),(27,117,73,65,99,107,35,58),(28,120,74,68,100,110,36,61),(29,115,75,71,101,105,37,64),(30,118,76,66,102,108,38,59),(31,113,77,69,103,111,39,62),(32,116,78,72,104,106,40,57)], [(1,34,5,38),(2,33,6,37),(3,40,7,36),(4,39,8,35),(9,32,13,28),(10,31,14,27),(11,30,15,26),(12,29,16,25),(17,105,21,109),(18,112,22,108),(19,111,23,107),(20,110,24,106),(41,63,45,59),(42,62,46,58),(43,61,47,57),(44,60,48,64),(49,77,53,73),(50,76,54,80),(51,75,55,79),(52,74,56,78),(65,81,69,85),(66,88,70,84),(67,87,71,83),(68,86,72,82),(89,119,93,115),(90,118,94,114),(91,117,95,113),(92,116,96,120),(97,121,101,125),(98,128,102,124),(99,127,103,123),(100,126,104,122)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 8A | ··· | 8L | 8M | 8N | 8O | 8P |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 8 | 8 | 8 | 8 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 8 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | M4(2) | C4○D4 | C8○D4 | C8.C22 | C8.26D4 |
| kernel | C8.M4(2) | C8⋊C8 | Q8⋊C8 | C8⋊2C8 | C4×Q16 | C8⋊4Q8 | Q8⋊C4 | C2.D8 | C2×Q16 | C2×C8 | C8 | C2×C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 4 | 2 | 4 | 2 | 2 |
Matrix representation of C8.M4(2) ►in GL6(𝔽17)
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 8 | 0 | 0 |
| 0 | 0 | 8 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 2 |
| 0 | 0 | 0 | 0 | 2 | 14 |
| 11 | 3 | 0 | 0 | 0 | 0 |
| 12 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 15 | 3 |
| 0 | 0 | 0 | 0 | 3 | 2 |
| 0 | 0 | 8 | 5 | 0 | 0 |
| 0 | 0 | 5 | 9 | 0 | 0 |
| 14 | 7 | 0 | 0 | 0 | 0 |
| 11 | 3 | 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))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,12,8,0,0,0,0,8,5,0,0,0,0,0,0,3,2,0,0,0,0,2,14],[11,12,0,0,0,0,3,6,0,0,0,0,0,0,0,0,8,5,0,0,0,0,5,9,0,0,15,3,0,0,0,0,3,2,0,0],[14,11,0,0,0,0,7,3,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,1,0,0,0,0,0,0,1,0,0] >;
C8.M4(2) in GAP, Magma, Sage, TeX
C_8.M_4(2)
% in TeX
G:=Group("C8.M4(2)"); // GroupNames label
G:=SmallGroup(128,325);
// by ID
G=gap.SmallGroup(128,325);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,232,2102,723,100,1123,570,136,172]);
// Polycyclic
G:=Group<a,b,c|a^8=b^8=1,c^2=a^4,b*a*b^-1=a^3,c*a*c^-1=a^-1,c*b*c^-1=a^6*b^5>;
// generators/relations