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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

C1C4 — C22×Q16
C1C2C4C2×C4C22×C4C22×Q8 — C22×Q16
C1C2C4 — C22×Q16
C1C23C22×C4 — C22×Q16
C1C2C2C4 — 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 12A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J4K4L8A8B8C8D8E8F8G8H
 size 1111111122224444444422222222
ρ11111111111111111111111111111    trivial
ρ21-11-11-11-1-11-1111-1-11-11-1-1-11-11-111    linear of order 2
ρ31-1-1-111-11-111-11-1-11-1-1111-111-1-11-1    linear of order 2
ρ411-111-1-1-111-1-11-11-1-111-1-111-1-111-1    linear of order 2
ρ51-1-1-111-11-111-1-111-111-1-11-111-1-11-1    linear of order 2
ρ61-11-11-11-1-11-11-1-111-11-11-1-11-11-111    linear of order 2
ρ711-111-1-1-111-1-1-11-111-1-11-111-1-111-1    linear of order 2
ρ8111111111111-1-1-1-1-1-1-1-111111111    linear of order 2
ρ91-11-11-11-1-11-1111-1-1-11-1111-11-11-1-1    linear of order 2
ρ101-1-1-111-11-111-11-1-1111-1-1-11-1-111-11    linear of order 2
ρ111111111111111111-1-1-1-1-1-1-1-1-1-1-1-1    linear of order 2
ρ1211-111-1-1-111-1-11-11-11-1-111-1-111-1-11    linear of order 2
ρ131-1-1-111-11-111-1-111-1-1-111-11-1-111-11    linear of order 2
ρ141-11-11-11-1-11-11-1-1111-11-111-11-11-1-1    linear of order 2
ρ1511-111-1-1-111-1-1-11-11-111-11-1-111-1-11    linear of order 2
ρ16111111111111-1-1-1-11111-1-1-1-1-1-1-1-1    linear of order 2
ρ1722-222-2-2-2-2-2220000000000000000    orthogonal lifted from D4
ρ1822222222-2-2-2-20000000000000000    orthogonal lifted from D4
ρ192-22-22-22-22-22-20000000000000000    orthogonal lifted from D4
ρ202-2-2-222-222-2-220000000000000000    orthogonal lifted from D4
ρ212-222-2-2-22000000000000-2222-2-2-22    symplectic lifted from Q16, Schur index 2
ρ222-2-22-222-2000000000000222-22-2-2-2    symplectic lifted from Q16, Schur index 2
ρ232-2-22-222-2000000000000-2-2-22-2222    symplectic lifted from Q16, Schur index 2
ρ242-222-2-2-220000000000002-2-2-2222-2    symplectic lifted from Q16, Schur index 2
ρ25222-2-22-2-2000000000000-22-222-22-2    symplectic lifted from Q16, Schur index 2
ρ2622-2-2-2-22200000000000022-2-2-2-222    symplectic lifted from Q16, Schur index 2
ρ2722-2-2-2-222000000000000-2-22222-2-2    symplectic lifted from Q16, Schur index 2
ρ28222-2-22-2-20000000000002-22-2-22-22    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)])

C22×Q16 is a maximal subgroup of
(C2×C4)⋊9Q16  (C2×C4)⋊6Q16  (C2×Q16)⋊10C4  M4(2).33D4  C232Q16  (C2×C8).41D4  (C2×C4)⋊2Q16  M4(2).6D4  C4⋊C4.98D4  (C2×C4)⋊3Q16  C23.41D8  Q167D4  Q16.8D4  C42.279C23  Q8.(C2×D4)  C42.17C23  C8.D4⋊C2  M4(2).20D4  Q169D4  Q1612D4
C22×Q16 is a maximal quotient of
C42.224D4  C42.367D4  C233Q16  C42.267D4  C42.282D4  C42.297D4  D45Q16  D46Q16  Q85Q16  Q86Q16

Matrix representation of C22×Q16 in GL6(𝔽17)

1600000
0160000
0016000
0001600
000010
000001
,
100000
010000
0016000
0001600
000010
000001
,
010000
1600000
000100
0016000
0000143
00001414
,
1600000
010000
001000
0001600
0000013
0000130

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

Export

Character table of C22×Q16 in TeX

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