p-group, metacyclic, nilpotent (class 2), monomial
Aliases: C4⋊C32, C16.7Q8, C16.29D4, C42.8C8, C2.3M6(2), C8.26M4(2), C4.12M5(2), C2.2(C2×C32), (C2×C32).2C2, (C4×C16).4C2, (C2×C4).4C16, (C4×C8).14C4, (C2×C8).12C8, C2.2(C4⋊C16), C4.25(C4⋊C8), C8.42(C4⋊C4), (C2×C16).10C4, C22.9(C2×C16), (C2×C16).110C22, (C2×C4).97(C2×C8), (C2×C8).261(C2×C4), SmallGroup(128,153)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C4⋊C32
G = < a,b | a4=b32=1, bab-1=a-1 >
Smallest permutation representation of C4⋊C32
►Regular action on 128 points
Generators in S128
(1 79 105 51)(2 52 106 80)(3 81 107 53)(4 54 108 82)(5 83 109 55)(6 56 110 84)(7 85 111 57)(8 58 112 86)(9 87 113 59)(10 60 114 88)(11 89 115 61)(12 62 116 90)(13 91 117 63)(14 64 118 92)(15 93 119 33)(16 34 120 94)(17 95 121 35)(18 36 122 96)(19 65 123 37)(20 38 124 66)(21 67 125 39)(22 40 126 68)(23 69 127 41)(24 42 128 70)(25 71 97 43)(26 44 98 72)(27 73 99 45)(28 46 100 74)(29 75 101 47)(30 48 102 76)(31 77 103 49)(32 50 104 78)
(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)
G:=sub<Sym(128)| (1,79,105,51)(2,52,106,80)(3,81,107,53)(4,54,108,82)(5,83,109,55)(6,56,110,84)(7,85,111,57)(8,58,112,86)(9,87,113,59)(10,60,114,88)(11,89,115,61)(12,62,116,90)(13,91,117,63)(14,64,118,92)(15,93,119,33)(16,34,120,94)(17,95,121,35)(18,36,122,96)(19,65,123,37)(20,38,124,66)(21,67,125,39)(22,40,126,68)(23,69,127,41)(24,42,128,70)(25,71,97,43)(26,44,98,72)(27,73,99,45)(28,46,100,74)(29,75,101,47)(30,48,102,76)(31,77,103,49)(32,50,104,78), (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)>;
G:=Group( (1,79,105,51)(2,52,106,80)(3,81,107,53)(4,54,108,82)(5,83,109,55)(6,56,110,84)(7,85,111,57)(8,58,112,86)(9,87,113,59)(10,60,114,88)(11,89,115,61)(12,62,116,90)(13,91,117,63)(14,64,118,92)(15,93,119,33)(16,34,120,94)(17,95,121,35)(18,36,122,96)(19,65,123,37)(20,38,124,66)(21,67,125,39)(22,40,126,68)(23,69,127,41)(24,42,128,70)(25,71,97,43)(26,44,98,72)(27,73,99,45)(28,46,100,74)(29,75,101,47)(30,48,102,76)(31,77,103,49)(32,50,104,78), (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) );
G=PermutationGroup([[(1,79,105,51),(2,52,106,80),(3,81,107,53),(4,54,108,82),(5,83,109,55),(6,56,110,84),(7,85,111,57),(8,58,112,86),(9,87,113,59),(10,60,114,88),(11,89,115,61),(12,62,116,90),(13,91,117,63),(14,64,118,92),(15,93,119,33),(16,34,120,94),(17,95,121,35),(18,36,122,96),(19,65,123,37),(20,38,124,66),(21,67,125,39),(22,40,126,68),(23,69,127,41),(24,42,128,70),(25,71,97,43),(26,44,98,72),(27,73,99,45),(28,46,100,74),(29,75,101,47),(30,48,102,76),(31,77,103,49),(32,50,104,78)], [(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)]])
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 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | |||||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | C8 | C16 | C32 | D4 | Q8 | M4(2) | M5(2) | M6(2) |
| kernel | C4⋊C32 | C4×C16 | C2×C32 | C4×C8 | C2×C16 | C42 | C2×C8 | C2×C4 | C4 | C16 | C16 | C8 | C4 | C2 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 16 | 32 | 1 | 1 | 2 | 4 | 8 |
Matrix representation of C4⋊C32 ►in GL3(𝔽97) generated by
| 96 | 0 | 0 |
| 0 | 1 | 22 |
| 0 | 44 | 96 |
| 51 | 0 | 0 |
| 0 | 76 | 49 |
| 0 | 45 | 21 |
G:=sub<GL(3,GF(97))| [96,0,0,0,1,44,0,22,96],[51,0,0,0,76,45,0,49,21] >;
C4⋊C32 in GAP, Magma, Sage, TeX
C_4\rtimes C_{32} % in TeX
G:=Group("C4:C32"); // GroupNames label
G:=SmallGroup(128,153);
// by ID
G=gap.SmallGroup(128,153);
# by ID
G:=PCGroup([7,-2,2,-2,2,-2,-2,-2,56,85,36,80,102,124]);
// Polycyclic
G:=Group<a,b|a^4=b^32=1,b*a*b^-1=a^-1>;
// generators/relations
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