p-group, metabelian, nilpotent (class 4), monomial
Aliases: C16:2D4, C23.18D8, (C2xC4).38D8, (C2xSD32):1C2, C16:3C4:12C2, C8.104(C2xD4), (C2xC8).216D4, C2.D16:18C2, C4.22(C4oD8), C8.49(C4oD4), C8:7D4.10C2, (C2xM5(2)):2C2, (C2xD8).8C22, C2.Q32:18C2, C8.18D4:33C2, C4.95(C4:D4), C2.23(C8:7D4), (C2xC8).525C23, (C2xC16).25C22, (C22xC4).351D4, C22.111(C2xD8), (C2xQ16).9C22, C2.13(C16:C22), C2.D8.10C22, C2.13(Q32:C2), (C22xC8).301C22, (C2xC4).793(C2xD4), SmallGroup(128,952)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
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, C2xC4, C2xC4, D4, Q8, C23, C23, C16, C16, C22:C4, C4:C4, C2xC8, C2xC8, D8, Q16, C22xC4, C2xD4, C2xQ8, D4:C4, Q8:C4, C2.D8, C2xC16, M5(2), SD32, C4:D4, C22:Q8, C22xC8, C2xD8, C2xQ16, C2.D16, C2.Q32, C16:3C4, C8:7D4, C8.18D4, C2xM5(2), C2xSD32, C16:2D4
Quotients: C1, C2, C22, D4, C23, D8, C2xD4, C4oD4, C4:D4, C2xD8, C4oD8, C8:7D4, C16:C22, Q32:C2, C16:2D4
Character table of C16:2D4
class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 8A | 8B | 8C | 8D | 8E | 8F | 16A | 16B | 16C | 16D | 16E | 16F | 16G | 16H | |
size | 1 | 1 | 1 | 1 | 4 | 16 | 2 | 2 | 4 | 16 | 16 | 16 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | |
ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
ρ2 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
ρ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
ρ4 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
ρ5 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | linear of order 2 |
ρ6 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | linear of order 2 |
ρ7 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | linear of order 2 |
ρ8 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | linear of order 2 |
ρ9 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | -2 | 0 | 0 | 2 | 0 | -2 | 0 | -2 | 0 | 0 | 2 | orthogonal lifted from D4 |
ρ10 | 2 | 2 | 2 | 2 | 2 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ11 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | -2 | 0 | 0 | -2 | 0 | 2 | 0 | 2 | 0 | 0 | -2 | orthogonal lifted from D4 |
ρ12 | 2 | 2 | 2 | 2 | -2 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ13 | 2 | 2 | 2 | 2 | 2 | 0 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√2 | -√2 | -√2 | -√2 | √2 | √2 | √2 | √2 | orthogonal lifted from D8 |
ρ14 | 2 | 2 | 2 | 2 | 2 | 0 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √2 | √2 | √2 | √2 | -√2 | -√2 | -√2 | -√2 | orthogonal lifted from D8 |
ρ15 | 2 | 2 | 2 | 2 | -2 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √2 | -√2 | √2 | -√2 | -√2 | √2 | √2 | -√2 | orthogonal lifted from D8 |
ρ16 | 2 | 2 | 2 | 2 | -2 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√2 | √2 | -√2 | √2 | √2 | -√2 | -√2 | √2 | orthogonal lifted from D8 |
ρ17 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | -2i | 0 | 2i | 0 | 2i | -2i | 0 | complex lifted from C4oD4 |
ρ18 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 2i | 0 | -2i | 0 | -2i | 2i | 0 | complex lifted from C4oD4 |
ρ19 | 2 | -2 | 2 | -2 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | -√2 | -√-2 | √2 | √-2 | -√2 | -√-2 | √-2 | √2 | complex lifted from C4oD8 |
ρ20 | 2 | -2 | 2 | -2 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | -√2 | √-2 | √2 | -√-2 | -√2 | √-2 | -√-2 | √2 | complex lifted from C4oD8 |
ρ21 | 2 | -2 | 2 | -2 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | √2 | -√-2 | -√2 | √-2 | √2 | -√-2 | √-2 | -√2 | complex lifted from C4oD8 |
ρ22 | 2 | -2 | 2 | -2 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | √2 | √-2 | -√2 | -√-2 | √2 | √-2 | -√-2 | -√2 | complex lifted from C4oD8 |
ρ23 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2√2 | 2√2 | -2√2 | 2√2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C16:C22 |
ρ24 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2√2 | -2√2 | 2√2 | -2√2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C16:C22 |
ρ25 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2√2 | -2√2 | 2√2 | 2√2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q32:C2, Schur index 2 |
ρ26 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2√2 | 2√2 | -2√2 | -2√2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q32:C2, Schur index 2 |
(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)
13 | 9 | 0 | 0 | 0 | 0 |
0 | 4 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 10 | 0 | 0 |
0 | 0 | 7 | 1 | 0 | 0 |
0 | 0 | 8 | 4 | 6 | 3 |
0 | 0 | 12 | 1 | 7 | 9 |
1 | 2 | 0 | 0 | 0 | 0 |
16 | 16 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 16 | 0 |
0 | 0 | 6 | 3 | 1 | 15 |
0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 1 | 5 | 7 | 14 |
1 | 0 | 0 | 0 | 0 | 0 |
16 | 16 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 0 | 16 | 0 | 0 |
0 | 0 | 0 | 0 | 16 | 0 |
0 | 0 | 0 | 14 | 16 | 1 |
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
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