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G = C2.Q32order 64 = 26

1st central extension by C2 of Q32

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

Aliases: Q161C4, C8.15D4, C2.1Q32, C4.2SD16, C2.2SD32, C22.9D8, C8.8(C2×C4), (C2×C16).1C2, (C2×C4).61D4, C2.D8.1C2, (C2×Q16).1C2, C4.2(C22⋊C4), (C2×C8).70C22, C2.7(D4⋊C4), SmallGroup(64,39)

Series: Derived Chief Lower central Upper central Jennings

C1C8 — C2.Q32
C1C2C4C2×C4C2×C8C2×Q16 — C2.Q32
C1C2C4C8 — C2.Q32
C1C22C2×C4C2×C8 — C2.Q32
C1C2C2C2C2C4C4C2×C8 — C2.Q32

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

4C4
4C4
8C4
2Q8
2Q8
4C2×C4
4Q8
4C2×C4
2C16
2Q16
2C2×Q8
2C4⋊C4

Character table of C2.Q32

 class 12A2B2C4A4B4C4D4E4F8A8B8C8D16A16B16C16D16E16F16G16H
 size 1111228888222222222222
ρ11111111111111111111111    trivial
ρ2111111-11-111111-1-1-1-1-1-1-1-1    linear of order 2
ρ3111111-1-1-1-1111111111111    linear of order 2
ρ41111111-11-11111-1-1-1-1-1-1-1-1    linear of order 2
ρ51-1-11-11i-1-i11-11-1i-i-iiii-i-i    linear of order 4
ρ61-1-11-11-i-1i11-11-1-iii-i-i-iii    linear of order 4
ρ71-1-11-11-i1i-11-11-1i-i-iiii-i-i    linear of order 4
ρ81-1-11-11i1-i-11-11-1-iii-i-i-iii    linear of order 4
ρ92222220000-2-2-2-200000000    orthogonal lifted from D4
ρ102-2-22-220000-22-2200000000    orthogonal lifted from D4
ρ112222-2-2000000002-22-22-22-2    orthogonal lifted from D8
ρ122222-2-200000000-22-22-22-22    orthogonal lifted from D8
ρ1322-2-20000002-2-22ζ1651631615169ζ1651631615169165163ζ1615169165163ζ1615169    symplectic lifted from Q32, Schur index 2
ρ1422-2-2000000-222-2ζ1615169ζ165163ζ1615169ζ16516316151691651631615169165163    symplectic lifted from Q32, Schur index 2
ρ1522-2-20000002-2-22165163ζ1615169165163ζ1615169ζ1651631615169ζ1651631615169    symplectic lifted from Q32, Schur index 2
ρ1622-2-2000000-222-216151691651631615169165163ζ1615169ζ165163ζ1615169ζ165163    symplectic lifted from Q32, Schur index 2
ρ172-2-222-200000000--2--2-2-2--2-2-2--2    complex lifted from SD16
ρ182-2-222-200000000-2-2--2--2-2--2--2-2    complex lifted from SD16
ρ192-22-2000000-2-222ζ16131611ζ1615169ζ165163ζ16716ζ165163ζ1615169ζ16131611ζ16716    complex lifted from SD32
ρ202-22-200000022-2-2ζ16716ζ16131611ζ1615169ζ165163ζ1615169ζ16131611ζ16716ζ165163    complex lifted from SD32
ρ212-22-200000022-2-2ζ1615169ζ165163ζ16716ζ16131611ζ16716ζ165163ζ1615169ζ16131611    complex lifted from SD32
ρ222-22-2000000-2-222ζ165163ζ16716ζ16131611ζ1615169ζ16131611ζ16716ζ165163ζ1615169    complex lifted from SD32

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

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

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

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

C2.Q32 is a maximal subgroup of
C23.24D8  C23.39D8  C23.41D8  C4×SD32  C4×Q32  SD323C4  Q324C4  Q167D4  D88D4  Q16.8D4  D8.10D4  D8.4D4  Q16.4D4  D8.5D4  Q16.5D4  C16.19D4  C168D4  C16.D4  C162D4  Q16⋊Q8  Q16.Q8  C23.50D8  C23.51D8  C23.20D8  C8.22SD16  C8.12SD16  C8.14SD16  D5.Q32
 C2p.Q32: C4.Q32  C4.SD32  C6.Q32  C2.Dic24  C6.5Q32  C10.Q32  C40.78D4  C40.15D4 ...
C2.Q32 is a maximal quotient of
C23.32D8  C8.7C42  D5.Q32
 C2p.Q32: Q161C8  C8.27D8  C4.6Q32  C6.Q32  C2.Dic24  C6.5Q32  C10.Q32  C40.78D4 ...

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

1600
010
001
,
1300
0140
0011
,
100
001
0160
G:=sub<GL(3,GF(17))| [16,0,0,0,1,0,0,0,1],[13,0,0,0,14,0,0,0,11],[1,0,0,0,0,16,0,1,0] >;

C2.Q32 in GAP, Magma, Sage, TeX

C_2.Q_{32}
% in TeX

G:=Group("C2.Q32");
// GroupNames label

G:=SmallGroup(64,39);
// by ID

G=gap.SmallGroup(64,39);
# by ID

G:=PCGroup([6,-2,2,-2,2,-2,-2,48,73,199,362,188,230,1444,730,88]);
// 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=a*b^-1>;
// generators/relations

Export

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

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