p-group, metacyclic, nilpotent (class 5), monomial
Aliases: C32⋊3C4, C2.2D32, C8.5Q16, C4.1Q32, C16.2Q8, C2.2Q64, C22.10D16, (C2×C32).3C2, (C2×C4).68D8, C8.15(C4⋊C4), C16.16(C2×C4), (C2×C8).239D4, C16⋊3C4.2C2, C2.3(C16⋊3C4), C4.10(C2.D8), (C2×C16).77C22, SmallGroup(128,155)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C32⋊3C4
G = < a,b | a32=b4=1, bab-1=a-1 >
Smallest permutation representation of C32⋊3C4
►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 71 97 39)(2 70 98 38)(3 69 99 37)(4 68 100 36)(5 67 101 35)(6 66 102 34)(7 65 103 33)(8 96 104 64)(9 95 105 63)(10 94 106 62)(11 93 107 61)(12 92 108 60)(13 91 109 59)(14 90 110 58)(15 89 111 57)(16 88 112 56)(17 87 113 55)(18 86 114 54)(19 85 115 53)(20 84 116 52)(21 83 117 51)(22 82 118 50)(23 81 119 49)(24 80 120 48)(25 79 121 47)(26 78 122 46)(27 77 123 45)(28 76 124 44)(29 75 125 43)(30 74 126 42)(31 73 127 41)(32 72 128 40)
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), (1,71,97,39)(2,70,98,38)(3,69,99,37)(4,68,100,36)(5,67,101,35)(6,66,102,34)(7,65,103,33)(8,96,104,64)(9,95,105,63)(10,94,106,62)(11,93,107,61)(12,92,108,60)(13,91,109,59)(14,90,110,58)(15,89,111,57)(16,88,112,56)(17,87,113,55)(18,86,114,54)(19,85,115,53)(20,84,116,52)(21,83,117,51)(22,82,118,50)(23,81,119,49)(24,80,120,48)(25,79,121,47)(26,78,122,46)(27,77,123,45)(28,76,124,44)(29,75,125,43)(30,74,126,42)(31,73,127,41)(32,72,128,40)>;
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), (1,71,97,39)(2,70,98,38)(3,69,99,37)(4,68,100,36)(5,67,101,35)(6,66,102,34)(7,65,103,33)(8,96,104,64)(9,95,105,63)(10,94,106,62)(11,93,107,61)(12,92,108,60)(13,91,109,59)(14,90,110,58)(15,89,111,57)(16,88,112,56)(17,87,113,55)(18,86,114,54)(19,85,115,53)(20,84,116,52)(21,83,117,51)(22,82,118,50)(23,81,119,49)(24,80,120,48)(25,79,121,47)(26,78,122,46)(27,77,123,45)(28,76,124,44)(29,75,125,43)(30,74,126,42)(31,73,127,41)(32,72,128,40) );
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)], [(1,71,97,39),(2,70,98,38),(3,69,99,37),(4,68,100,36),(5,67,101,35),(6,66,102,34),(7,65,103,33),(8,96,104,64),(9,95,105,63),(10,94,106,62),(11,93,107,61),(12,92,108,60),(13,91,109,59),(14,90,110,58),(15,89,111,57),(16,88,112,56),(17,87,113,55),(18,86,114,54),(19,85,115,53),(20,84,116,52),(21,83,117,51),(22,82,118,50),(23,81,119,49),(24,80,120,48),(25,79,121,47),(26,78,122,46),(27,77,123,45),(28,76,124,44),(29,75,125,43),(30,74,126,42),(31,73,127,41),(32,72,128,40)]])
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 | 2 |
| type | + | + | + | - | + | - | + | - | + | + | - | |
| image | C1 | C2 | C2 | C4 | Q8 | D4 | Q16 | D8 | Q32 | D16 | D32 | Q64 |
| kernel | C32⋊3C4 | C16⋊3C4 | C2×C32 | C32 | C16 | C2×C8 | C8 | C2×C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 4 | 1 | 1 | 2 | 2 | 4 | 4 | 8 | 8 |
Matrix representation of C32⋊3C4 ►in GL3(𝔽97) generated by
| 1 | 0 | 0 |
| 0 | 63 | 0 |
| 0 | 0 | 77 |
| 22 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 96 | 0 |
G:=sub<GL(3,GF(97))| [1,0,0,0,63,0,0,0,77],[22,0,0,0,0,96,0,1,0] >;
C32⋊3C4 in GAP, Magma, Sage, TeX
C_{32}\rtimes_3C_4 % in TeX
G:=Group("C32:3C4"); // GroupNames label
G:=SmallGroup(128,155);
// by ID
G=gap.SmallGroup(128,155);
# by ID
G:=PCGroup([7,-2,2,-2,2,-2,-2,-2,56,85,148,422,268,1684,242,4037,124]);
// Polycyclic
G:=Group<a,b|a^32=b^4=1,b*a*b^-1=a^-1>;
// generators/relations
Export