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

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

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

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

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

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