Copied to
clipboard

G = Q64⋊C2order 128 = 27

2nd semidirect product of Q64 and C2 acting faithfully

p-group, metabelian, nilpotent (class 5), monomial

Aliases: C8.9D8, Q642C2, C16.4D4, C32.C22, SD642C2, C4.15D16, M6(2)⋊2C2, C22.6D16, C16.11C23, D16.3C22, Q32.3C22, C8.50(C2×D4), C4.18(C2×D8), (C2×C4).52D8, (C2×Q32)⋊11C2, C4○D16.4C2, C2.17(C2×D16), (C2×C8).142D4, (C2×C16).33C22, SmallGroup(128,996)

Series: Derived Chief Lower central Upper central Jennings

C1C16 — Q64⋊C2
C1C2C4C8C16C2×C16C2×Q32 — Q64⋊C2
C1C2C4C8C16 — Q64⋊C2
C1C2C2×C4C2×C8C2×C16 — Q64⋊C2
C1C2C2C2C2C2C2C2C2C4C4C4C4C8C8C16 — Q64⋊C2

Generators and relations for Q64⋊C2
 G = < a,b,c | a32=c2=1, b2=a16, bab-1=a-1, cac=a17, bc=cb >

2C2
16C2
8C4
8C22
8C4
8C4
4D4
4Q8
4Q8
4Q8
8D4
8C2×C4
8C2×C4
8Q8
2Q16
2D8
2Q16
2Q16
4C2×Q8
4SD16
4Q16
4C4○D4
2C4○D8
2SD32
2C2×Q16
2Q32

Character table of Q64⋊C2

 class 12A2B2C4A4B4C4D4E8A8B8C16A16B16C16D16E16F32A32B32C32D32E32F32G32H
 size 112162216161622422224444444444
ρ111111111111111111111111111    trivial
ρ211-1-11-111-111-11111-1-1-11-11-11-11    linear of order 2
ρ3111-111-111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ411-111-1-11-111-11111-1-11-11-11-11-1    linear of order 2
ρ51111111-1-1111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ611-1-11-11-1111-11111-1-11-11-11-11-1    linear of order 2
ρ7111-111-1-1-111111111111111111    linear of order 2
ρ811-111-1-1-1111-11111-1-1-11-11-11-11    linear of order 2
ρ9222022000222-2-2-2-2-2-200000000    orthogonal lifted from D4
ρ1022-202-200022-2-2-2-2-22200000000    orthogonal lifted from D4
ρ1122-202-2000-2-22000000-222-22-2-22    orthogonal lifted from D8
ρ1222-202-2000-2-220000002-2-22-222-2    orthogonal lifted from D8
ρ13222022000-2-2-200000022-2-2-2-222    orthogonal lifted from D8
ρ14222022000-2-2-2000000-2-22222-2-2    orthogonal lifted from D8
ρ152220-2-2000000-22-222-2ζ16716ζ16716165163165163ζ165163ζ1651631671616716    orthogonal lifted from D16
ρ162220-2-20000002-22-2-22ζ165163ζ165163ζ16716ζ167161671616716165163165163    orthogonal lifted from D16
ρ1722-20-220000002-22-22-2ζ165163165163ζ167161671616716ζ16716165163ζ165163    orthogonal lifted from D16
ρ182220-2-20000002-22-2-221651631651631671616716ζ16716ζ16716ζ165163ζ165163    orthogonal lifted from D16
ρ1922-20-220000002-22-22-2165163ζ16516316716ζ16716ζ1671616716ζ165163165163    orthogonal lifted from D16
ρ202220-2-2000000-22-222-21671616716ζ165163ζ165163165163165163ζ16716ζ16716    orthogonal lifted from D16
ρ2122-20-22000000-22-22-2216716ζ16716ζ165163165163165163ζ165163ζ1671616716    orthogonal lifted from D16
ρ2222-20-22000000-22-22-22ζ1671616716165163ζ165163ζ16516316516316716ζ16716    orthogonal lifted from D16
ρ234-40000000-22220-2ζ165+2ζ163-2ζ1615+2ζ169165-2ζ1631615-2ζ1690000000000    symplectic faithful, Schur index 2
ρ244-4000000022-2201615-2ζ169-2ζ165+2ζ163-2ζ1615+2ζ169165-2ζ1630000000000    symplectic faithful, Schur index 2
ρ254-40000000-22220165-2ζ1631615-2ζ169-2ζ165+2ζ163-2ζ1615+2ζ1690000000000    symplectic faithful, Schur index 2
ρ264-4000000022-220-2ζ1615+2ζ169165-2ζ1631615-2ζ169-2ζ165+2ζ1630000000000    symplectic faithful, Schur index 2

Smallest permutation representation of Q64⋊C2
On 64 points
Generators in S64
(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 39 17 55)(2 38 18 54)(3 37 19 53)(4 36 20 52)(5 35 21 51)(6 34 22 50)(7 33 23 49)(8 64 24 48)(9 63 25 47)(10 62 26 46)(11 61 27 45)(12 60 28 44)(13 59 29 43)(14 58 30 42)(15 57 31 41)(16 56 32 40)
(1 54)(2 39)(3 56)(4 41)(5 58)(6 43)(7 60)(8 45)(9 62)(10 47)(11 64)(12 49)(13 34)(14 51)(15 36)(16 53)(17 38)(18 55)(19 40)(20 57)(21 42)(22 59)(23 44)(24 61)(25 46)(26 63)(27 48)(28 33)(29 50)(30 35)(31 52)(32 37)

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

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

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

Matrix representation of Q64⋊C2 in GL4(𝔽97) generated by

4144261
76309626
49299139
873732
,
8279126
20662696
1926590
76843178
,
2048950
7648095
35807749
62172149
G:=sub<GL(4,GF(97))| [41,76,49,8,44,30,29,73,26,96,91,7,1,26,39,32],[82,20,1,76,79,66,92,84,1,26,65,31,26,96,90,78],[20,76,35,62,48,48,80,17,95,0,77,21,0,95,49,49] >;

Q64⋊C2 in GAP, Magma, Sage, TeX

Q_{64}\rtimes C_2
% in TeX

G:=Group("Q64:C2");
// GroupNames label

G:=SmallGroup(128,996);
// by ID

G=gap.SmallGroup(128,996);
# by ID

G:=PCGroup([7,-2,2,2,-2,-2,-2,-2,141,456,1430,675,346,192,1684,851,242,4037,2028,124]);
// Polycyclic

G:=Group<a,b,c|a^32=c^2=1,b^2=a^16,b*a*b^-1=a^-1,c*a*c=a^17,b*c=c*b>;
// generators/relations

Export

Subgroup lattice of Q64⋊C2 in TeX
Character table of Q64⋊C2 in TeX

׿
×
𝔽