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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

C1C8 — C2×Q32
C1C2C4C8C2×C8C2×Q16 — C2×Q32
C1C2C4C8 — C2×Q32
C1C22C2×C4C2×C8 — C2×Q32
C1C2C2C2C2C4C4C8 — 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 >

4C4
4C4
4C4
4C4
2Q8
2Q8
2Q8
2Q8
4C2×C4
4Q8
4C2×C4
4Q8
2Q16
2C2×Q8
2C2×Q8
2Q16

Character table of C2×Q32

 class 12A2B2C4A4B4C4D4E4F8A8B8C8D16A16B16C16D16E16F16G16H
 size 1111228888222222222222
ρ11111111111111111111111    trivial
ρ21-1-11-11-11-11-11-11111-1-1-11-1    linear of order 2
ρ31-1-11-11-1-111-11-11-1-1-1111-11    linear of order 2
ρ41111111-1-111111-1-1-1-1-1-1-1-1    linear of order 2
ρ5111111-1-1-1-1111111111111    linear of order 2
ρ61-1-11-111-11-1-11-11111-1-1-11-1    linear of order 2
ρ71-1-11-1111-1-1-11-11-1-1-1111-11    linear of order 2
ρ8111111-111-11111-1-1-1-1-1-1-1-1    linear of order 2
ρ92-2-22-2200002-22-200000000    orthogonal lifted from D4
ρ102222220000-2-2-2-200000000    orthogonal lifted from D4
ρ112222-2-2000000002-222-22-2-2    orthogonal lifted from D8
ρ122-2-222-2000000002-22-22-2-22    orthogonal lifted from D8
ρ132222-2-200000000-22-2-22-222    orthogonal lifted from D8
ρ142-2-222-200000000-22-22-222-2    orthogonal lifted from D8
ρ152-22-2000000-2-222ζ1651631615169165163ζ165163ζ1615169165163ζ16151691615169    symplectic lifted from Q32, Schur index 2
ρ1622-2-20000002-2-22165163ζ1615169ζ165163ζ165163ζ161516916516316151691615169    symplectic lifted from Q32, Schur index 2
ρ1722-2-2000000-222-21615169165163ζ1615169ζ16151691651631615169ζ165163ζ165163    symplectic lifted from Q32, Schur index 2
ρ182-22-200000022-2-2ζ1615169ζ1651631615169ζ16151691651631615169165163ζ165163    symplectic lifted from Q32, Schur index 2
ρ1922-2-2000000-222-2ζ1615169ζ16516316151691615169ζ165163ζ1615169165163165163    symplectic lifted from Q32, Schur index 2
ρ202-22-2000000-2-222165163ζ1615169ζ1651631651631615169ζ1651631615169ζ1615169    symplectic lifted from Q32, Schur index 2
ρ2122-2-20000002-2-22ζ16516316151691651631651631615169ζ165163ζ1615169ζ1615169    symplectic lifted from Q32, Schur index 2
ρ222-22-200000022-2-21615169165163ζ16151691615169ζ165163ζ1615169ζ165163165163    symplectic lifted from Q32, Schur index 2

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

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

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

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

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

1600
0160
0016
,
100
028
01310
,
100
0710
01210
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

Export

Subgroup lattice of C2×Q32 in TeX
Character table of C2×Q32 in TeX

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