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G = C4×C32order 128 = 27

Abelian group of type [4,32]

direct product, p-group, abelian, monomial

Aliases: C4×C32, SmallGroup(128,128)

Series: Derived Chief Lower central Upper central Jennings

C1 — C4×C32
C1C2C4C8C2×C8C2×C16C4×C16 — C4×C32
C1 — C4×C32
C1 — C4×C32
C1C2C2C2C2C2C2C2C2C4C4C4C4C8C8C2×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 85 60 123)(2 86 61 124)(3 87 62 125)(4 88 63 126)(5 89 64 127)(6 90 33 128)(7 91 34 97)(8 92 35 98)(9 93 36 99)(10 94 37 100)(11 95 38 101)(12 96 39 102)(13 65 40 103)(14 66 41 104)(15 67 42 105)(16 68 43 106)(17 69 44 107)(18 70 45 108)(19 71 46 109)(20 72 47 110)(21 73 48 111)(22 74 49 112)(23 75 50 113)(24 76 51 114)(25 77 52 115)(26 78 53 116)(27 79 54 117)(28 80 55 118)(29 81 56 119)(30 82 57 120)(31 83 58 121)(32 84 59 122)
(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)| (1,85,60,123)(2,86,61,124)(3,87,62,125)(4,88,63,126)(5,89,64,127)(6,90,33,128)(7,91,34,97)(8,92,35,98)(9,93,36,99)(10,94,37,100)(11,95,38,101)(12,96,39,102)(13,65,40,103)(14,66,41,104)(15,67,42,105)(16,68,43,106)(17,69,44,107)(18,70,45,108)(19,71,46,109)(20,72,47,110)(21,73,48,111)(22,74,49,112)(23,75,50,113)(24,76,51,114)(25,77,52,115)(26,78,53,116)(27,79,54,117)(28,80,55,118)(29,81,56,119)(30,82,57,120)(31,83,58,121)(32,84,59,122), (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( (1,85,60,123)(2,86,61,124)(3,87,62,125)(4,88,63,126)(5,89,64,127)(6,90,33,128)(7,91,34,97)(8,92,35,98)(9,93,36,99)(10,94,37,100)(11,95,38,101)(12,96,39,102)(13,65,40,103)(14,66,41,104)(15,67,42,105)(16,68,43,106)(17,69,44,107)(18,70,45,108)(19,71,46,109)(20,72,47,110)(21,73,48,111)(22,74,49,112)(23,75,50,113)(24,76,51,114)(25,77,52,115)(26,78,53,116)(27,79,54,117)(28,80,55,118)(29,81,56,119)(30,82,57,120)(31,83,58,121)(32,84,59,122), (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([(1,85,60,123),(2,86,61,124),(3,87,62,125),(4,88,63,126),(5,89,64,127),(6,90,33,128),(7,91,34,97),(8,92,35,98),(9,93,36,99),(10,94,37,100),(11,95,38,101),(12,96,39,102),(13,65,40,103),(14,66,41,104),(15,67,42,105),(16,68,43,106),(17,69,44,107),(18,70,45,108),(19,71,46,109),(20,72,47,110),(21,73,48,111),(22,74,49,112),(23,75,50,113),(24,76,51,114),(25,77,52,115),(26,78,53,116),(27,79,54,117),(28,80,55,118),(29,81,56,119),(30,82,57,120),(31,83,58,121),(32,84,59,122)], [(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 2A2B2C4A···4L8A···8P16A···16AF32A···32BL
order12224···48···816···1632···32
size11111···11···11···11···1

128 irreducible representations

dim111111111111
type+++
imageC1C2C2C4C4C4C8C8C8C16C16C32
kernelC4×C32C4×C16C2×C32C32C4×C8C2×C16C16C42C2×C8C8C2×C4C4
# reps112822844161664

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

10
022
,
460
042
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

Export

Subgroup lattice of C4×C32 in TeX

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