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## G = C22×D16order 128 = 27

### Direct product of C22 and D16

direct product, p-group, metabelian, nilpotent (class 4), monomial

Aliases: C22×D16, C162C23, D81C23, C8.9C24, C23.63D8, C4.21(C2×D8), (C2×C4).94D8, C8.54(C2×D4), (C22×C16)⋊9C2, (C2×C8).262D4, (C2×C16)⋊18C22, (C2×D8)⋊45C22, (C22×D8)⋊14C2, C2.24(C22×D8), C4.15(C22×D4), C22.75(C2×D8), (C2×C8).571C23, (C22×C4).621D4, (C22×C8).541C22, (C2×C4).872(C2×D4), 2-Sylow(GO-(4,17)), SmallGroup(128,2140)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C8 — C22×D16
 Chief series C1 — C2 — C4 — C8 — C2×C8 — C22×C8 — C22×D8 — C22×D16
 Lower central C1 — C2 — C4 — C8 — C22×D16
 Upper central C1 — C23 — C22×C4 — C22×C8 — C22×D16
 Jennings C1 — C2 — C2 — C2 — C2 — C4 — C4 — C8 — C22×D16

Generators and relations for C22×D16
G = < a,b,c,d | a2=b2=c16=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 660 in 220 conjugacy classes, 100 normal (9 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, D4, C23, C23, C16, C2×C8, D8, D8, C22×C4, C2×D4, C24, C2×C16, D16, C22×C8, C2×D8, C2×D8, C22×D4, C22×C16, C2×D16, C22×D8, C22×D16
Quotients: C1, C2, C22, D4, C23, D8, C2×D4, C24, D16, C2×D8, C22×D4, C2×D16, C22×D8, C22×D16

Smallest permutation representation of C22×D16
On 64 points
Generators in S64
(1 44)(2 45)(3 46)(4 47)(5 48)(6 33)(7 34)(8 35)(9 36)(10 37)(11 38)(12 39)(13 40)(14 41)(15 42)(16 43)(17 61)(18 62)(19 63)(20 64)(21 49)(22 50)(23 51)(24 52)(25 53)(26 54)(27 55)(28 56)(29 57)(30 58)(31 59)(32 60)
(1 63)(2 64)(3 49)(4 50)(5 51)(6 52)(7 53)(8 54)(9 55)(10 56)(11 57)(12 58)(13 59)(14 60)(15 61)(16 62)(17 42)(18 43)(19 44)(20 45)(21 46)(22 47)(23 48)(24 33)(25 34)(26 35)(27 36)(28 37)(29 38)(30 39)(31 40)(32 41)
(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)
(1 63)(2 62)(3 61)(4 60)(5 59)(6 58)(7 57)(8 56)(9 55)(10 54)(11 53)(12 52)(13 51)(14 50)(15 49)(16 64)(17 46)(18 45)(19 44)(20 43)(21 42)(22 41)(23 40)(24 39)(25 38)(26 37)(27 36)(28 35)(29 34)(30 33)(31 48)(32 47)

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

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

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

44 conjugacy classes

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

44 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 type + + + + + + + + + image C1 C2 C2 C2 D4 D4 D8 D8 D16 kernel C22×D16 C22×C16 C2×D16 C22×D8 C2×C8 C22×C4 C2×C4 C23 C22 # reps 1 1 12 2 3 1 6 2 16

Matrix representation of C22×D16 in GL4(𝔽17) generated by

 1 0 0 0 0 16 0 0 0 0 16 0 0 0 0 16
,
 16 0 0 0 0 16 0 0 0 0 16 0 0 0 0 16
,
 1 0 0 0 0 16 0 0 0 0 13 11 0 0 6 13
,
 1 0 0 0 0 1 0 0 0 0 16 0 0 0 0 1
G:=sub<GL(4,GF(17))| [1,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[1,0,0,0,0,16,0,0,0,0,13,6,0,0,11,13],[1,0,0,0,0,1,0,0,0,0,16,0,0,0,0,1] >;

C22×D16 in GAP, Magma, Sage, TeX

C_2^2\times D_{16}
% in TeX

G:=Group("C2^2xD16");
// GroupNames label

G:=SmallGroup(128,2140);
// by ID

G=gap.SmallGroup(128,2140);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,-2,-2,253,1684,851,242,4037,2028,124]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^16=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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