p-group, metabelian, nilpotent (class 3), monomial
Aliases: (C2×C8).41D4, (C2×D4).94D4, (C2×Q8).84D4, C4.11C22≀C2, (C22×Q16)⋊2C2, C4.47(C4⋊D4), (C22×C4).142D4, C23.903(C2×D4), C2.14(C8.2D4), C22.205C22≀C2, C2.29(D4.7D4), C2.17(C8.12D4), C22.104(C4○D8), C22.76(C4⋊1D4), C2.20(C23⋊2D4), (C22×C8).320C22, (C2×C42).351C22, C2.16(Q8.D4), (C22×SD16).10C2, (C22×D4).67C22, (C22×Q8).56C22, C22.224(C4⋊D4), C23.67C23⋊8C2, (C22×C4).1437C23, C22.7C42⋊21C2, C22.119(C8.C22), (C2×Q8⋊C4)⋊34C2, (C2×C4).1028(C2×D4), (C2×D4⋊C4).21C2, (C2×C4).876(C4○D4), (C2×C4⋊C4).110C22, (C2×C4.4D4).11C2, SmallGroup(128,747)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for (C2×C8).41D4
G = < a,b,c,d | a2=b8=1, c4=d2=b4, ab=ba, ac=ca, ad=da, cbc-1=ab5, dbd-1=b-1, dcd-1=c3 >
Subgroups: 440 in 195 conjugacy classes, 54 normal (34 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, Q16, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C24, C2.C42, D4⋊C4, Q8⋊C4, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C4.4D4, C22×C8, C2×SD16, C2×Q16, C22×D4, C22×Q8, C22.7C42, C23.67C23, C2×D4⋊C4, C2×Q8⋊C4, C2×C4.4D4, C22×SD16, C22×Q16, (C2×C8).41D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C22≀C2, C4⋊D4, C4⋊1D4, C4○D8, C8.C22, C23⋊2D4, D4.7D4, Q8.D4, C8.12D4, C8.2D4, (C2×C8).41D4
►On 64 points
(1 62)(2 63)(3 64)(4 57)(5 58)(6 59)(7 60)(8 61)(9 19)(10 20)(11 21)(12 22)(13 23)(14 24)(15 17)(16 18)(25 55)(26 56)(27 49)(28 50)(29 51)(30 52)(31 53)(32 54)(33 44)(34 45)(35 46)(36 47)(37 48)(38 41)(39 42)(40 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 48 11 27 5 44 15 31)(2 34 12 54 6 38 16 50)(3 42 13 29 7 46 9 25)(4 36 14 56 8 40 10 52)(17 53 62 37 21 49 58 33)(18 28 63 45 22 32 59 41)(19 55 64 39 23 51 60 35)(20 30 57 47 24 26 61 43)
(1 27 5 31)(2 26 6 30)(3 25 7 29)(4 32 8 28)(9 42 13 46)(10 41 14 45)(11 48 15 44)(12 47 16 43)(17 33 21 37)(18 40 22 36)(19 39 23 35)(20 38 24 34)(49 58 53 62)(50 57 54 61)(51 64 55 60)(52 63 56 59)
G:=sub<Sym(64)| (1,62)(2,63)(3,64)(4,57)(5,58)(6,59)(7,60)(8,61)(9,19)(10,20)(11,21)(12,22)(13,23)(14,24)(15,17)(16,18)(25,55)(26,56)(27,49)(28,50)(29,51)(30,52)(31,53)(32,54)(33,44)(34,45)(35,46)(36,47)(37,48)(38,41)(39,42)(40,43), (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), (1,48,11,27,5,44,15,31)(2,34,12,54,6,38,16,50)(3,42,13,29,7,46,9,25)(4,36,14,56,8,40,10,52)(17,53,62,37,21,49,58,33)(18,28,63,45,22,32,59,41)(19,55,64,39,23,51,60,35)(20,30,57,47,24,26,61,43), (1,27,5,31)(2,26,6,30)(3,25,7,29)(4,32,8,28)(9,42,13,46)(10,41,14,45)(11,48,15,44)(12,47,16,43)(17,33,21,37)(18,40,22,36)(19,39,23,35)(20,38,24,34)(49,58,53,62)(50,57,54,61)(51,64,55,60)(52,63,56,59)>;
G:=Group( (1,62)(2,63)(3,64)(4,57)(5,58)(6,59)(7,60)(8,61)(9,19)(10,20)(11,21)(12,22)(13,23)(14,24)(15,17)(16,18)(25,55)(26,56)(27,49)(28,50)(29,51)(30,52)(31,53)(32,54)(33,44)(34,45)(35,46)(36,47)(37,48)(38,41)(39,42)(40,43), (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), (1,48,11,27,5,44,15,31)(2,34,12,54,6,38,16,50)(3,42,13,29,7,46,9,25)(4,36,14,56,8,40,10,52)(17,53,62,37,21,49,58,33)(18,28,63,45,22,32,59,41)(19,55,64,39,23,51,60,35)(20,30,57,47,24,26,61,43), (1,27,5,31)(2,26,6,30)(3,25,7,29)(4,32,8,28)(9,42,13,46)(10,41,14,45)(11,48,15,44)(12,47,16,43)(17,33,21,37)(18,40,22,36)(19,39,23,35)(20,38,24,34)(49,58,53,62)(50,57,54,61)(51,64,55,60)(52,63,56,59) );
G=PermutationGroup([[(1,62),(2,63),(3,64),(4,57),(5,58),(6,59),(7,60),(8,61),(9,19),(10,20),(11,21),(12,22),(13,23),(14,24),(15,17),(16,18),(25,55),(26,56),(27,49),(28,50),(29,51),(30,52),(31,53),(32,54),(33,44),(34,45),(35,46),(36,47),(37,48),(38,41),(39,42),(40,43)], [(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)], [(1,48,11,27,5,44,15,31),(2,34,12,54,6,38,16,50),(3,42,13,29,7,46,9,25),(4,36,14,56,8,40,10,52),(17,53,62,37,21,49,58,33),(18,28,63,45,22,32,59,41),(19,55,64,39,23,51,60,35),(20,30,57,47,24,26,61,43)], [(1,27,5,31),(2,26,6,30),(3,25,7,29),(4,32,8,28),(9,42,13,46),(10,41,14,45),(11,48,15,44),(12,47,16,43),(17,33,21,37),(18,40,22,36),(19,39,23,35),(20,38,24,34),(49,58,53,62),(50,57,54,61),(51,64,55,60),(52,63,56,59)]])
32 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4N | 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 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D4 | C4○D4 | C4○D8 | C8.C22 |
| kernel | (C2×C8).41D4 | C22.7C42 | C23.67C23 | C2×D4⋊C4 | C2×Q8⋊C4 | C2×C4.4D4 | C22×SD16 | C22×Q16 | C2×C8 | C22×C4 | C2×D4 | C2×Q8 | C2×C4 | C22 | C22 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 2 | 2 | 4 | 2 | 8 | 2 |
Matrix representation of (C2×C8).41D4 ►in GL6(𝔽17)
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 11 | 0 | 0 |
| 0 | 0 | 3 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 3 |
| 0 | 0 | 0 | 0 | 14 | 12 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 10 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 14 | 12 |
| 0 | 0 | 0 | 0 | 5 | 3 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 10 | 0 | 0 |
| 0 | 0 | 12 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 14 | 12 |
| 0 | 0 | 0 | 0 | 5 | 3 |
G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[4,0,0,0,0,0,0,13,0,0,0,0,0,0,0,3,0,0,0,0,11,6,0,0,0,0,0,0,5,14,0,0,0,0,3,12],[0,13,0,0,0,0,13,0,0,0,0,0,0,0,10,12,0,0,0,0,10,0,0,0,0,0,0,0,14,5,0,0,0,0,12,3],[0,4,0,0,0,0,13,0,0,0,0,0,0,0,10,12,0,0,0,0,10,7,0,0,0,0,0,0,14,5,0,0,0,0,12,3] >;
(C2×C8).41D4 in GAP, Magma, Sage, TeX
(C_2\times C_8)._{41}D_4 % in TeX
G:=Group("(C2xC8).41D4"); // GroupNames label
G:=SmallGroup(128,747);
// by ID
G=gap.SmallGroup(128,747);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,448,141,456,422,387,2019,1018,521,248,2804,1411,718,172]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^8=1,c^4=d^2=b^4,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=a*b^5,d*b*d^-1=b^-1,d*c*d^-1=c^3>;
// generators/relations