direct product, p-group, abelian, monomial
Aliases: C2×C32, SmallGroup(64,50)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
C1 — C2×C32 |
C1 — C2×C32 |
C1 — C2×C32 |
Generators and relations for C2×C32
G = < a,b | a2=b32=1, ab=ba >
(1 63)(2 64)(3 33)(4 34)(5 35)(6 36)(7 37)(8 38)(9 39)(10 40)(11 41)(12 42)(13 43)(14 44)(15 45)(16 46)(17 47)(18 48)(19 49)(20 50)(21 51)(22 52)(23 53)(24 54)(25 55)(26 56)(27 57)(28 58)(29 59)(30 60)(31 61)(32 62)
(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)
G:=sub<Sym(64)| (1,63)(2,64)(3,33)(4,34)(5,35)(6,36)(7,37)(8,38)(9,39)(10,40)(11,41)(12,42)(13,43)(14,44)(15,45)(16,46)(17,47)(18,48)(19,49)(20,50)(21,51)(22,52)(23,53)(24,54)(25,55)(26,56)(27,57)(28,58)(29,59)(30,60)(31,61)(32,62), (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)>;
G:=Group( (1,63)(2,64)(3,33)(4,34)(5,35)(6,36)(7,37)(8,38)(9,39)(10,40)(11,41)(12,42)(13,43)(14,44)(15,45)(16,46)(17,47)(18,48)(19,49)(20,50)(21,51)(22,52)(23,53)(24,54)(25,55)(26,56)(27,57)(28,58)(29,59)(30,60)(31,61)(32,62), (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) );
G=PermutationGroup([[(1,63),(2,64),(3,33),(4,34),(5,35),(6,36),(7,37),(8,38),(9,39),(10,40),(11,41),(12,42),(13,43),(14,44),(15,45),(16,46),(17,47),(18,48),(19,49),(20,50),(21,51),(22,52),(23,53),(24,54),(25,55),(26,56),(27,57),(28,58),(29,59),(30,60),(31,61),(32,62)], [(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)]])
C2×C32 is a maximal subgroup of
C32⋊5C4 C22⋊C32 D4.C16 D16⋊2C4 Q32⋊2C4 D16.C4 C4⋊C32 C32⋊3C4 C32⋊4C4 C32.C4 M7(2) D4○C32 C4○D32 D5⋊C32
C2×C32 is a maximal quotient of
C22⋊C32 C4⋊C32 M7(2) D5⋊C32
64 conjugacy classes
class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 8A | ··· | 8H | 16A | ··· | 16P | 32A | ··· | 32AF |
order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 16 | ··· | 16 | 32 | ··· | 32 |
size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
64 irreducible representations
dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
type | + | + | + | |||||||
image | C1 | C2 | C2 | C4 | C4 | C8 | C8 | C16 | C16 | C32 |
kernel | C2×C32 | C32 | C2×C16 | C16 | C2×C8 | C8 | C2×C4 | C4 | C22 | C2 |
# reps | 1 | 2 | 1 | 2 | 2 | 4 | 4 | 8 | 8 | 32 |
Matrix representation of C2×C32 ►in GL2(𝔽97) generated by
96 | 0 |
0 | 1 |
19 | 0 |
0 | 45 |
G:=sub<GL(2,GF(97))| [96,0,0,1],[19,0,0,45] >;
C2×C32 in GAP, Magma, Sage, TeX
C_2\times C_{32}
% in TeX
G:=Group("C2xC32");
// GroupNames label
G:=SmallGroup(64,50);
// by ID
G=gap.SmallGroup(64,50);
# by ID
G:=PCGroup([6,-2,2,-2,-2,-2,-2,24,50,69,88]);
// Polycyclic
G:=Group<a,b|a^2=b^32=1,a*b=b*a>;
// generators/relations
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