p-group, metabelian, nilpotent (class 3), monomial
Aliases: (C2×C4)⋊3D8, (C2×C8).49D4, C4.76C22≀C2, (C2×D4).115D4, (C22×D8).5C2, C22.85(C2×D8), C2.14(C8⋊7D4), C2.14(C4⋊D8), C2.14(C8⋊2D4), C23.916(C2×D4), (C22×C4).150D4, C22.4Q16⋊38C2, C4.72(C4.4D4), (C22×C8).77C22, C2.18(D4.2D4), C22.112(C4○D8), (C2×C42).368C22, (C22×D4).86C22, C22.238(C4⋊D4), C22.140(C8⋊C22), C24.3C22⋊10C2, (C22×C4).1450C23, C4.76(C22.D4), C2.27(C23.10D4), (C2×C4⋊C8)⋊20C2, (C2×D4⋊C4)⋊15C2, (C2×C4).1045(C2×D4), (C2×C4).881(C4○D4), (C2×C4⋊C4).131C22, SmallGroup(128,786)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for (C2×C4)⋊3D8
G = < a,b,c,d | a2=b4=c8=d2=1, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd=ab-1, dcd=c-1 >
Subgroups: 480 in 180 conjugacy classes, 52 normal (22 characteristic)
C1, C2, C2, C2, C4, 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, C2×D4, C2×D4, C24, D4⋊C4, C4⋊C8, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C22×C8, C2×D8, C22×D4, C22.4Q16, C24.3C22, C2×D4⋊C4, C2×C4⋊C8, C22×D8, (C2×C4)⋊3D8
Quotients: C1, C2, C22, D4, C23, D8, C2×D4, C4○D4, C22≀C2, C4⋊D4, C22.D4, C4.4D4, C2×D8, C4○D8, C8⋊C22, C23.10D4, C4⋊D8, D4.2D4, C8⋊7D4, C8⋊2D4, (C2×C4)⋊3D8
►On 64 points
(1 50)(2 51)(3 52)(4 53)(5 54)(6 55)(7 56)(8 49)(9 21)(10 22)(11 23)(12 24)(13 17)(14 18)(15 19)(16 20)(25 40)(26 33)(27 34)(28 35)(29 36)(30 37)(31 38)(32 39)(41 59)(42 60)(43 61)(44 62)(45 63)(46 64)(47 57)(48 58)
(1 62 23 36)(2 37 24 63)(3 64 17 38)(4 39 18 57)(5 58 19 40)(6 33 20 59)(7 60 21 34)(8 35 22 61)(9 27 56 42)(10 43 49 28)(11 29 50 44)(12 45 51 30)(13 31 52 46)(14 47 53 32)(15 25 54 48)(16 41 55 26)
(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 12)(10 11)(13 16)(14 15)(17 20)(18 19)(21 24)(22 23)(25 39)(26 38)(27 37)(28 36)(29 35)(30 34)(31 33)(32 40)(41 64)(42 63)(43 62)(44 61)(45 60)(46 59)(47 58)(48 57)(49 50)(51 56)(52 55)(53 54)
G:=sub<Sym(64)| (1,50)(2,51)(3,52)(4,53)(5,54)(6,55)(7,56)(8,49)(9,21)(10,22)(11,23)(12,24)(13,17)(14,18)(15,19)(16,20)(25,40)(26,33)(27,34)(28,35)(29,36)(30,37)(31,38)(32,39)(41,59)(42,60)(43,61)(44,62)(45,63)(46,64)(47,57)(48,58), (1,62,23,36)(2,37,24,63)(3,64,17,38)(4,39,18,57)(5,58,19,40)(6,33,20,59)(7,60,21,34)(8,35,22,61)(9,27,56,42)(10,43,49,28)(11,29,50,44)(12,45,51,30)(13,31,52,46)(14,47,53,32)(15,25,54,48)(16,41,55,26), (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)(9,12)(10,11)(13,16)(14,15)(17,20)(18,19)(21,24)(22,23)(25,39)(26,38)(27,37)(28,36)(29,35)(30,34)(31,33)(32,40)(41,64)(42,63)(43,62)(44,61)(45,60)(46,59)(47,58)(48,57)(49,50)(51,56)(52,55)(53,54)>;
G:=Group( (1,50)(2,51)(3,52)(4,53)(5,54)(6,55)(7,56)(8,49)(9,21)(10,22)(11,23)(12,24)(13,17)(14,18)(15,19)(16,20)(25,40)(26,33)(27,34)(28,35)(29,36)(30,37)(31,38)(32,39)(41,59)(42,60)(43,61)(44,62)(45,63)(46,64)(47,57)(48,58), (1,62,23,36)(2,37,24,63)(3,64,17,38)(4,39,18,57)(5,58,19,40)(6,33,20,59)(7,60,21,34)(8,35,22,61)(9,27,56,42)(10,43,49,28)(11,29,50,44)(12,45,51,30)(13,31,52,46)(14,47,53,32)(15,25,54,48)(16,41,55,26), (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)(9,12)(10,11)(13,16)(14,15)(17,20)(18,19)(21,24)(22,23)(25,39)(26,38)(27,37)(28,36)(29,35)(30,34)(31,33)(32,40)(41,64)(42,63)(43,62)(44,61)(45,60)(46,59)(47,58)(48,57)(49,50)(51,56)(52,55)(53,54) );
G=PermutationGroup([[(1,50),(2,51),(3,52),(4,53),(5,54),(6,55),(7,56),(8,49),(9,21),(10,22),(11,23),(12,24),(13,17),(14,18),(15,19),(16,20),(25,40),(26,33),(27,34),(28,35),(29,36),(30,37),(31,38),(32,39),(41,59),(42,60),(43,61),(44,62),(45,63),(46,64),(47,57),(48,58)], [(1,62,23,36),(2,37,24,63),(3,64,17,38),(4,39,18,57),(5,58,19,40),(6,33,20,59),(7,60,21,34),(8,35,22,61),(9,27,56,42),(10,43,49,28),(11,29,50,44),(12,45,51,30),(13,31,52,46),(14,47,53,32),(15,25,54,48),(16,41,55,26)], [(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),(9,12),(10,11),(13,16),(14,15),(17,20),(18,19),(21,24),(22,23),(25,39),(26,38),(27,37),(28,36),(29,35),(30,34),(31,33),(32,40),(41,64),(42,63),(43,62),(44,61),(45,60),(46,59),(47,58),(48,57),(49,50),(51,56),(52,55),(53,54)]])
32 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 8A | ··· | 8H |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 8 | 8 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D8 | C4○D4 | C4○D8 | C8⋊C22 |
| kernel | (C2×C4)⋊3D8 | C22.4Q16 | C24.3C22 | C2×D4⋊C4 | C2×C4⋊C8 | C22×D8 | C2×C8 | C22×C4 | C2×D4 | C2×C4 | C2×C4 | C22 | C22 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 4 | 4 | 6 | 4 | 2 |
Matrix representation of (C2×C4)⋊3D8 ►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 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 7 | 13 |
| 14 | 3 | 0 | 0 | 0 | 0 |
| 14 | 14 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 9 |
| 0 | 0 | 0 | 0 | 2 | 10 |
| 3 | 14 | 0 | 0 | 0 | 0 |
| 14 | 14 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 9 |
| 0 | 0 | 0 | 0 | 6 | 10 |
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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,4,7,0,0,0,0,0,13],[14,14,0,0,0,0,3,14,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,7,2,0,0,0,0,9,10],[3,14,0,0,0,0,14,14,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,7,6,0,0,0,0,9,10] >;
(C2×C4)⋊3D8 in GAP, Magma, Sage, TeX
(C_2\times C_4)\rtimes_3D_8
% in TeX
G:=Group("(C2xC4):3D8"); // GroupNames label
G:=SmallGroup(128,786);
// by ID
G=gap.SmallGroup(128,786);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,141,422,387,58,2019,1018,248,2804,1411,172,4037,2028,124]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^8=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,d*b*d=a*b^-1,d*c*d=c^-1>;
// generators/relations