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G = C16⋊2D4  order 128 = 27

2nd semidirect product of C16 and D4 acting via D4/C2=C22

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

Aliases: C16⋊2D4, C23.18D8, (C2×C4).38D8, (C2×SD32)⋊1C2, C16⋊3C4⋊12C2, C8.104(C2×D4), (C2×C8).216D4, C2.D16⋊18C2, C4.22(C4○D8), C8.49(C4○D4), C8⋊7D4.10C2, (C2×M5(2))⋊2C2, (C2×D8).8C22, C2.Q32⋊18C2, C8.18D4⋊33C2, C4.95(C4⋊D4), C2.23(C8⋊7D4), (C2×C8).525C23, (C2×C16).25C22, (C22×C4).351D4, C22.111(C2×D8), (C2×Q16).9C22, C2.13(C16⋊C22), C2.D8.10C22, C2.13(Q32⋊C2), (C22×C8).301C22, (C2×C4).793(C2×D4), SmallGroup(128,952)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C2×C8 — C16⋊2D4
Chief series
C1 — C2 — C4 — C2×C4 — C2×C8 — C22×C8 — C2×M5(2) — C16⋊2D4
Lower central
C1 — C2 — C4 — C2×C8 — C16⋊2D4
Upper central
C1 — C22 — C22×C4 — C22×C8 — C16⋊2D4
Jennings
C1 — C2 — C2 — C2 — C2 — C4 — C4 — C2×C8 — C16⋊2D4

Generators and relations for C16⋊2D4
 G = < a,b,c | a16=b4=c2=1, bab-1=a-1, cac=a7, cbc=b-1 >

Subgroups: 216 in 80 conjugacy classes, 32 normal (30 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C16, C16, C22⋊C4, C4⋊C4, C2×C8, C2×C8, D8, Q16, C22×C4, C2×D4, C2×Q8, D4⋊C4, Q8⋊C4, C2.D8, C2×C16, M5(2), SD32, C4⋊D4, C22⋊Q8, C22×C8, C2×D8, C2×Q16, C2.D16, C2.Q32, C16⋊3C4, C8⋊7D4, C8.18D4, C2×M5(2), C2×SD32, C16⋊2D4
Quotients: C1, C2, C22, D4, C23, D8, C2×D4, C4○D4, C4⋊D4, C2×D8, C4○D8, C8⋊7D4, C16⋊C22, Q32⋊C2, C16⋊2D4

Character table of C16⋊2D4

 class 12A2B2C2D2E4A4B4C4D4E4F8A8B8C8D8E8F16A16B16C16D16E16F16G16H
 size 111141622416161622224444444444
ρ111111111111111111111111111    trivial
ρ211111-1111-1-1-111111111111111    linear of order 2
ρ3111111111-1-11111111-1-1-1-1-1-1-1-1    linear of order 2
ρ411111-111111-1111111-1-1-1-1-1-1-1-1    linear of order 2
ρ51111-1111-11-1-11111-1-11-11-11-1-11    linear of order 2
ρ61111-1-111-1-1111111-1-11-11-11-1-11    linear of order 2
ρ71111-1111-1-11-11111-1-1-11-11-111-1    linear of order 2
ρ81111-1-111-11-111111-1-1-11-11-111-1    linear of order 2
ρ92-22-2002-20000-222-20020-20-2002    orthogonal lifted from D4
ρ10222220222000-2-2-2-2-2-200000000    orthogonal lifted from D4
ρ112-22-2002-20000-222-200-2020200-2    orthogonal lifted from D4
ρ122222-2022-2000-2-2-2-22200000000    orthogonal lifted from D4
ρ13222220-2-2-2000000000-√2-√2-√2-√2√2√2√2√2    orthogonal lifted from D8
ρ14222220-2-2-2000000000√2√2√2√2-√2-√2-√2-√2    orthogonal lifted from D8
ρ152222-20-2-22000000000√2-√2√2-√2-√2√2√2-√2    orthogonal lifted from D8
ρ162222-20-2-22000000000-√2√2-√2√2√2-√2-√2√2    orthogonal lifted from D8
ρ172-22-2002-200002-2-22000-2i02i02i-2i0    complex lifted from C4○D4
ρ182-22-2002-200002-2-220002i0-2i0-2i2i0    complex lifted from C4○D4
ρ192-22-200-22000000002i-2i-√2-√-2√2√-2-√2-√-2√-2√2    complex lifted from C4○D8
ρ202-22-200-2200000000-2i2i-√2√-2√2-√-2-√2√-2-√-2√2    complex lifted from C4○D8
ρ212-22-200-2200000000-2i2i√2-√-2-√2√-2√2-√-2√-2-√2    complex lifted from C4○D8
ρ222-22-200-22000000002i-2i√2√-2-√2-√-2√2√-2-√-2-√2    complex lifted from C4○D8
ρ234-4-4400000000-2√22√2-2√22√20000000000    orthogonal lifted from C16⋊C22
ρ244-4-44000000002√2-2√22√2-2√20000000000    orthogonal lifted from C16⋊C22
ρ2544-4-400000000-2√2-2√22√22√20000000000    symplectic lifted from Q32⋊C2, Schur index 2
ρ2644-4-4000000002√22√2-2√2-2√20000000000    symplectic lifted from Q32⋊C2, Schur index 2

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

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

Matrix representation of C16⋊2D4 ►in GL6(𝔽17)

1390000
040000
0011000
007100
008463
0012179
,
120000
16160000
0000160
0063115
001000
0015714
,
100000
16160000
001000
0001600
0000160
00014161

G:=sub<GL(6,GF(17))| [13,0,0,0,0,0,9,4,0,0,0,0,0,0,1,7,8,12,0,0,10,1,4,1,0,0,0,0,6,7,0,0,0,0,3,9],[1,16,0,0,0,0,2,16,0,0,0,0,0,0,0,6,1,1,0,0,0,3,0,5,0,0,16,1,0,7,0,0,0,15,0,14],[1,16,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,14,0,0,0,0,16,16,0,0,0,0,0,1] >;
 

C16⋊2D4 in GAP, Magma, Sage, TeX 

C_{16}\rtimes_2D_4
 
% in TeX
 
G:=Group("C16:2D4");
 
// GroupNames label
 
G:=SmallGroup(128,952);
 
// by ID
 
G=gap.SmallGroup(128,952);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,141,736,422,723,1123,360,4037,124]);
 
// Polycyclic
 
G:=Group<a,b,c|a^16=b^4=c^2=1,b*a*b^-1=a^-1,c*a*c=a^7,c*b*c=b^-1>;
 
// generators/relations
 

Export

Character table of C16⋊2D4 in TeX

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