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G = C16⋊D4order 128 = 27

1st semidirect product of C16 and D4 acting via D4/C2=C22

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

Aliases: C161D4, C23.16D8, (C2×D16)⋊8C2, C164C44C2, (C2×C4).36D8, C87D433C2, (C2×C8).214D4, C8.102(C2×D4), C2.D1617C2, C8.47(C4○D4), C4.20(C4○D8), (C2×M5(2))⋊1C2, (C2×D8).7C22, C2.21(C87D4), C4.93(C4⋊D4), C2.D8.8C22, (C2×C16).23C22, (C2×C8).523C23, C22.109(C2×D8), (C22×C4).349D4, C2.12(C16⋊C22), (C22×C8).299C22, (C2×C4).791(C2×D4), SmallGroup(128,950)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C8 — C16⋊D4
C1C2C4C2×C4C2×C8C22×C8C2×M5(2) — C16⋊D4
C1C2C4C2×C8 — C16⋊D4
C1C22C22×C4C22×C8 — C16⋊D4
C1C2C2C2C2C4C4C2×C8 — C16⋊D4

Generators and relations for C16⋊D4
 G = < a,b,c | a16=b4=c2=1, bab-1=a7, cac=a-1, cbc=b-1 >

Subgroups: 264 in 85 conjugacy classes, 32 normal (20 characteristic)
C1, C2, C2 [×2], C2 [×3], C4 [×2], C4 [×3], C22, C22 [×9], C8 [×2], C8, C2×C4 [×2], C2×C4 [×4], D4 [×8], C23, C23 [×2], C16 [×2], C16, C22⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], C2×C8 [×2], D8 [×4], C22×C4, C2×D4 [×4], D4⋊C4 [×2], C2.D8 [×2], C2×C16 [×2], M5(2) [×2], D16 [×2], C4⋊D4 [×2], C22×C8, C2×D8 [×2], C2.D16 [×2], C164C4, C87D4 [×2], C2×M5(2), C2×D16, C16⋊D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D8 [×2], C2×D4 [×2], C4○D4, C4⋊D4, C2×D8, C4○D8, C87D4, C16⋊C22 [×2], C16⋊D4

Character table of C16⋊D4

 class 12A2B2C2D2E2F4A4B4C4D4E8A8B8C8D8E8F16A16B16C16D16E16F16G16H
 size 111141616224161622224444444444
ρ111111111111111111111111111    trivial
ρ21111-1-1-111-1111111-1-11-11-11-1-11    linear of order 2
ρ311111-1-1111-1-111111111111111    linear of order 2
ρ41111-11111-1-1-11111-1-11-11-11-1-11    linear of order 2
ρ51111-11-111-11-11111-1-1-11-11-111-1    linear of order 2
ρ611111-111111-1111111-1-1-1-1-1-1-1-1    linear of order 2
ρ71111-1-1111-1-111111-1-1-11-11-111-1    linear of order 2
ρ8111111-1111-11111111-1-1-1-1-1-1-1-1    linear of order 2
ρ92222-20022-200-2-2-2-22200000000    orthogonal lifted from D4
ρ10222220022200-2-2-2-2-2-200000000    orthogonal lifted from D4
ρ112-22-20002-2000-222-200-2020200-2    orthogonal lifted from D4
ρ122-22-20002-2000-222-20020-20-2002    orthogonal lifted from D4
ρ132222-200-2-2200000000-22-222-2-22    orthogonal lifted from D8
ρ142222200-2-2-2000000002222-2-2-2-2    orthogonal lifted from D8
ρ152222200-2-2-200000000-2-2-2-22222    orthogonal lifted from D8
ρ162222-200-2-22000000002-22-2-222-2    orthogonal lifted from D8
ρ172-22-20002-20002-2-22000-2i02i02i-2i0    complex lifted from C4○D4
ρ182-22-20002-20002-2-220002i0-2i0-2i2i0    complex lifted from C4○D4
ρ192-22-2000-220000000-2i2i-2-22--2-2-2--22    complex lifted from C4○D8
ρ202-22-2000-2200000002i-2i-2--22-2-2--2-22    complex lifted from C4○D8
ρ212-22-2000-2200000002i-2i2-2-2--22-2--2-2    complex lifted from C4○D8
ρ222-22-2000-220000000-2i2i2--2-2-22--2-2-2    complex lifted from C4○D8
ρ234-4-4400000000-2222-22220000000000    orthogonal lifted from C16⋊C22
ρ2444-4-400000000-22-2222220000000000    orthogonal lifted from C16⋊C22
ρ254-4-440000000022-2222-220000000000    orthogonal lifted from C16⋊C22
ρ2644-4-4000000002222-22-220000000000    orthogonal lifted from C16⋊C22

Smallest permutation representation of C16⋊D4
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 62 38 21)(2 53 39 28)(3 60 40 19)(4 51 41 26)(5 58 42 17)(6 49 43 24)(7 56 44 31)(8 63 45 22)(9 54 46 29)(10 61 47 20)(11 52 48 27)(12 59 33 18)(13 50 34 25)(14 57 35 32)(15 64 36 23)(16 55 37 30)
(1 16)(2 15)(3 14)(4 13)(5 12)(6 11)(7 10)(8 9)(17 59)(18 58)(19 57)(20 56)(21 55)(22 54)(23 53)(24 52)(25 51)(26 50)(27 49)(28 64)(29 63)(30 62)(31 61)(32 60)(33 42)(34 41)(35 40)(36 39)(37 38)(43 48)(44 47)(45 46)

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

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

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

Matrix representation of C16⋊D4 in GL6(𝔽17)

1600000
0160000
0094106
00011168
00671110
0011173
,
1620000
1610000
00671110
00621315
0094106
0067916
,
1600000
1610000
0094106
000619
00671110
005108

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,9,0,6,11,0,0,4,11,7,1,0,0,10,16,11,7,0,0,6,8,10,3],[16,16,0,0,0,0,2,1,0,0,0,0,0,0,6,6,9,6,0,0,7,2,4,7,0,0,11,13,10,9,0,0,10,15,6,16],[16,16,0,0,0,0,0,1,0,0,0,0,0,0,9,0,6,5,0,0,4,6,7,1,0,0,10,1,11,0,0,0,6,9,10,8] >;

C16⋊D4 in GAP, Magma, Sage, TeX

C_{16}\rtimes D_4
% in TeX

G:=Group("C16:D4");
// GroupNames label

G:=SmallGroup(128,950);
// by ID

G=gap.SmallGroup(128,950);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,141,288,422,723,1123,360,4037,124]);
// Polycyclic

G:=Group<a,b,c|a^16=b^4=c^2=1,b*a*b^-1=a^7,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations

Export

Character table of C16⋊D4 in TeX

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