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G = C2×D32order 128 = 27

Direct product of C2 and D32

direct product, p-group, metabelian, nilpotent (class 5), monomial

Aliases: C2×D32, C8.18D8, C4.6D16, C16.9D4, C322C22, D161C22, C16.6C23, C22.14D16, (C2×C32)⋊5C2, (C2×D16)⋊7C2, C8.45(C2×D4), C4.13(C2×D8), (C2×C4).88D8, C2.12(C2×D16), (C2×C8).257D4, (C2×C16).88C22, 2-Sylow(SO-(4,31)), SmallGroup(128,991)

Series: Derived Chief Lower central Upper central Jennings

C1C16 — C2×D32
C1C2C4C8C16C2×C16C2×D16 — C2×D32
C1C2C4C8C16 — C2×D32
C1C22C2×C4C2×C8C2×C16 — C2×D32
C1C2C2C2C2C2C2C2C2C4C4C4C4C8C8C16 — C2×D32

Generators and relations for C2×D32
 G = < a,b,c | a2=b32=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 268 in 60 conjugacy classes, 28 normal (14 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C22, C22 [×8], C8 [×2], C2×C4, D4 [×6], C23 [×2], C16 [×2], C2×C8, D8 [×6], C2×D4 [×2], C32 [×2], C2×C16, D16 [×4], D16 [×2], C2×D8 [×2], C2×C32, D32 [×4], C2×D16 [×2], C2×D32
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D8 [×2], C2×D4, D16 [×2], C2×D8, D32 [×2], C2×D16, C2×D32

Smallest permutation representation of C2×D32
On 64 points
Generators in S64
(1 56)(2 57)(3 58)(4 59)(5 60)(6 61)(7 62)(8 63)(9 64)(10 33)(11 34)(12 35)(13 36)(14 37)(15 38)(16 39)(17 40)(18 41)(19 42)(20 43)(21 44)(22 45)(23 46)(24 47)(25 48)(26 49)(27 50)(28 51)(29 52)(30 53)(31 54)(32 55)
(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)
(2 32)(3 31)(4 30)(5 29)(6 28)(7 27)(8 26)(9 25)(10 24)(11 23)(12 22)(13 21)(14 20)(15 19)(16 18)(33 47)(34 46)(35 45)(36 44)(37 43)(38 42)(39 41)(48 64)(49 63)(50 62)(51 61)(52 60)(53 59)(54 58)(55 57)

G:=sub<Sym(64)| (1,56)(2,57)(3,58)(4,59)(5,60)(6,61)(7,62)(8,63)(9,64)(10,33)(11,34)(12,35)(13,36)(14,37)(15,38)(16,39)(17,40)(18,41)(19,42)(20,43)(21,44)(22,45)(23,46)(24,47)(25,48)(26,49)(27,50)(28,51)(29,52)(30,53)(31,54)(32,55), (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), (2,32)(3,31)(4,30)(5,29)(6,28)(7,27)(8,26)(9,25)(10,24)(11,23)(12,22)(13,21)(14,20)(15,19)(16,18)(33,47)(34,46)(35,45)(36,44)(37,43)(38,42)(39,41)(48,64)(49,63)(50,62)(51,61)(52,60)(53,59)(54,58)(55,57)>;

G:=Group( (1,56)(2,57)(3,58)(4,59)(5,60)(6,61)(7,62)(8,63)(9,64)(10,33)(11,34)(12,35)(13,36)(14,37)(15,38)(16,39)(17,40)(18,41)(19,42)(20,43)(21,44)(22,45)(23,46)(24,47)(25,48)(26,49)(27,50)(28,51)(29,52)(30,53)(31,54)(32,55), (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), (2,32)(3,31)(4,30)(5,29)(6,28)(7,27)(8,26)(9,25)(10,24)(11,23)(12,22)(13,21)(14,20)(15,19)(16,18)(33,47)(34,46)(35,45)(36,44)(37,43)(38,42)(39,41)(48,64)(49,63)(50,62)(51,61)(52,60)(53,59)(54,58)(55,57) );

G=PermutationGroup([(1,56),(2,57),(3,58),(4,59),(5,60),(6,61),(7,62),(8,63),(9,64),(10,33),(11,34),(12,35),(13,36),(14,37),(15,38),(16,39),(17,40),(18,41),(19,42),(20,43),(21,44),(22,45),(23,46),(24,47),(25,48),(26,49),(27,50),(28,51),(29,52),(30,53),(31,54),(32,55)], [(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)], [(2,32),(3,31),(4,30),(5,29),(6,28),(7,27),(8,26),(9,25),(10,24),(11,23),(12,22),(13,21),(14,20),(15,19),(16,18),(33,47),(34,46),(35,45),(36,44),(37,43),(38,42),(39,41),(48,64),(49,63),(50,62),(51,61),(52,60),(53,59),(54,58),(55,57)])

38 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B8A8B8C8D16A···16H32A···32P
order1222222244888816···1632···32
size1111161616162222222···22···2

38 irreducible representations

dim11112222222
type+++++++++++
imageC1C2C2C2D4D4D8D8D16D16D32
kernelC2×D32C2×C32D32C2×D16C16C2×C8C8C2×C4C4C22C2
# reps114211224416

Matrix representation of C2×D32 in GL3(𝔽97) generated by

9600
010
001
,
100
05770
02757
,
9600
010
0096
G:=sub<GL(3,GF(97))| [96,0,0,0,1,0,0,0,1],[1,0,0,0,57,27,0,70,57],[96,0,0,0,1,0,0,0,96] >;

C2×D32 in GAP, Magma, Sage, TeX

C_2\times D_{32}
% in TeX

G:=Group("C2xD32");
// GroupNames label

G:=SmallGroup(128,991);
// by ID

G=gap.SmallGroup(128,991);
# by ID

G:=PCGroup([7,-2,2,2,-2,-2,-2,-2,141,675,346,192,1684,851,242,4037,2028,124]);
// Polycyclic

G:=Group<a,b,c|a^2=b^32=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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