p-group, metacyclic, nilpotent (class 2), monomial
Aliases: C32:5C4, C16.4C8, C8.2C16, C42.4C8, C8.20C42, C2.1M6(2), (C2xC8).15C8, (C2xC4).2C16, (C4xC8).33C4, (C2xC32).7C2, C2.3(C4xC16), C4.12(C4xC8), C8.28(C2xC8), (C4xC16).16C2, C16.24(C2xC4), (C2xC16).17C4, C4.12(C2xC16), C22.7(C2xC16), (C2xC16).108C22, (C2xC4).95(C2xC8), (C2xC8).259(C2xC4), SmallGroup(128,129)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C32:5C4
G = < a,b | a32=b4=1, bab-1=a17 >
Quotients: C1, C2, C4, C22, C8, C2xC4, C16, C42, C2xC8, C4xC8, C2xC16, C4xC16, M6(2), C32:5C4
Smallest permutation representation of C32:5C4
►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 51 84 109)(2 36 85 126)(3 53 86 111)(4 38 87 128)(5 55 88 113)(6 40 89 98)(7 57 90 115)(8 42 91 100)(9 59 92 117)(10 44 93 102)(11 61 94 119)(12 46 95 104)(13 63 96 121)(14 48 65 106)(15 33 66 123)(16 50 67 108)(17 35 68 125)(18 52 69 110)(19 37 70 127)(20 54 71 112)(21 39 72 97)(22 56 73 114)(23 41 74 99)(24 58 75 116)(25 43 76 101)(26 60 77 118)(27 45 78 103)(28 62 79 120)(29 47 80 105)(30 64 81 122)(31 49 82 107)(32 34 83 124)
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), (1,51,84,109)(2,36,85,126)(3,53,86,111)(4,38,87,128)(5,55,88,113)(6,40,89,98)(7,57,90,115)(8,42,91,100)(9,59,92,117)(10,44,93,102)(11,61,94,119)(12,46,95,104)(13,63,96,121)(14,48,65,106)(15,33,66,123)(16,50,67,108)(17,35,68,125)(18,52,69,110)(19,37,70,127)(20,54,71,112)(21,39,72,97)(22,56,73,114)(23,41,74,99)(24,58,75,116)(25,43,76,101)(26,60,77,118)(27,45,78,103)(28,62,79,120)(29,47,80,105)(30,64,81,122)(31,49,82,107)(32,34,83,124)>;
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), (1,51,84,109)(2,36,85,126)(3,53,86,111)(4,38,87,128)(5,55,88,113)(6,40,89,98)(7,57,90,115)(8,42,91,100)(9,59,92,117)(10,44,93,102)(11,61,94,119)(12,46,95,104)(13,63,96,121)(14,48,65,106)(15,33,66,123)(16,50,67,108)(17,35,68,125)(18,52,69,110)(19,37,70,127)(20,54,71,112)(21,39,72,97)(22,56,73,114)(23,41,74,99)(24,58,75,116)(25,43,76,101)(26,60,77,118)(27,45,78,103)(28,62,79,120)(29,47,80,105)(30,64,81,122)(31,49,82,107)(32,34,83,124) );
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)], [(1,51,84,109),(2,36,85,126),(3,53,86,111),(4,38,87,128),(5,55,88,113),(6,40,89,98),(7,57,90,115),(8,42,91,100),(9,59,92,117),(10,44,93,102),(11,61,94,119),(12,46,95,104),(13,63,96,121),(14,48,65,106),(15,33,66,123),(16,50,67,108),(17,35,68,125),(18,52,69,110),(19,37,70,127),(20,54,71,112),(21,39,72,97),(22,56,73,114),(23,41,74,99),(24,58,75,116),(25,43,76,101),(26,60,77,118),(27,45,78,103),(28,62,79,120),(29,47,80,105),(30,64,81,122),(31,49,82,107),(32,34,83,124)]])
80 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 16A | ··· | 16P | 16Q | ··· | 16X | 32A | ··· | 32AF |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 16 | ··· | 16 | 16 | ··· | 16 | 32 | ··· | 32 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 |
| type | + | + | + | |||||||||
| image | C1 | C2 | C2 | C4 | C4 | C4 | C8 | C8 | C8 | C16 | C16 | M6(2) |
| kernel | C32:5C4 | C4xC16 | C2xC32 | C32 | C4xC8 | C2xC16 | C16 | C42 | C2xC8 | C8 | C2xC4 | C2 |
| # reps | 1 | 1 | 2 | 8 | 2 | 2 | 8 | 4 | 4 | 16 | 16 | 16 |
Matrix representation of C32:5C4 ►in GL3(F97) generated by
| 96 | 0 | 0 |
| 0 | 25 | 30 |
| 0 | 30 | 72 |
| 75 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 96 | 0 |
G:=sub<GL(3,GF(97))| [96,0,0,0,25,30,0,30,72],[75,0,0,0,0,96,0,1,0] >;
C32:5C4 in GAP, Magma, Sage, TeX
C_{32}\rtimes_5C_4 % in TeX
G:=Group("C32:5C4"); // GroupNames label
G:=SmallGroup(128,129);
// by ID
G=gap.SmallGroup(128,129);
# by ID
G:=PCGroup([7,-2,2,-2,2,-2,-2,-2,28,925,64,100,102,124]);
// Polycyclic
G:=Group<a,b|a^32=b^4=1,b*a*b^-1=a^17>;
// generators/relations
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