p-group, metabelian, nilpotent (class 3), monomial
Aliases: (C2×C4)⋊2D8, (C2×C8)⋊6D4, (C2×D4)⋊10D4, C4.7C22≀C2, (C22×D8)⋊2C2, C22.81(C2×D8), C2.13(C8⋊3D4), C2.12(C4⋊D8), C4.43(C4⋊D4), C2.10(C8⋊4D4), (C22×C4).307D4, C23.899(C2×D4), C2.19(C22⋊D8), C22.201C22≀C2, C22.72(C4⋊1D4), C2.16(C23⋊2D4), (C22×C8).107C22, C24.3C22⋊7C2, (C2×C42).347C22, (C22×D4).63C22, C22.220(C4⋊D4), C22.129(C8⋊C22), C22.7C42⋊18C2, (C22×C4).1433C23, (C2×C4⋊1D4)⋊2C2, (C2×C4).744(C2×D4), (C2×D4⋊C4)⋊19C2, (C2×C4).872(C4○D4), (C2×C4⋊C4).106C22, SmallGroup(128,743)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for (C2×C4)⋊2D8
G = < a,b,c,d | a2=b4=c8=d2=1, dbd=ab=ba, ac=ca, ad=da, cbc-1=ab-1, dcd=c-1 >
Subgroups: 712 in 255 conjugacy classes, 58 normal (18 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, C23, C23, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, D8, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, D4⋊C4, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C4⋊1D4, C22×C8, C2×D8, C22×D4, C22×D4, C22×D4, C22.7C42, C24.3C22, C2×D4⋊C4, C2×C4⋊1D4, C22×D8, (C2×C4)⋊2D8
Quotients: C1, C2, C22, D4, C23, D8, C2×D4, C4○D4, C22≀C2, C4⋊D4, C4⋊1D4, C2×D8, C8⋊C22, C23⋊2D4, C22⋊D8, C4⋊D8, C8⋊4D4, C8⋊3D4, (C2×C4)⋊2D8
►On 64 points
(1 13)(2 14)(3 15)(4 16)(5 9)(6 10)(7 11)(8 12)(17 46)(18 47)(19 48)(20 41)(21 42)(22 43)(23 44)(24 45)(25 53)(26 54)(27 55)(28 56)(29 49)(30 50)(31 51)(32 52)(33 57)(34 58)(35 59)(36 60)(37 61)(38 62)(39 63)(40 64)
(1 48 51 57)(2 34 52 20)(3 42 53 59)(4 36 54 22)(5 44 55 61)(6 38 56 24)(7 46 49 63)(8 40 50 18)(9 23 27 37)(10 62 28 45)(11 17 29 39)(12 64 30 47)(13 19 31 33)(14 58 32 41)(15 21 25 35)(16 60 26 43)
(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 8)(2 7)(3 6)(4 5)(9 16)(10 15)(11 14)(12 13)(17 34)(18 33)(19 40)(20 39)(21 38)(22 37)(23 36)(24 35)(25 28)(26 27)(29 32)(30 31)(41 63)(42 62)(43 61)(44 60)(45 59)(46 58)(47 57)(48 64)(49 52)(50 51)(53 56)(54 55)
G:=sub<Sym(64)| (1,13)(2,14)(3,15)(4,16)(5,9)(6,10)(7,11)(8,12)(17,46)(18,47)(19,48)(20,41)(21,42)(22,43)(23,44)(24,45)(25,53)(26,54)(27,55)(28,56)(29,49)(30,50)(31,51)(32,52)(33,57)(34,58)(35,59)(36,60)(37,61)(38,62)(39,63)(40,64), (1,48,51,57)(2,34,52,20)(3,42,53,59)(4,36,54,22)(5,44,55,61)(6,38,56,24)(7,46,49,63)(8,40,50,18)(9,23,27,37)(10,62,28,45)(11,17,29,39)(12,64,30,47)(13,19,31,33)(14,58,32,41)(15,21,25,35)(16,60,26,43), (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), (1,8)(2,7)(3,6)(4,5)(9,16)(10,15)(11,14)(12,13)(17,34)(18,33)(19,40)(20,39)(21,38)(22,37)(23,36)(24,35)(25,28)(26,27)(29,32)(30,31)(41,63)(42,62)(43,61)(44,60)(45,59)(46,58)(47,57)(48,64)(49,52)(50,51)(53,56)(54,55)>;
G:=Group( (1,13)(2,14)(3,15)(4,16)(5,9)(6,10)(7,11)(8,12)(17,46)(18,47)(19,48)(20,41)(21,42)(22,43)(23,44)(24,45)(25,53)(26,54)(27,55)(28,56)(29,49)(30,50)(31,51)(32,52)(33,57)(34,58)(35,59)(36,60)(37,61)(38,62)(39,63)(40,64), (1,48,51,57)(2,34,52,20)(3,42,53,59)(4,36,54,22)(5,44,55,61)(6,38,56,24)(7,46,49,63)(8,40,50,18)(9,23,27,37)(10,62,28,45)(11,17,29,39)(12,64,30,47)(13,19,31,33)(14,58,32,41)(15,21,25,35)(16,60,26,43), (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), (1,8)(2,7)(3,6)(4,5)(9,16)(10,15)(11,14)(12,13)(17,34)(18,33)(19,40)(20,39)(21,38)(22,37)(23,36)(24,35)(25,28)(26,27)(29,32)(30,31)(41,63)(42,62)(43,61)(44,60)(45,59)(46,58)(47,57)(48,64)(49,52)(50,51)(53,56)(54,55) );
G=PermutationGroup([[(1,13),(2,14),(3,15),(4,16),(5,9),(6,10),(7,11),(8,12),(17,46),(18,47),(19,48),(20,41),(21,42),(22,43),(23,44),(24,45),(25,53),(26,54),(27,55),(28,56),(29,49),(30,50),(31,51),(32,52),(33,57),(34,58),(35,59),(36,60),(37,61),(38,62),(39,63),(40,64)], [(1,48,51,57),(2,34,52,20),(3,42,53,59),(4,36,54,22),(5,44,55,61),(6,38,56,24),(7,46,49,63),(8,40,50,18),(9,23,27,37),(10,62,28,45),(11,17,29,39),(12,64,30,47),(13,19,31,33),(14,58,32,41),(15,21,25,35),(16,60,26,43)], [(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)], [(1,8),(2,7),(3,6),(4,5),(9,16),(10,15),(11,14),(12,13),(17,34),(18,33),(19,40),(20,39),(21,38),(22,37),(23,36),(24,35),(25,28),(26,27),(29,32),(30,31),(41,63),(42,62),(43,61),(44,60),(45,59),(46,58),(47,57),(48,64),(49,52),(50,51),(53,56),(54,55)]])
32 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | ··· | 2M | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 8A | ··· | 8H |
| order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 8 | ··· | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D8 | C4○D4 | C8⋊C22 |
| kernel | (C2×C4)⋊2D8 | C22.7C42 | C24.3C22 | C2×D4⋊C4 | C2×C4⋊1D4 | C22×D8 | C2×C8 | C22×C4 | C2×D4 | C2×C4 | C2×C4 | C22 |
| # reps | 1 | 1 | 1 | 2 | 1 | 2 | 4 | 2 | 6 | 8 | 2 | 2 |
Matrix representation of (C2×C4)⋊2D8 ►in GL6(𝔽17)
| 1 | 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 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 16 | 0 | 0 |
| 0 | 0 | 2 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 3 | 3 | 0 | 0 | 0 | 0 |
| 14 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 14 | 14 | 0 | 0 | 0 | 0 |
| 14 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 15 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,2,0,0,0,0,16,16,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[3,14,0,0,0,0,3,3,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16],[14,14,0,0,0,0,14,3,0,0,0,0,0,0,16,15,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
(C2×C4)⋊2D8 in GAP, Magma, Sage, TeX
(C_2\times C_4)\rtimes_2D_8
% in TeX
G:=Group("(C2xC4):2D8"); // GroupNames label
G:=SmallGroup(128,743);
// by ID
G=gap.SmallGroup(128,743);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,141,422,387,2019,1018,521,248,2804,1411,718,172]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^8=d^2=1,d*b*d=a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=a*b^-1,d*c*d=c^-1>;
// generators/relations