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## G = C4×C32order 128 = 27

### Abelian group of type [4,32]

Aliases: C4×C32, SmallGroup(128,128)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C4×C32
 Chief series C1 — C2 — C4 — C8 — C2×C8 — C2×C16 — C4×C16 — C4×C32
 Lower central C1 — C4×C32
 Upper central C1 — C4×C32
 Jennings C1 — C2 — C2 — C2 — C2 — C2 — C2 — C2 — C2 — C4 — C4 — C4 — C4 — C8 — C8 — C2×C16 — C4×C32

Generators and relations for C4×C32
G = < a,b | a4=b32=1, ab=ba >

Smallest permutation representation of C4×C32
Regular action on 128 points
Generators in S128
(1 52 92 108)(2 53 93 109)(3 54 94 110)(4 55 95 111)(5 56 96 112)(6 57 65 113)(7 58 66 114)(8 59 67 115)(9 60 68 116)(10 61 69 117)(11 62 70 118)(12 63 71 119)(13 64 72 120)(14 33 73 121)(15 34 74 122)(16 35 75 123)(17 36 76 124)(18 37 77 125)(19 38 78 126)(20 39 79 127)(21 40 80 128)(22 41 81 97)(23 42 82 98)(24 43 83 99)(25 44 84 100)(26 45 85 101)(27 46 86 102)(28 47 87 103)(29 48 88 104)(30 49 89 105)(31 50 90 106)(32 51 91 107)
(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)| (1,52,92,108)(2,53,93,109)(3,54,94,110)(4,55,95,111)(5,56,96,112)(6,57,65,113)(7,58,66,114)(8,59,67,115)(9,60,68,116)(10,61,69,117)(11,62,70,118)(12,63,71,119)(13,64,72,120)(14,33,73,121)(15,34,74,122)(16,35,75,123)(17,36,76,124)(18,37,77,125)(19,38,78,126)(20,39,79,127)(21,40,80,128)(22,41,81,97)(23,42,82,98)(24,43,83,99)(25,44,84,100)(26,45,85,101)(27,46,86,102)(28,47,87,103)(29,48,88,104)(30,49,89,105)(31,50,90,106)(32,51,91,107), (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( (1,52,92,108)(2,53,93,109)(3,54,94,110)(4,55,95,111)(5,56,96,112)(6,57,65,113)(7,58,66,114)(8,59,67,115)(9,60,68,116)(10,61,69,117)(11,62,70,118)(12,63,71,119)(13,64,72,120)(14,33,73,121)(15,34,74,122)(16,35,75,123)(17,36,76,124)(18,37,77,125)(19,38,78,126)(20,39,79,127)(21,40,80,128)(22,41,81,97)(23,42,82,98)(24,43,83,99)(25,44,84,100)(26,45,85,101)(27,46,86,102)(28,47,87,103)(29,48,88,104)(30,49,89,105)(31,50,90,106)(32,51,91,107), (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([[(1,52,92,108),(2,53,93,109),(3,54,94,110),(4,55,95,111),(5,56,96,112),(6,57,65,113),(7,58,66,114),(8,59,67,115),(9,60,68,116),(10,61,69,117),(11,62,70,118),(12,63,71,119),(13,64,72,120),(14,33,73,121),(15,34,74,122),(16,35,75,123),(17,36,76,124),(18,37,77,125),(19,38,78,126),(20,39,79,127),(21,40,80,128),(22,41,81,97),(23,42,82,98),(24,43,83,99),(25,44,84,100),(26,45,85,101),(27,46,86,102),(28,47,87,103),(29,48,88,104),(30,49,89,105),(31,50,90,106),(32,51,91,107)], [(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)]])

128 conjugacy classes

 class 1 2A 2B 2C 4A ··· 4L 8A ··· 8P 16A ··· 16AF 32A ··· 32BL order 1 2 2 2 4 ··· 4 8 ··· 8 16 ··· 16 32 ··· 32 size 1 1 1 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1

128 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 type + + + image C1 C2 C2 C4 C4 C4 C8 C8 C8 C16 C16 C32 kernel C4×C32 C4×C16 C2×C32 C32 C4×C8 C2×C16 C16 C42 C2×C8 C8 C2×C4 C4 # reps 1 1 2 8 2 2 8 4 4 16 16 64

Matrix representation of C4×C32 in GL2(𝔽97) generated by

 1 0 0 22
,
 46 0 0 42
G:=sub<GL(2,GF(97))| [1,0,0,22],[46,0,0,42] >;

C4×C32 in GAP, Magma, Sage, TeX

C_4\times C_{32}
% in TeX

G:=Group("C4xC32");
// GroupNames label

G:=SmallGroup(128,128);
// by ID

G=gap.SmallGroup(128,128);
# by ID

G:=PCGroup([7,-2,2,-2,2,-2,-2,-2,28,64,100,102,124]);
// Polycyclic

G:=Group<a,b|a^4=b^32=1,a*b=b*a>;
// generators/relations

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