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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

C1C8 — C22×D16
C1C2C4C8C2×C8C22×C8C22×D8 — C22×D16
C1C2C4C8 — C22×D16
C1C23C22×C4C22×C8 — C22×D16
C1C2C2C2C2C4C4C8 — 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 [×6], C2 [×8], C4, C4 [×3], C22 [×7], C22 [×32], C8, C8 [×3], C2×C4 [×6], D4 [×20], C23, C23 [×20], C16 [×4], C2×C8 [×6], D8 [×8], D8 [×12], C22×C4, C2×D4 [×18], C24 [×2], C2×C16 [×6], D16 [×16], C22×C8, C2×D8 [×12], C2×D8 [×6], C22×D4 [×2], C22×C16, C2×D16 [×12], C22×D8 [×2], C22×D16
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D8 [×4], C2×D4 [×6], C24, D16 [×4], C2×D8 [×6], C22×D4, C2×D16 [×6], C22×D8, C22×D16

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

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

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

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

44 conjugacy classes

class 1 2A···2G2H···2O4A4B4C4D8A···8H16A···16P
order12···22···244448···816···16
size11···18···822222···22···2

44 irreducible representations

dim111122222
type+++++++++
imageC1C2C2C2D4D4D8D8D16
kernelC22×D16C22×C16C2×D16C22×D8C2×C8C22×C4C2×C4C23C22
# reps11122316216

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

1000
01600
00160
00016
,
16000
01600
00160
00016
,
1000
01600
001311
00613
,
1000
0100
00160
0001
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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