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## G = C22×Q16order 64 = 26

### Direct product of C22 and Q16

direct product, p-group, metabelian, nilpotent (class 3), monomial

Aliases: C22×Q16, C4.3C24, C8.12C23, C23.62D4, Q8.1C23, (C2×C4).89D4, C4.18(C2×D4), (C2×C8).85C22, (C22×C8).10C2, C2.25(C22×D4), C22.66(C2×D4), (C22×Q8).9C2, (C2×C4).137C23, (C2×Q8).68C22, (C22×C4).131C22, SmallGroup(64,252)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C4 — C22×Q16
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C22×Q8 — C22×Q16
 Lower central C1 — C2 — C4 — C22×Q16
 Upper central C1 — C23 — C22×C4 — C22×Q16
 Jennings C1 — C2 — C2 — C4 — C22×Q16

Generators and relations for C22×Q16
G = < a,b,c,d | a2=b2=c8=1, d2=c4, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 169 in 129 conjugacy classes, 89 normal (7 characteristic)
C1, C2, C2 [×6], C4, C4 [×3], C4 [×8], C22 [×7], C8 [×4], C2×C4 [×6], C2×C4 [×12], Q8 [×8], Q8 [×12], C23, C2×C8 [×6], Q16 [×16], C22×C4, C22×C4 [×2], C2×Q8 [×12], C2×Q8 [×6], C22×C8, C2×Q16 [×12], C22×Q8 [×2], C22×Q16
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], Q16 [×4], C2×D4 [×6], C24, C2×Q16 [×6], C22×D4, C22×Q16

Character table of C22×Q16

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 8A 8B 8C 8D 8E 8F 8G 8H size 1 1 1 1 1 1 1 1 2 2 2 2 4 4 4 4 4 4 4 4 2 2 2 2 2 2 2 2 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 -1 1 -1 1 -1 1 -1 -1 1 -1 1 1 1 -1 -1 1 -1 1 -1 -1 -1 1 -1 1 -1 1 1 linear of order 2 ρ3 1 -1 -1 -1 1 1 -1 1 -1 1 1 -1 1 -1 -1 1 -1 -1 1 1 1 -1 1 1 -1 -1 1 -1 linear of order 2 ρ4 1 1 -1 1 1 -1 -1 -1 1 1 -1 -1 1 -1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 1 -1 linear of order 2 ρ5 1 -1 -1 -1 1 1 -1 1 -1 1 1 -1 -1 1 1 -1 1 1 -1 -1 1 -1 1 1 -1 -1 1 -1 linear of order 2 ρ6 1 -1 1 -1 1 -1 1 -1 -1 1 -1 1 -1 -1 1 1 -1 1 -1 1 -1 -1 1 -1 1 -1 1 1 linear of order 2 ρ7 1 1 -1 1 1 -1 -1 -1 1 1 -1 -1 -1 1 -1 1 1 -1 -1 1 -1 1 1 -1 -1 1 1 -1 linear of order 2 ρ8 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 linear of order 2 ρ9 1 -1 1 -1 1 -1 1 -1 -1 1 -1 1 1 1 -1 -1 -1 1 -1 1 1 1 -1 1 -1 1 -1 -1 linear of order 2 ρ10 1 -1 -1 -1 1 1 -1 1 -1 1 1 -1 1 -1 -1 1 1 1 -1 -1 -1 1 -1 -1 1 1 -1 1 linear of order 2 ρ11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ12 1 1 -1 1 1 -1 -1 -1 1 1 -1 -1 1 -1 1 -1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 linear of order 2 ρ13 1 -1 -1 -1 1 1 -1 1 -1 1 1 -1 -1 1 1 -1 -1 -1 1 1 -1 1 -1 -1 1 1 -1 1 linear of order 2 ρ14 1 -1 1 -1 1 -1 1 -1 -1 1 -1 1 -1 -1 1 1 1 -1 1 -1 1 1 -1 1 -1 1 -1 -1 linear of order 2 ρ15 1 1 -1 1 1 -1 -1 -1 1 1 -1 -1 -1 1 -1 1 -1 1 1 -1 1 -1 -1 1 1 -1 -1 1 linear of order 2 ρ16 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ17 2 2 -2 2 2 -2 -2 -2 -2 -2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ18 2 2 2 2 2 2 2 2 -2 -2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ19 2 -2 2 -2 2 -2 2 -2 2 -2 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ20 2 -2 -2 -2 2 2 -2 2 2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ21 2 -2 2 2 -2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 -√2 √2 √2 √2 -√2 -√2 -√2 √2 symplectic lifted from Q16, Schur index 2 ρ22 2 -2 -2 2 -2 2 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 √2 √2 √2 -√2 √2 -√2 -√2 -√2 symplectic lifted from Q16, Schur index 2 ρ23 2 -2 -2 2 -2 2 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 -√2 -√2 -√2 √2 -√2 √2 √2 √2 symplectic lifted from Q16, Schur index 2 ρ24 2 -2 2 2 -2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 √2 -√2 -√2 -√2 √2 √2 √2 -√2 symplectic lifted from Q16, Schur index 2 ρ25 2 2 2 -2 -2 2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 -√2 √2 -√2 √2 √2 -√2 √2 -√2 symplectic lifted from Q16, Schur index 2 ρ26 2 2 -2 -2 -2 -2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 √2 √2 -√2 -√2 -√2 -√2 √2 √2 symplectic lifted from Q16, Schur index 2 ρ27 2 2 -2 -2 -2 -2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 -√2 -√2 √2 √2 √2 √2 -√2 -√2 symplectic lifted from Q16, Schur index 2 ρ28 2 2 2 -2 -2 2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 √2 -√2 √2 -√2 -√2 √2 -√2 √2 symplectic lifted from Q16, Schur index 2

Smallest permutation representation of C22×Q16
Regular action on 64 points
Generators in S64
(1 50)(2 51)(3 52)(4 53)(5 54)(6 55)(7 56)(8 49)(9 32)(10 25)(11 26)(12 27)(13 28)(14 29)(15 30)(16 31)(17 57)(18 58)(19 59)(20 60)(21 61)(22 62)(23 63)(24 64)(33 46)(34 47)(35 48)(36 41)(37 42)(38 43)(39 44)(40 45)
(1 59)(2 60)(3 61)(4 62)(5 63)(6 64)(7 57)(8 58)(9 40)(10 33)(11 34)(12 35)(13 36)(14 37)(15 38)(16 39)(17 56)(18 49)(19 50)(20 51)(21 52)(22 53)(23 54)(24 55)(25 46)(26 47)(27 48)(28 41)(29 42)(30 43)(31 44)(32 45)
(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 38 5 34)(2 37 6 33)(3 36 7 40)(4 35 8 39)(9 61 13 57)(10 60 14 64)(11 59 15 63)(12 58 16 62)(17 32 21 28)(18 31 22 27)(19 30 23 26)(20 29 24 25)(41 56 45 52)(42 55 46 51)(43 54 47 50)(44 53 48 49)

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

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

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

Matrix representation of C22×Q16 in GL6(𝔽17)

 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 16 0 0 0 0 0 0 0 0 1 0 0 0 0 16 0 0 0 0 0 0 0 14 3 0 0 0 0 14 14
,
 16 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 0 13 0 0 0 0 13 0

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,14,14,0,0,0,0,3,14],[16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,13,0,0,0,0,13,0] >;

C22×Q16 in GAP, Magma, Sage, TeX

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

G:=Group("C2^2xQ16");
// GroupNames label

G:=SmallGroup(64,252);
// by ID

G=gap.SmallGroup(64,252);
# by ID

G:=PCGroup([6,-2,2,2,2,-2,-2,192,217,199,1444,730,88]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^8=1,d^2=c^4,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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