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G = Q16⋊D6order 192 = 26·3

2nd semidirect product of Q16 and D6 acting via D6/C6=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D88D6, Q167D6, C24.32D4, C12.51D8, C24.29C23, D24.11C22, C4○D82S3, (C2×C6).9D8, C3⋊D166C2, C3⋊C164C22, C6.68(C2×D8), (C2×C8).98D6, (C2×D24)⋊22C2, C35(C16⋊C22), C8.6D66C2, (C3×D8)⋊8C22, C8.7(C3⋊D4), C12.C86C2, C4.24(D4⋊S3), (C2×C12).185D4, C12.191(C2×D4), C8.35(C22×S3), (C3×Q16)⋊7C22, C22.5(D4⋊S3), (C2×C24).104C22, (C3×C4○D8)⋊4C2, C2.23(C2×D4⋊S3), C4.17(C2×C3⋊D4), (C2×C4).81(C3⋊D4), SmallGroup(192,752)

Series: Derived Chief Lower central Upper central

C1C24 — Q16⋊D6
C1C3C6C12C24D24C2×D24 — Q16⋊D6
C3C6C12C24 — Q16⋊D6
C1C2C2×C4C2×C8C4○D8

Generators and relations for Q16⋊D6
 G = < a,b,c,d | a8=c6=d2=1, b2=a4, bab-1=dad=a-1, ac=ca, cbc-1=a4b, dbd=a3b, dcd=c-1 >

Subgroups: 360 in 90 conjugacy classes, 35 normal (27 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C23, C12, C12, D6, C2×C6, C2×C6, C16, C2×C8, D8, D8, SD16, Q16, C2×D4, C4○D4, C24, D12, C2×C12, C2×C12, C3×D4, C3×Q8, C22×S3, M5(2), D16, SD32, C2×D8, C4○D8, C3⋊C16, D24, D24, C2×C24, C3×D8, C3×SD16, C3×Q16, C2×D12, C3×C4○D4, C16⋊C22, C12.C8, C3⋊D16, C8.6D6, C2×D24, C3×C4○D8, Q16⋊D6
Quotients: C1, C2, C22, S3, D4, C23, D6, D8, C2×D4, C3⋊D4, C22×S3, C2×D8, D4⋊S3, C2×C3⋊D4, C16⋊C22, C2×D4⋊S3, Q16⋊D6

Character table of Q16⋊D6

 class 12A2B2C2D2E34A4B4C6A6B6C6D8A8B8C12A12B12C12D12E16A16B16C16D24A24B24C24D
 size 112824242228248822422488121212124444
ρ1111111111111111111111111111111    trivial
ρ211-1-1-111-1111-1-1-111-1-1-11111-1-11-11-11    linear of order 2
ρ311-1-11-11-1111-1-1-111-1-1-1111-111-1-11-11    linear of order 2
ρ41111-1-11111111111111111-1-1-1-11111    linear of order 2
ρ5111-1-1-1111-111-1-1111111-1-111111111    linear of order 2
ρ611-111-11-11-11-11111-1-1-11-1-11-1-11-11-11    linear of order 2
ρ711-11-111-11-11-11111-1-1-11-1-1-111-1-11-11    linear of order 2
ρ8111-111111-111-1-1111111-1-1-1-1-1-11111    linear of order 2
ρ922200022202200-2-2-2222000000-2-2-2-2    orthogonal lifted from D4
ρ10222200-1222-1-1-1-1222-1-1-1-1-10000-1-1-1-1    orthogonal lifted from S3
ρ11222-200-122-2-1-111222-1-1-1110000-1-1-1-1    orthogonal lifted from D6
ρ1222-20002-2202-200-2-22-2-220000002-22-2    orthogonal lifted from D4
ρ1322-2200-1-22-2-11-1-122-211-11100001-11-1    orthogonal lifted from D6
ρ1422-2-200-1-222-111122-211-1-1-100001-11-1    orthogonal lifted from D6
ρ1522-200022-202-20000022-200-22-220000    orthogonal lifted from D8
ρ1622-200022-202-20000022-2002-22-20000    orthogonal lifted from D8
ρ172220002-2-202200000-2-2-200-2-2220000    orthogonal lifted from D8
ρ182220002-2-202200000-2-2-20022-2-20000    orthogonal lifted from D8
ρ1922-2000-1-220-11-3--3-2-2211-1-3--30000-11-11    complex lifted from C3⋊D4
ρ20222000-1220-1-1--3-3-2-2-2-1-1-1-3--300001111    complex lifted from C3⋊D4
ρ2122-2000-1-220-11--3-3-2-2211-1--3-30000-11-11    complex lifted from C3⋊D4
ρ22222000-1220-1-1-3--3-2-2-2-1-1-1--3-300001111    complex lifted from C3⋊D4
ρ23444000-2-4-40-2-2000002220000000000    orthogonal lifted from D4⋊S3, Schur index 2
ρ2444-4000-24-40-2200000-2-220000000000    orthogonal lifted from D4⋊S3, Schur index 2
ρ254-400004000-400022-2200000000000-22022    orthogonal lifted from C16⋊C22
ρ264-400004000-4000-222200000000000220-22    orthogonal lifted from C16⋊C22
ρ274-40000-2000200022-220-2323000000062-6-2    orthogonal faithful
ρ284-40000-2000200022-22023-230000000-626-2    orthogonal faithful
ρ294-40000-20002000-2222023-2300000006-2-62    orthogonal faithful
ρ304-40000-20002000-22220-23230000000-6-262    orthogonal faithful

Smallest permutation representation of Q16⋊D6
On 48 points
Generators in S48
(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)
(1 13 5 9)(2 12 6 16)(3 11 7 15)(4 10 8 14)(17 28 21 32)(18 27 22 31)(19 26 23 30)(20 25 24 29)(33 47 37 43)(34 46 38 42)(35 45 39 41)(36 44 40 48)
(1 38 25)(2 39 26)(3 40 27)(4 33 28)(5 34 29)(6 35 30)(7 36 31)(8 37 32)(9 42 20 13 46 24)(10 43 21 14 47 17)(11 44 22 15 48 18)(12 45 23 16 41 19)
(1 25)(2 32)(3 31)(4 30)(5 29)(6 28)(7 27)(8 26)(9 17)(10 24)(11 23)(12 22)(13 21)(14 20)(15 19)(16 18)(33 35)(36 40)(37 39)(41 48)(42 47)(43 46)(44 45)

G:=sub<Sym(48)| (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), (1,13,5,9)(2,12,6,16)(3,11,7,15)(4,10,8,14)(17,28,21,32)(18,27,22,31)(19,26,23,30)(20,25,24,29)(33,47,37,43)(34,46,38,42)(35,45,39,41)(36,44,40,48), (1,38,25)(2,39,26)(3,40,27)(4,33,28)(5,34,29)(6,35,30)(7,36,31)(8,37,32)(9,42,20,13,46,24)(10,43,21,14,47,17)(11,44,22,15,48,18)(12,45,23,16,41,19), (1,25)(2,32)(3,31)(4,30)(5,29)(6,28)(7,27)(8,26)(9,17)(10,24)(11,23)(12,22)(13,21)(14,20)(15,19)(16,18)(33,35)(36,40)(37,39)(41,48)(42,47)(43,46)(44,45)>;

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), (1,13,5,9)(2,12,6,16)(3,11,7,15)(4,10,8,14)(17,28,21,32)(18,27,22,31)(19,26,23,30)(20,25,24,29)(33,47,37,43)(34,46,38,42)(35,45,39,41)(36,44,40,48), (1,38,25)(2,39,26)(3,40,27)(4,33,28)(5,34,29)(6,35,30)(7,36,31)(8,37,32)(9,42,20,13,46,24)(10,43,21,14,47,17)(11,44,22,15,48,18)(12,45,23,16,41,19), (1,25)(2,32)(3,31)(4,30)(5,29)(6,28)(7,27)(8,26)(9,17)(10,24)(11,23)(12,22)(13,21)(14,20)(15,19)(16,18)(33,35)(36,40)(37,39)(41,48)(42,47)(43,46)(44,45) );

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)], [(1,13,5,9),(2,12,6,16),(3,11,7,15),(4,10,8,14),(17,28,21,32),(18,27,22,31),(19,26,23,30),(20,25,24,29),(33,47,37,43),(34,46,38,42),(35,45,39,41),(36,44,40,48)], [(1,38,25),(2,39,26),(3,40,27),(4,33,28),(5,34,29),(6,35,30),(7,36,31),(8,37,32),(9,42,20,13,46,24),(10,43,21,14,47,17),(11,44,22,15,48,18),(12,45,23,16,41,19)], [(1,25),(2,32),(3,31),(4,30),(5,29),(6,28),(7,27),(8,26),(9,17),(10,24),(11,23),(12,22),(13,21),(14,20),(15,19),(16,18),(33,35),(36,40),(37,39),(41,48),(42,47),(43,46),(44,45)]])

Matrix representation of Q16⋊D6 in GL4(𝔽97) generated by

161800
799500
009579
001816
,
00960
00096
1000
0100
,
969600
1000
0011
00960
,
969600
0100
009516
00182
G:=sub<GL(4,GF(97))| [16,79,0,0,18,95,0,0,0,0,95,18,0,0,79,16],[0,0,1,0,0,0,0,1,96,0,0,0,0,96,0,0],[96,1,0,0,96,0,0,0,0,0,1,96,0,0,1,0],[96,0,0,0,96,1,0,0,0,0,95,18,0,0,16,2] >;

Q16⋊D6 in GAP, Magma, Sage, TeX

Q_{16}\rtimes D_6
% in TeX

G:=Group("Q16:D6");
// GroupNames label

G:=SmallGroup(192,752);
// by ID

G=gap.SmallGroup(192,752);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,254,387,675,185,192,1684,438,102,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^8=c^6=d^2=1,b^2=a^4,b*a*b^-1=d*a*d=a^-1,a*c=c*a,c*b*c^-1=a^4*b,d*b*d=a^3*b,d*c*d=c^-1>;
// generators/relations

Export

Character table of Q16⋊D6 in TeX

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