direct product, p-group, metabelian, nilpotent (class 3), monomial
Aliases: C2×D4⋊C8, C42.313D4, C42.595C23, C4○(D4⋊C8), D4⋊3(C2×C8), (C2×D4)⋊5C8, C4.78(C2×D8), C4⋊C8⋊54C22, (C4×C8)⋊66C22, (C4×D4).11C4, (C2×C4).164D8, C4.1(C22×C8), C22.34C4≀C2, C4.9(C22⋊C8), C4.89(C2×SD16), C4.1(C2×M4(2)), C42.254(C2×C4), (C2×C4).125SD16, (C22×D4).22C4, (C22×C4).652D4, (C2×C4).42M4(2), C4.53(D4⋊C4), (C4×D4).260C22, C22.40(C22⋊C8), C22.48(D4⋊C4), C23.217(C22⋊C4), (C2×C42).1032C22, (C2×C4×C8)⋊1C2, (C2×C4⋊C8)⋊1C2, (C2×C4×D4).5C2, C2.1(C2×C4≀C2), (C2×C4)○(D4⋊C8), (C2×C4⋊C4).36C4, (C2×C4).50(C2×C8), C4⋊C4.173(C2×C4), C2.1(C2×D4⋊C4), C2.10(C2×C22⋊C8), (C2×D4).188(C2×C4), (C2×C4).1135(C2×D4), (C22×C4).393(C2×C4), (C2×C4).300(C22×C4), C22.94(C2×C22⋊C4), (C2×C4).235(C22⋊C4), SmallGroup(128,206)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2×D4⋊C8
G = < a,b,c,d | a2=b4=c2=d8=1, ab=ba, ac=ca, ad=da, cbc=dbd-1=b-1, dcd-1=bc >
Subgroups: 348 in 172 conjugacy classes, 76 normal (30 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, D4, C23, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×D4, C2×D4, C24, C4×C8, C4×C8, C4⋊C8, C4⋊C8, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C4×D4, C4×D4, C22×C8, C23×C4, C22×D4, D4⋊C8, C2×C4×C8, C2×C4⋊C8, C2×C4×D4, C2×D4⋊C8
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, C23, C22⋊C4, C2×C8, M4(2), D8, SD16, C22×C4, C2×D4, C22⋊C8, D4⋊C4, C4≀C2, C2×C22⋊C4, C22×C8, C2×M4(2), C2×D8, C2×SD16, D4⋊C8, C2×C22⋊C8, C2×D4⋊C4, C2×C4≀C2, C2×D4⋊C8
►On 64 points
(1 43)(2 44)(3 45)(4 46)(5 47)(6 48)(7 41)(8 42)(9 23)(10 24)(11 17)(12 18)(13 19)(14 20)(15 21)(16 22)(25 61)(26 62)(27 63)(28 64)(29 57)(30 58)(31 59)(32 60)(33 54)(34 55)(35 56)(36 49)(37 50)(38 51)(39 52)(40 53)
(1 53 63 17)(2 18 64 54)(3 55 57 19)(4 20 58 56)(5 49 59 21)(6 22 60 50)(7 51 61 23)(8 24 62 52)(9 41 38 25)(10 26 39 42)(11 43 40 27)(12 28 33 44)(13 45 34 29)(14 30 35 46)(15 47 36 31)(16 32 37 48)
(1 21)(2 60)(3 23)(4 62)(5 17)(6 64)(7 19)(8 58)(9 45)(10 14)(11 47)(12 16)(13 41)(15 43)(18 22)(20 24)(25 34)(26 46)(27 36)(28 48)(29 38)(30 42)(31 40)(32 44)(33 37)(35 39)(49 63)(50 54)(51 57)(52 56)(53 59)(55 61)
(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)
G:=sub<Sym(64)| (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,41)(8,42)(9,23)(10,24)(11,17)(12,18)(13,19)(14,20)(15,21)(16,22)(25,61)(26,62)(27,63)(28,64)(29,57)(30,58)(31,59)(32,60)(33,54)(34,55)(35,56)(36,49)(37,50)(38,51)(39,52)(40,53), (1,53,63,17)(2,18,64,54)(3,55,57,19)(4,20,58,56)(5,49,59,21)(6,22,60,50)(7,51,61,23)(8,24,62,52)(9,41,38,25)(10,26,39,42)(11,43,40,27)(12,28,33,44)(13,45,34,29)(14,30,35,46)(15,47,36,31)(16,32,37,48), (1,21)(2,60)(3,23)(4,62)(5,17)(6,64)(7,19)(8,58)(9,45)(10,14)(11,47)(12,16)(13,41)(15,43)(18,22)(20,24)(25,34)(26,46)(27,36)(28,48)(29,38)(30,42)(31,40)(32,44)(33,37)(35,39)(49,63)(50,54)(51,57)(52,56)(53,59)(55,61), (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)>;
G:=Group( (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,41)(8,42)(9,23)(10,24)(11,17)(12,18)(13,19)(14,20)(15,21)(16,22)(25,61)(26,62)(27,63)(28,64)(29,57)(30,58)(31,59)(32,60)(33,54)(34,55)(35,56)(36,49)(37,50)(38,51)(39,52)(40,53), (1,53,63,17)(2,18,64,54)(3,55,57,19)(4,20,58,56)(5,49,59,21)(6,22,60,50)(7,51,61,23)(8,24,62,52)(9,41,38,25)(10,26,39,42)(11,43,40,27)(12,28,33,44)(13,45,34,29)(14,30,35,46)(15,47,36,31)(16,32,37,48), (1,21)(2,60)(3,23)(4,62)(5,17)(6,64)(7,19)(8,58)(9,45)(10,14)(11,47)(12,16)(13,41)(15,43)(18,22)(20,24)(25,34)(26,46)(27,36)(28,48)(29,38)(30,42)(31,40)(32,44)(33,37)(35,39)(49,63)(50,54)(51,57)(52,56)(53,59)(55,61), (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) );
G=PermutationGroup([[(1,43),(2,44),(3,45),(4,46),(5,47),(6,48),(7,41),(8,42),(9,23),(10,24),(11,17),(12,18),(13,19),(14,20),(15,21),(16,22),(25,61),(26,62),(27,63),(28,64),(29,57),(30,58),(31,59),(32,60),(33,54),(34,55),(35,56),(36,49),(37,50),(38,51),(39,52),(40,53)], [(1,53,63,17),(2,18,64,54),(3,55,57,19),(4,20,58,56),(5,49,59,21),(6,22,60,50),(7,51,61,23),(8,24,62,52),(9,41,38,25),(10,26,39,42),(11,43,40,27),(12,28,33,44),(13,45,34,29),(14,30,35,46),(15,47,36,31),(16,32,37,48)], [(1,21),(2,60),(3,23),(4,62),(5,17),(6,64),(7,19),(8,58),(9,45),(10,14),(11,47),(12,16),(13,41),(15,43),(18,22),(20,24),(25,34),(26,46),(27,36),(28,48),(29,38),(30,42),(31,40),(32,44),(33,37),(35,39),(49,63),(50,54),(51,57),(52,56),(53,59),(55,61)], [(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)]])
56 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4H | 4I | ··· | 4P | 4Q | 4R | 4S | 4T | 8A | ··· | 8P | 8Q | ··· | 8X |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C8 | D4 | D4 | M4(2) | D8 | SD16 | C4≀C2 |
| kernel | C2×D4⋊C8 | D4⋊C8 | C2×C4×C8 | C2×C4⋊C8 | C2×C4×D4 | C2×C4⋊C4 | C4×D4 | C22×D4 | C2×D4 | C42 | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C22 |
| # reps | 1 | 4 | 1 | 1 | 1 | 2 | 4 | 2 | 16 | 2 | 2 | 4 | 4 | 4 | 8 |
Matrix representation of C2×D4⋊C8 ►in GL4(𝔽17) generated by
| 1 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 16 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 9 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 3 | 14 |
| 0 | 0 | 14 | 14 |
G:=sub<GL(4,GF(17))| [1,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,0,16,0,0,1,0],[1,0,0,0,0,16,0,0,0,0,0,1,0,0,1,0],[9,0,0,0,0,4,0,0,0,0,3,14,0,0,14,14] >;
C2×D4⋊C8 in GAP, Magma, Sage, TeX
C_2\times D_4\rtimes C_8
% in TeX
G:=Group("C2xD4:C8"); // GroupNames label
G:=SmallGroup(128,206);
// by ID
G=gap.SmallGroup(128,206);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,1123,570,136,172]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^2=d^8=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=d*b*d^-1=b^-1,d*c*d^-1=b*c>;
// generators/relations