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G = Q1610D4order 128 = 27

4th semidirect product of Q16 and D4 acting via D4/C22=C2

p-group, metabelian, nilpotent (class 3), monomial

Aliases: Q1610D4, C42.38C23, C4.1252+ 1+4, C2.55D42, C89D48C2, C8.34(C2×D4), C82D420C2, C88D420C2, C83D419C2, C4⋊C4.358D4, Q86D43C2, Q85D43C2, Q8.21(C2×D4), D4⋊D436C2, Q8⋊D417C2, C4⋊SD1618C2, (C2×D4).163D4, C4⋊C8.94C22, (C2×C8).88C23, C4.85(C22×D4), Q16⋊C420C2, D4.7D437C2, C4⋊C4.210C23, (C2×C4).469C24, Q8.D437C2, C22⋊C4.159D4, (C2×D8).80C22, C23.102(C2×D4), C4.Q8.53C22, C8⋊C4.38C22, C2.59(D4○SD16), (C2×D4).209C23, (C4×D4).145C22, C41D4.75C22, C4⋊D4.60C22, C22⋊C8.72C22, (C4×Q8).139C22, (C2×Q8).388C23, C22⋊Q8.59C22, D4⋊C4.66C22, (C22×C8).285C22, (C2×Q16).160C22, Q8⋊C4.67C22, (C2×SD16).49C22, C4.4D4.54C22, C22.729(C22×D4), C2.78(D8⋊C22), (C22×C4).1120C23, (C22×Q8).330C22, (C2×M4(2)).104C22, (C2×C4○D8)⋊26C2, (C2×C4).593(C2×D4), (C2×C8.C22)⋊29C2, (C2×C4○D4).186C22, SmallGroup(128,2003)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — Q1610D4
C1C2C22C2×C4C2×Q8C2×C4○D4C2×C4○D8 — Q1610D4
C1C2C2×C4 — Q1610D4
C1C22C4×D4 — Q1610D4
C1C2C2C2×C4 — Q1610D4

Generators and relations for Q1610D4
 G = < a,b,c,d | a8=c4=d2=1, b2=a4, bab-1=a-1, cac-1=dad=a3, cbc-1=a4b, bd=db, dcd=c-1 >

Subgroups: 512 in 244 conjugacy classes, 94 normal (84 characteristic)
C1, C2 [×3], C2 [×5], C4 [×2], C4 [×11], C22, C22 [×15], C8 [×2], C8 [×3], C2×C4 [×5], C2×C4 [×17], D4 [×20], Q8 [×4], Q8 [×6], C23 [×2], C23 [×3], C42, C42 [×2], C22⋊C4 [×2], C22⋊C4 [×6], C4⋊C4 [×3], C4⋊C4 [×3], C2×C8 [×4], C2×C8 [×2], M4(2) [×2], D8 [×3], SD16 [×9], Q16 [×4], Q16 [×2], C22×C4 [×2], C22×C4 [×4], C2×D4 [×4], C2×D4 [×8], C2×Q8 [×3], C2×Q8 [×4], C4○D4 [×9], C8⋊C4, C22⋊C8 [×2], D4⋊C4 [×3], Q8⋊C4 [×3], C4⋊C8, C4.Q8, C4×D4, C4×D4 [×2], C4×Q8 [×2], C4⋊D4 [×3], C4⋊D4 [×3], C22⋊Q8, C22⋊Q8, C4.4D4, C4.4D4, C41D4, C41D4, C22×C8, C2×M4(2), C2×D8 [×2], C2×SD16 [×5], C2×Q16 [×2], C4○D8 [×4], C8.C22 [×4], C22×Q8, C2×C4○D4 [×3], C89D4, Q16⋊C4, Q8⋊D4, D4⋊D4 [×2], D4.7D4, C4⋊SD16, Q8.D4, C88D4, C82D4, C83D4, Q85D4, Q86D4, C2×C4○D8, C2×C8.C22, Q1610D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C24, C22×D4 [×2], 2+ 1+4, D42, D8⋊C22, D4○SD16, Q1610D4

Character table of Q1610D4

 class 12A2B2C2D2E2F2G2H4A4B4C4D4E4F4G4H4I4J4K4L4M4N8A8B8C8D8E8F
 size 11114488822224444444888444488
ρ111111111111111111111111111111    trivial
ρ2111111-1-1-11111-1-1111-1-1-1-1-1111111    linear of order 2
ρ3111111-1-111111-11111-11-111-1-1-1-1-1-1    linear of order 2
ρ411111111-111111-11111-11-1-1-1-1-1-1-1-1    linear of order 2
ρ51111-11-11-1-11-11-111-1-1-111-111-1-111-1    linear of order 2
ρ61111-111-11-11-111-11-1-11-1-11-11-1-111-1    linear of order 2
ρ71111-111-1-1-11-11111-1-111-1-11-111-1-11    linear of order 2
ρ81111-11-111-11-11-1-11-1-1-1-111-1-111-1-11    linear of order 2
ρ91111-1-111-1-11-11-11-111-11-11-11-1-11-11    linear of order 2
ρ101111-1-1-1-11-11-111-1-1111-11-111-1-11-11    linear of order 2
ρ111111-1-1-1-1-1-11-1111-1111111-1-111-11-1    linear of order 2
ρ121111-1-1111-11-11-1-1-111-1-1-1-11-111-11-1    linear of order 2
ρ1311111-1-111111111-1-1-111-1-1-11111-1-1    linear of order 2
ρ1411111-11-1-11111-1-1-1-1-1-1-11111111-1-1    linear of order 2
ρ1511111-11-111111-11-1-1-1-111-1-1-1-1-1-111    linear of order 2
ρ1611111-1-11-111111-1-1-1-11-1-111-1-1-1-111    linear of order 2
ρ172-22-2000000-202-220002-20000-22000    orthogonal lifted from D4
ρ182222-220002-22-200-22-200000000000    orthogonal lifted from D4
ρ192-22-2000000-20222000-2-200002-2000    orthogonal lifted from D4
ρ202222-2-20002-22-2002-2200000000000    orthogonal lifted from D4
ρ212-22-2000000-2022-2000-220000-22000    orthogonal lifted from D4
ρ2222222-2000-2-2-2-20022-200000000000    orthogonal lifted from D4
ρ232-22-2000000-202-2-20002200002-2000    orthogonal lifted from D4
ρ24222222000-2-2-2-200-2-2200000000000    orthogonal lifted from D4
ρ254-44-400000040-40000000000000000    orthogonal lifted from 2+ 1+4
ρ264-4-4400000-4i04i00000000000000000    complex lifted from D8⋊C22
ρ274-4-44000004i0-4i00000000000000000    complex lifted from D8⋊C22
ρ2844-4-40000000000000000000-2-2002-200    complex lifted from D4○SD16
ρ2944-4-400000000000000000002-200-2-200    complex lifted from D4○SD16

Smallest permutation representation of Q1610D4
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 15 5 11)(2 14 6 10)(3 13 7 9)(4 12 8 16)(17 46 21 42)(18 45 22 41)(19 44 23 48)(20 43 24 47)(25 57 29 61)(26 64 30 60)(27 63 31 59)(28 62 32 58)(33 50 37 54)(34 49 38 53)(35 56 39 52)(36 55 40 51)
(1 64 48 36)(2 59 41 39)(3 62 42 34)(4 57 43 37)(5 60 44 40)(6 63 45 35)(7 58 46 38)(8 61 47 33)(9 32 21 49)(10 27 22 52)(11 30 23 55)(12 25 24 50)(13 28 17 53)(14 31 18 56)(15 26 19 51)(16 29 20 54)
(1 50)(2 53)(3 56)(4 51)(5 54)(6 49)(7 52)(8 55)(9 35)(10 38)(11 33)(12 36)(13 39)(14 34)(15 37)(16 40)(17 59)(18 62)(19 57)(20 60)(21 63)(22 58)(23 61)(24 64)(25 48)(26 43)(27 46)(28 41)(29 44)(30 47)(31 42)(32 45)

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

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

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

Matrix representation of Q1610D4 in GL6(𝔽17)

1600000
0160000
004009
0000413
0040013
0004013
,
100000
010000
0010150
0000161
0010160
00116160
,
320000
12140000
001630
00146107
00113133
0014101314
,
14150000
430000
006705
00101111
00661216
001212516

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,4,0,4,0,0,0,0,0,0,4,0,0,0,4,0,0,0,0,9,13,13,13],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,1,1,0,0,0,0,0,16,0,0,15,16,16,16,0,0,0,1,0,0],[3,12,0,0,0,0,2,14,0,0,0,0,0,0,1,14,11,14,0,0,6,6,3,10,0,0,3,10,13,13,0,0,0,7,3,14],[14,4,0,0,0,0,15,3,0,0,0,0,0,0,6,1,6,12,0,0,7,0,6,12,0,0,0,11,12,5,0,0,5,11,16,16] >;

Q1610D4 in GAP, Magma, Sage, TeX

Q_{16}\rtimes_{10}D_4
% in TeX

G:=Group("Q16:10D4");
// GroupNames label

G:=SmallGroup(128,2003);
// by ID

G=gap.SmallGroup(128,2003);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,456,758,352,346,248,2804,1411,375,172]);
// Polycyclic

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

Export

Character table of Q1610D4 in TeX

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