p-group, metacyclic, nilpotent (class 5), monomial
Aliases: C32⋊4C4, C8.6Q16, C4.2Q32, C16.3Q8, C2.3SD64, C22.11D16, (C2×C32).6C2, (C2×C4).69D8, C8.16(C4⋊C4), C16.17(C2×C4), (C2×C8).240D4, C16⋊3C4.3C2, C2.4(C16⋊3C4), C4.11(C2.D8), (C2×C16).78C22, SmallGroup(128,156)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C32⋊4C4
G = < a,b | a32=b4=1, bab-1=a15 >
Smallest permutation representation of C32⋊4C4
►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 61 82 115)(2 44 83 98)(3 59 84 113)(4 42 85 128)(5 57 86 111)(6 40 87 126)(7 55 88 109)(8 38 89 124)(9 53 90 107)(10 36 91 122)(11 51 92 105)(12 34 93 120)(13 49 94 103)(14 64 95 118)(15 47 96 101)(16 62 65 116)(17 45 66 99)(18 60 67 114)(19 43 68 97)(20 58 69 112)(21 41 70 127)(22 56 71 110)(23 39 72 125)(24 54 73 108)(25 37 74 123)(26 52 75 106)(27 35 76 121)(28 50 77 104)(29 33 78 119)(30 48 79 102)(31 63 80 117)(32 46 81 100)
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), (1,61,82,115)(2,44,83,98)(3,59,84,113)(4,42,85,128)(5,57,86,111)(6,40,87,126)(7,55,88,109)(8,38,89,124)(9,53,90,107)(10,36,91,122)(11,51,92,105)(12,34,93,120)(13,49,94,103)(14,64,95,118)(15,47,96,101)(16,62,65,116)(17,45,66,99)(18,60,67,114)(19,43,68,97)(20,58,69,112)(21,41,70,127)(22,56,71,110)(23,39,72,125)(24,54,73,108)(25,37,74,123)(26,52,75,106)(27,35,76,121)(28,50,77,104)(29,33,78,119)(30,48,79,102)(31,63,80,117)(32,46,81,100)>;
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), (1,61,82,115)(2,44,83,98)(3,59,84,113)(4,42,85,128)(5,57,86,111)(6,40,87,126)(7,55,88,109)(8,38,89,124)(9,53,90,107)(10,36,91,122)(11,51,92,105)(12,34,93,120)(13,49,94,103)(14,64,95,118)(15,47,96,101)(16,62,65,116)(17,45,66,99)(18,60,67,114)(19,43,68,97)(20,58,69,112)(21,41,70,127)(22,56,71,110)(23,39,72,125)(24,54,73,108)(25,37,74,123)(26,52,75,106)(27,35,76,121)(28,50,77,104)(29,33,78,119)(30,48,79,102)(31,63,80,117)(32,46,81,100) );
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)], [(1,61,82,115),(2,44,83,98),(3,59,84,113),(4,42,85,128),(5,57,86,111),(6,40,87,126),(7,55,88,109),(8,38,89,124),(9,53,90,107),(10,36,91,122),(11,51,92,105),(12,34,93,120),(13,49,94,103),(14,64,95,118),(15,47,96,101),(16,62,65,116),(17,45,66,99),(18,60,67,114),(19,43,68,97),(20,58,69,112),(21,41,70,127),(22,56,71,110),(23,39,72,125),(24,54,73,108),(25,37,74,123),(26,52,75,106),(27,35,76,121),(28,50,77,104),(29,33,78,119),(30,48,79,102),(31,63,80,117),(32,46,81,100)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 8A | 8B | 8C | 8D | 16A | ··· | 16H | 32A | ··· | 32P |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 16 | ··· | 16 | 32 | ··· | 32 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 16 | 16 | 16 | 16 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | + | - | + | - | + | ||
| image | C1 | C2 | C2 | C4 | Q8 | D4 | Q16 | D8 | Q32 | D16 | SD64 |
| kernel | C32⋊4C4 | C16⋊3C4 | C2×C32 | C32 | C16 | C2×C8 | C8 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 4 | 1 | 1 | 2 | 2 | 4 | 4 | 16 |
Matrix representation of C32⋊4C4 ►in GL4(𝔽97) generated by
| 26 | 95 | 0 | 0 |
| 2 | 26 | 0 | 0 |
| 0 | 0 | 92 | 73 |
| 0 | 0 | 12 | 19 |
| 87 | 53 | 0 | 0 |
| 53 | 10 | 0 | 0 |
| 0 | 0 | 12 | 69 |
| 0 | 0 | 71 | 85 |
G:=sub<GL(4,GF(97))| [26,2,0,0,95,26,0,0,0,0,92,12,0,0,73,19],[87,53,0,0,53,10,0,0,0,0,12,71,0,0,69,85] >;
C32⋊4C4 in GAP, Magma, Sage, TeX
C_{32}\rtimes_4C_4 % in TeX
G:=Group("C32:4C4"); // GroupNames label
G:=SmallGroup(128,156);
// by ID
G=gap.SmallGroup(128,156);
# by ID
G:=PCGroup([7,-2,2,-2,2,-2,-2,-2,56,85,596,422,268,1684,242,4037,124]);
// Polycyclic
G:=Group<a,b|a^32=b^4=1,b*a*b^-1=a^15>;
// generators/relations
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