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## G = C2×Q32order 64 = 26

### Direct product of C2 and Q32

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

Aliases: C2×Q32, C4.8D8, C8.11D4, C8.8C23, C16.5C22, C22.16D8, Q16.1C22, C4.9(C2×D4), (C2×C16).4C2, (C2×C4).83D4, C2.14(C2×D8), (C2×Q16).4C2, (C2×C8).84C22, SmallGroup(64,188)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C8 — C2×Q32
 Chief series C1 — C2 — C4 — C8 — C2×C8 — C2×Q16 — C2×Q32
 Lower central C1 — C2 — C4 — C8 — C2×Q32
 Upper central C1 — C22 — C2×C4 — C2×C8 — C2×Q32
 Jennings C1 — C2 — C2 — C2 — C2 — C4 — C4 — C8 — C2×Q32

Generators and relations for C2×Q32
G = < a,b,c | a2=b16=1, c2=b8, ab=ba, ac=ca, cbc-1=b-1 >

Character table of C2×Q32

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 8A 8B 8C 8D 16A 16B 16C 16D 16E 16F 16G 16H size 1 1 1 1 2 2 8 8 8 8 2 2 2 2 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 trivial ρ2 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 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 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 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 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 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 linear of order 2 ρ9 2 -2 -2 2 -2 2 0 0 0 0 2 -2 2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 2 2 2 0 0 0 0 -2 -2 -2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 2 2 -2 -2 0 0 0 0 0 0 0 0 √2 -√2 √2 √2 -√2 √2 -√2 -√2 orthogonal lifted from D8 ρ12 2 -2 -2 2 2 -2 0 0 0 0 0 0 0 0 √2 -√2 √2 -√2 √2 -√2 -√2 √2 orthogonal lifted from D8 ρ13 2 2 2 2 -2 -2 0 0 0 0 0 0 0 0 -√2 √2 -√2 -√2 √2 -√2 √2 √2 orthogonal lifted from D8 ρ14 2 -2 -2 2 2 -2 0 0 0 0 0 0 0 0 -√2 √2 -√2 √2 -√2 √2 √2 -√2 orthogonal lifted from D8 ρ15 2 -2 2 -2 0 0 0 0 0 0 -√2 -√2 √2 √2 ζ165-ζ163 -ζ1615+ζ169 -ζ165+ζ163 ζ165-ζ163 ζ1615-ζ169 -ζ165+ζ163 ζ1615-ζ169 -ζ1615+ζ169 symplectic lifted from Q32, Schur index 2 ρ16 2 2 -2 -2 0 0 0 0 0 0 √2 -√2 -√2 √2 -ζ165+ζ163 ζ1615-ζ169 ζ165-ζ163 ζ165-ζ163 ζ1615-ζ169 -ζ165+ζ163 -ζ1615+ζ169 -ζ1615+ζ169 symplectic lifted from Q32, Schur index 2 ρ17 2 2 -2 -2 0 0 0 0 0 0 -√2 √2 √2 -√2 -ζ1615+ζ169 -ζ165+ζ163 ζ1615-ζ169 ζ1615-ζ169 -ζ165+ζ163 -ζ1615+ζ169 ζ165-ζ163 ζ165-ζ163 symplectic lifted from Q32, Schur index 2 ρ18 2 -2 2 -2 0 0 0 0 0 0 √2 √2 -√2 -√2 ζ1615-ζ169 ζ165-ζ163 -ζ1615+ζ169 ζ1615-ζ169 -ζ165+ζ163 -ζ1615+ζ169 -ζ165+ζ163 ζ165-ζ163 symplectic lifted from Q32, Schur index 2 ρ19 2 2 -2 -2 0 0 0 0 0 0 -√2 √2 √2 -√2 ζ1615-ζ169 ζ165-ζ163 -ζ1615+ζ169 -ζ1615+ζ169 ζ165-ζ163 ζ1615-ζ169 -ζ165+ζ163 -ζ165+ζ163 symplectic lifted from Q32, Schur index 2 ρ20 2 -2 2 -2 0 0 0 0 0 0 -√2 -√2 √2 √2 -ζ165+ζ163 ζ1615-ζ169 ζ165-ζ163 -ζ165+ζ163 -ζ1615+ζ169 ζ165-ζ163 -ζ1615+ζ169 ζ1615-ζ169 symplectic lifted from Q32, Schur index 2 ρ21 2 2 -2 -2 0 0 0 0 0 0 √2 -√2 -√2 √2 ζ165-ζ163 -ζ1615+ζ169 -ζ165+ζ163 -ζ165+ζ163 -ζ1615+ζ169 ζ165-ζ163 ζ1615-ζ169 ζ1615-ζ169 symplectic lifted from Q32, Schur index 2 ρ22 2 -2 2 -2 0 0 0 0 0 0 √2 √2 -√2 -√2 -ζ1615+ζ169 -ζ165+ζ163 ζ1615-ζ169 -ζ1615+ζ169 ζ165-ζ163 ζ1615-ζ169 ζ165-ζ163 -ζ165+ζ163 symplectic lifted from Q32, Schur index 2

Smallest permutation representation of C2×Q32
Regular action on 64 points
Generators in S64
(1 18)(2 19)(3 20)(4 21)(5 22)(6 23)(7 24)(8 25)(9 26)(10 27)(11 28)(12 29)(13 30)(14 31)(15 32)(16 17)(33 49)(34 50)(35 51)(36 52)(37 53)(38 54)(39 55)(40 56)(41 57)(42 58)(43 59)(44 60)(45 61)(46 62)(47 63)(48 64)
(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 47 9 39)(2 46 10 38)(3 45 11 37)(4 44 12 36)(5 43 13 35)(6 42 14 34)(7 41 15 33)(8 40 16 48)(17 64 25 56)(18 63 26 55)(19 62 27 54)(20 61 28 53)(21 60 29 52)(22 59 30 51)(23 58 31 50)(24 57 32 49)

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

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

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

C2×Q32 is a maximal subgroup of
Q322C4  C16.18D4  Q324C4  Q16.8D4  D8.10D4  D8.12D4  Q16.4D4  Q16.5D4  C16.19D4  C16.D4  D4.4D8  C4⋊Q32  C8.21D8  C8.7D8  Q64⋊C2  Q8○D16
C2×Q32 is a maximal quotient of
Q16.8D4  Q16.4D4  C16.19D4  C4.Q32  C23.51D8  C4.SD32  C4⋊Q32  C162Q8

Matrix representation of C2×Q32 in GL3(𝔽17) generated by

 16 0 0 0 16 0 0 0 16
,
 1 0 0 0 2 8 0 13 10
,
 1 0 0 0 7 10 0 12 10
G:=sub<GL(3,GF(17))| [16,0,0,0,16,0,0,0,16],[1,0,0,0,2,13,0,8,10],[1,0,0,0,7,12,0,10,10] >;

C2×Q32 in GAP, Magma, Sage, TeX

C_2\times Q_{32}
% in TeX

G:=Group("C2xQ32");
// GroupNames label

G:=SmallGroup(64,188);
// by ID

G=gap.SmallGroup(64,188);
# by ID

G:=PCGroup([6,-2,2,2,-2,-2,-2,192,121,199,579,297,165,1444,730,88]);
// Polycyclic

G:=Group<a,b,c|a^2=b^16=1,c^2=b^8,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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