direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C2×C8⋊C4, C4.8C42, C42.5C4, C22.9C42, C42.58C22, C22.8M4(2), C8⋊9(C2×C4), (C2×C8)⋊8C4, C4○(C8⋊C4), C2.5(C2×C42), (C22×C4).9C4, (C2×C42).5C2, C23.38(C2×C4), (C22×C8).14C2, (C2×C8).97C22, C4.32(C22×C4), C2.1(C2×M4(2)), (C2×C4).141C23, C22.17(C22×C4), (C22×C4).135C22, (C2×C4)○(C8⋊C4), (C2×C4).83(C2×C4), SmallGroup(64,84)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2×C8⋊C4
G = < a,b,c | a2=b8=c4=1, ab=ba, ac=ca, cbc-1=b5 >
Subgroups: 81 in 73 conjugacy classes, 65 normal (9 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, C23, C42, C2×C8, C22×C4, C22×C4, C8⋊C4, C2×C42, C22×C8, C2×C8⋊C4
Quotients: C1, C2, C4, C22, C2×C4, C23, C42, M4(2), C22×C4, C8⋊C4, C2×C42, C2×M4(2), C2×C8⋊C4
►Regular action on 64 points
(1 52)(2 53)(3 54)(4 55)(5 56)(6 49)(7 50)(8 51)(9 33)(10 34)(11 35)(12 36)(13 37)(14 38)(15 39)(16 40)(17 62)(18 63)(19 64)(20 57)(21 58)(22 59)(23 60)(24 61)(25 46)(26 47)(27 48)(28 41)(29 42)(30 43)(31 44)(32 45)
(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 26 59 9)(2 31 60 14)(3 28 61 11)(4 25 62 16)(5 30 63 13)(6 27 64 10)(7 32 57 15)(8 29 58 12)(17 40 55 46)(18 37 56 43)(19 34 49 48)(20 39 50 45)(21 36 51 42)(22 33 52 47)(23 38 53 44)(24 35 54 41)
G:=sub<Sym(64)| (1,52)(2,53)(3,54)(4,55)(5,56)(6,49)(7,50)(8,51)(9,33)(10,34)(11,35)(12,36)(13,37)(14,38)(15,39)(16,40)(17,62)(18,63)(19,64)(20,57)(21,58)(22,59)(23,60)(24,61)(25,46)(26,47)(27,48)(28,41)(29,42)(30,43)(31,44)(32,45), (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), (1,26,59,9)(2,31,60,14)(3,28,61,11)(4,25,62,16)(5,30,63,13)(6,27,64,10)(7,32,57,15)(8,29,58,12)(17,40,55,46)(18,37,56,43)(19,34,49,48)(20,39,50,45)(21,36,51,42)(22,33,52,47)(23,38,53,44)(24,35,54,41)>;
G:=Group( (1,52)(2,53)(3,54)(4,55)(5,56)(6,49)(7,50)(8,51)(9,33)(10,34)(11,35)(12,36)(13,37)(14,38)(15,39)(16,40)(17,62)(18,63)(19,64)(20,57)(21,58)(22,59)(23,60)(24,61)(25,46)(26,47)(27,48)(28,41)(29,42)(30,43)(31,44)(32,45), (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), (1,26,59,9)(2,31,60,14)(3,28,61,11)(4,25,62,16)(5,30,63,13)(6,27,64,10)(7,32,57,15)(8,29,58,12)(17,40,55,46)(18,37,56,43)(19,34,49,48)(20,39,50,45)(21,36,51,42)(22,33,52,47)(23,38,53,44)(24,35,54,41) );
G=PermutationGroup([[(1,52),(2,53),(3,54),(4,55),(5,56),(6,49),(7,50),(8,51),(9,33),(10,34),(11,35),(12,36),(13,37),(14,38),(15,39),(16,40),(17,62),(18,63),(19,64),(20,57),(21,58),(22,59),(23,60),(24,61),(25,46),(26,47),(27,48),(28,41),(29,42),(30,43),(31,44),(32,45)], [(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)], [(1,26,59,9),(2,31,60,14),(3,28,61,11),(4,25,62,16),(5,30,63,13),(6,27,64,10),(7,32,57,15),(8,29,58,12),(17,40,55,46),(18,37,56,43),(19,34,49,48),(20,39,50,45),(21,36,51,42),(22,33,52,47),(23,38,53,44),(24,35,54,41)]])
C2×C8⋊C4 is a maximal subgroup of
C42.20D4 C42.2Q8 C42.7Q8 C42.370D4 C42.2C8 M5(2)⋊C4 C42.66D4 C42.67D4 C42.68D4 C42.69D4 C2.C43 C43.C2 (C4×C8)⋊12C4 C42.378D4 C42.379D4 C23.36C42 C23.17C42 D4⋊C42 Q8⋊C42 D4.3C42 C43.7C2 C42.45Q8 C4⋊C8⋊13C4 C4⋊C8⋊14C4 C8.5C42 C8⋊C42 C8.6C42 C42.24Q8 C42.104D4 C42.26Q8 C42.106D4 (C2×C8).Q8 C23.9M4(2) C2.(C8⋊D4) C2.(C8⋊2D4) C42.107D4 C42.27Q8 C4.(C4×Q8) C8⋊(C4⋊C4) C42.28Q8 C42.109D4 C42.110D4 C42.111D4 C42.112D4 (C2×Q16)⋊10C4 (C2×D8)⋊10C4 C8⋊(C22⋊C4) C42.116D4 C42.120D4 C42.124D4 C42.125D4 C2×C4×M4(2) D4.5C42 C42.261C23 C42.266C23 C42.383D4 C42.287C23 C42.293C23 D4⋊6M4(2) C42.239D4 C42.247D4 C42.252D4 C42.257D4 C42.258D4 C42.5F5
C2×C8⋊C4 is a maximal quotient of
C8⋊9M4(2) C23.27C42 C42⋊4C8 C42.378D4 C23.36C42 C43.7C2 C4⋊C8⋊13C4 C8.23C42 C42.5F5
40 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4H | 4I | ··· | 4P | 8A | ··· | 8P |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
40 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 |
| type | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | M4(2) |
| kernel | C2×C8⋊C4 | C8⋊C4 | C2×C42 | C22×C8 | C42 | C2×C8 | C22×C4 | C22 |
| # reps | 1 | 4 | 1 | 2 | 4 | 16 | 4 | 8 |
Matrix representation of C2×C8⋊C4 ►in GL4(𝔽17) generated by
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 16 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 10 | 10 |
| 0 | 0 | 10 | 7 |
| 1 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 16 | 0 |
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[16,0,0,0,0,4,0,0,0,0,10,10,0,0,10,7],[1,0,0,0,0,13,0,0,0,0,0,16,0,0,1,0] >;
C2×C8⋊C4 in GAP, Magma, Sage, TeX
C_2\times C_8\rtimes C_4
% in TeX
G:=Group("C2xC8:C4"); // GroupNames label
G:=SmallGroup(64,84);
// by ID
G=gap.SmallGroup(64,84);
# by ID
G:=PCGroup([6,-2,2,2,-2,2,-2,48,409,103,117]);
// Polycyclic
G:=Group<a,b,c|a^2=b^8=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^5>;
// generators/relations