p-group, metabelian, nilpotent (class 3), monomial
Aliases: (C2×C8)⋊20D4, (C2×Q8)⋊11D4, (C2×C4)⋊4SD16, (C2×D4).93D4, C4.10C22≀C2, C2.15(C8⋊3D4), C2.13(C8⋊5D4), C4.46(C4⋊D4), C23.902(C2×D4), (C22×C4).309D4, C2.12(C4⋊SD16), (C22×SD16)⋊13C2, C22.92(C2×SD16), C2.19(C22⋊SD16), C22.204C22≀C2, C2.19(C23⋊2D4), C22.75(C4⋊1D4), (C22×C8).319C22, (C2×C42).350C22, (C22×D4).66C22, (C22×Q8).55C22, C22.223(C4⋊D4), C22.131(C8⋊C22), (C22×C4).1436C23, C22.7C42⋊31C2, C23.67C23⋊7C2, (C2×C4).746(C2×D4), (C2×C4⋊1D4).9C2, (C2×D4⋊C4)⋊34C2, (C2×C4).875(C4○D4), (C2×C4⋊C4).109C22, SmallGroup(128,746)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for (C2×C8)⋊20D4
G = < a,b,c,d | a2=b8=c4=d2=1, cbc-1=ab=ba, ac=ca, ad=da, dbd=b3, dcd=c-1 >
Subgroups: 616 in 237 conjugacy classes, 58 normal (18 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C4⋊C4, C2×C8, C2×C8, SD16, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C24, C2.C42, D4⋊C4, C2×C42, C2×C4⋊C4, C4⋊1D4, C22×C8, C2×SD16, C22×D4, C22×D4, C22×Q8, C22.7C42, C23.67C23, C2×D4⋊C4, C2×C4⋊1D4, C22×SD16, (C2×C8)⋊20D4
Quotients: C1, C2, C22, D4, C23, SD16, C2×D4, C4○D4, C22≀C2, C4⋊D4, C4⋊1D4, C2×SD16, C8⋊C22, C23⋊2D4, C22⋊SD16, C4⋊SD16, C8⋊5D4, C8⋊3D4, (C2×C8)⋊20D4
►On 64 points
(1 59)(2 60)(3 61)(4 62)(5 63)(6 64)(7 57)(8 58)(9 22)(10 23)(11 24)(12 17)(13 18)(14 19)(15 20)(16 21)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(31 33)(32 34)(41 55)(42 56)(43 49)(44 50)(45 51)(46 52)(47 53)(48 54)
(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 33 18 45)(2 32 19 52)(3 35 20 47)(4 26 21 54)(5 37 22 41)(6 28 23 56)(7 39 24 43)(8 30 17 50)(9 55 63 27)(10 42 64 38)(11 49 57 29)(12 44 58 40)(13 51 59 31)(14 46 60 34)(15 53 61 25)(16 48 62 36)
(1 45)(2 48)(3 43)(4 46)(5 41)(6 44)(7 47)(8 42)(9 27)(10 30)(11 25)(12 28)(13 31)(14 26)(15 29)(16 32)(17 38)(18 33)(19 36)(20 39)(21 34)(22 37)(23 40)(24 35)(49 61)(50 64)(51 59)(52 62)(53 57)(54 60)(55 63)(56 58)
G:=sub<Sym(64)| (1,59)(2,60)(3,61)(4,62)(5,63)(6,64)(7,57)(8,58)(9,22)(10,23)(11,24)(12,17)(13,18)(14,19)(15,20)(16,21)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(31,33)(32,34)(41,55)(42,56)(43,49)(44,50)(45,51)(46,52)(47,53)(48,54), (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), (1,33,18,45)(2,32,19,52)(3,35,20,47)(4,26,21,54)(5,37,22,41)(6,28,23,56)(7,39,24,43)(8,30,17,50)(9,55,63,27)(10,42,64,38)(11,49,57,29)(12,44,58,40)(13,51,59,31)(14,46,60,34)(15,53,61,25)(16,48,62,36), (1,45)(2,48)(3,43)(4,46)(5,41)(6,44)(7,47)(8,42)(9,27)(10,30)(11,25)(12,28)(13,31)(14,26)(15,29)(16,32)(17,38)(18,33)(19,36)(20,39)(21,34)(22,37)(23,40)(24,35)(49,61)(50,64)(51,59)(52,62)(53,57)(54,60)(55,63)(56,58)>;
G:=Group( (1,59)(2,60)(3,61)(4,62)(5,63)(6,64)(7,57)(8,58)(9,22)(10,23)(11,24)(12,17)(13,18)(14,19)(15,20)(16,21)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(31,33)(32,34)(41,55)(42,56)(43,49)(44,50)(45,51)(46,52)(47,53)(48,54), (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), (1,33,18,45)(2,32,19,52)(3,35,20,47)(4,26,21,54)(5,37,22,41)(6,28,23,56)(7,39,24,43)(8,30,17,50)(9,55,63,27)(10,42,64,38)(11,49,57,29)(12,44,58,40)(13,51,59,31)(14,46,60,34)(15,53,61,25)(16,48,62,36), (1,45)(2,48)(3,43)(4,46)(5,41)(6,44)(7,47)(8,42)(9,27)(10,30)(11,25)(12,28)(13,31)(14,26)(15,29)(16,32)(17,38)(18,33)(19,36)(20,39)(21,34)(22,37)(23,40)(24,35)(49,61)(50,64)(51,59)(52,62)(53,57)(54,60)(55,63)(56,58) );
G=PermutationGroup([[(1,59),(2,60),(3,61),(4,62),(5,63),(6,64),(7,57),(8,58),(9,22),(10,23),(11,24),(12,17),(13,18),(14,19),(15,20),(16,21),(25,35),(26,36),(27,37),(28,38),(29,39),(30,40),(31,33),(32,34),(41,55),(42,56),(43,49),(44,50),(45,51),(46,52),(47,53),(48,54)], [(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)], [(1,33,18,45),(2,32,19,52),(3,35,20,47),(4,26,21,54),(5,37,22,41),(6,28,23,56),(7,39,24,43),(8,30,17,50),(9,55,63,27),(10,42,64,38),(11,49,57,29),(12,44,58,40),(13,51,59,31),(14,46,60,34),(15,53,61,25),(16,48,62,36)], [(1,45),(2,48),(3,43),(4,46),(5,41),(6,44),(7,47),(8,42),(9,27),(10,30),(11,25),(12,28),(13,31),(14,26),(15,29),(16,32),(17,38),(18,33),(19,36),(20,39),(21,34),(22,37),(23,40),(24,35),(49,61),(50,64),(51,59),(52,62),(53,57),(54,60),(55,63),(56,58)]])
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 | D4 | SD16 | C4○D4 | C8⋊C22 |
| kernel | (C2×C8)⋊20D4 | C22.7C42 | C23.67C23 | C2×D4⋊C4 | C2×C4⋊1D4 | C22×SD16 | C2×C8 | C22×C4 | C2×D4 | C2×Q8 | C2×C4 | C2×C4 | C22 |
| # reps | 1 | 1 | 1 | 2 | 1 | 2 | 4 | 2 | 4 | 2 | 8 | 2 | 2 |
Matrix representation of (C2×C8)⋊20D4 ►in GL6(𝔽17)
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 5 | 14 | 0 | 0 | 0 | 0 |
| 3 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 7 | 0 | 0 |
| 0 | 0 | 5 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 12 |
| 0 | 0 | 0 | 0 | 5 | 12 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[5,3,0,0,0,0,14,12,0,0,0,0,0,0,0,5,0,0,0,0,7,7,0,0,0,0,0,0,12,5,0,0,0,0,12,12],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,1,0,0,0,0,0,16,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
(C2×C8)⋊20D4 in GAP, Magma, Sage, TeX
(C_2\times C_8)\rtimes_{20}D_4 % in TeX
G:=Group("(C2xC8):20D4"); // GroupNames label
G:=SmallGroup(128,746);
// by ID
G=gap.SmallGroup(128,746);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,448,141,422,387,2019,1018,521,248,2804,1411,718,172]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^8=c^4=d^2=1,c*b*c^-1=a*b=b*a,a*c=c*a,a*d=d*a,d*b*d=b^3,d*c*d=c^-1>;
// generators/relations