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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

 Derived series C1 — C16 — C2×D32
 Chief series C1 — C2 — C4 — C8 — C16 — C2×C16 — C2×D16 — C2×D32
 Lower central C1 — C2 — C4 — C8 — C16 — C2×D32
 Upper central C1 — C22 — C2×C4 — C2×C8 — C2×C16 — C2×D32
 Jennings C1 — C2 — C2 — C2 — C2 — C2 — C2 — C2 — C2 — C4 — C4 — C4 — C4 — C8 — C8 — C16 — 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, C2, C4, C22, C22, C8, C2×C4, D4, C23, C16, C2×C8, D8, C2×D4, C32, C2×C16, D16, D16, C2×D8, C2×C32, D32, C2×D16, C2×D32
Quotients: C1, C2, C22, D4, C23, D8, C2×D4, D16, C2×D8, D32, C2×D16, C2×D32

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

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

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

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

38 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 8A 8B 8C 8D 16A ··· 16H 32A ··· 32P order 1 2 2 2 2 2 2 2 4 4 8 8 8 8 16 ··· 16 32 ··· 32 size 1 1 1 1 16 16 16 16 2 2 2 2 2 2 2 ··· 2 2 ··· 2

38 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 type + + + + + + + + + + + image C1 C2 C2 C2 D4 D4 D8 D8 D16 D16 D32 kernel C2×D32 C2×C32 D32 C2×D16 C16 C2×C8 C8 C2×C4 C4 C22 C2 # reps 1 1 4 2 1 1 2 2 4 4 16

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

 96 0 0 0 1 0 0 0 1
,
 1 0 0 0 57 70 0 27 57
,
 96 0 0 0 1 0 0 0 96
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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