metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C20.48D4, C22⋊2Dic10, C23.20D10, (C2×C10)⋊3Q8, C4⋊Dic5⋊8C2, C5⋊4(C22⋊Q8), C10.8(C2×Q8), (C2×C4).83D10, C10.39(C2×D4), (C22×C4).5D5, (C2×Dic10)⋊6C2, C10.D4⋊2C2, C4.23(C5⋊D4), (C22×C20).6C2, C2.9(C2×Dic10), C23.D5.4C2, C10.15(C4○D4), C2.17(C4○D20), (C2×C20).91C22, (C2×C10).42C23, C22.54(C22×D5), (C22×C10).34C22, (C2×Dic5).14C22, C2.5(C2×C5⋊D4), SmallGroup(160,145)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C20.48D4
G = < a,b,c | a20=b4=1, c2=a10, bab-1=cac-1=a-1, cbc-1=a10b-1 >
Subgroups: 192 in 74 conjugacy classes, 37 normal (21 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, Q8, C23, C10, C10, C22⋊C4, C4⋊C4, C22×C4, C2×Q8, Dic5, C20, C20, C2×C10, C2×C10, C2×C10, C22⋊Q8, Dic10, C2×Dic5, C2×C20, C2×C20, C22×C10, C10.D4, C4⋊Dic5, C23.D5, C2×Dic10, C22×C20, C20.48D4
Quotients: C1, C2, C22, D4, Q8, C23, D5, C2×D4, C2×Q8, C4○D4, D10, C22⋊Q8, Dic10, C5⋊D4, C22×D5, C2×Dic10, C4○D20, C2×C5⋊D4, C20.48D4
►On 80 points
(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 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80)
(1 50 40 76)(2 49 21 75)(3 48 22 74)(4 47 23 73)(5 46 24 72)(6 45 25 71)(7 44 26 70)(8 43 27 69)(9 42 28 68)(10 41 29 67)(11 60 30 66)(12 59 31 65)(13 58 32 64)(14 57 33 63)(15 56 34 62)(16 55 35 61)(17 54 36 80)(18 53 37 79)(19 52 38 78)(20 51 39 77)
(1 66 11 76)(2 65 12 75)(3 64 13 74)(4 63 14 73)(5 62 15 72)(6 61 16 71)(7 80 17 70)(8 79 18 69)(9 78 19 68)(10 77 20 67)(21 59 31 49)(22 58 32 48)(23 57 33 47)(24 56 34 46)(25 55 35 45)(26 54 36 44)(27 53 37 43)(28 52 38 42)(29 51 39 41)(30 50 40 60)
G:=sub<Sym(80)| (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)(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80), (1,50,40,76)(2,49,21,75)(3,48,22,74)(4,47,23,73)(5,46,24,72)(6,45,25,71)(7,44,26,70)(8,43,27,69)(9,42,28,68)(10,41,29,67)(11,60,30,66)(12,59,31,65)(13,58,32,64)(14,57,33,63)(15,56,34,62)(16,55,35,61)(17,54,36,80)(18,53,37,79)(19,52,38,78)(20,51,39,77), (1,66,11,76)(2,65,12,75)(3,64,13,74)(4,63,14,73)(5,62,15,72)(6,61,16,71)(7,80,17,70)(8,79,18,69)(9,78,19,68)(10,77,20,67)(21,59,31,49)(22,58,32,48)(23,57,33,47)(24,56,34,46)(25,55,35,45)(26,54,36,44)(27,53,37,43)(28,52,38,42)(29,51,39,41)(30,50,40,60)>;
G:=Group( (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)(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80), (1,50,40,76)(2,49,21,75)(3,48,22,74)(4,47,23,73)(5,46,24,72)(6,45,25,71)(7,44,26,70)(8,43,27,69)(9,42,28,68)(10,41,29,67)(11,60,30,66)(12,59,31,65)(13,58,32,64)(14,57,33,63)(15,56,34,62)(16,55,35,61)(17,54,36,80)(18,53,37,79)(19,52,38,78)(20,51,39,77), (1,66,11,76)(2,65,12,75)(3,64,13,74)(4,63,14,73)(5,62,15,72)(6,61,16,71)(7,80,17,70)(8,79,18,69)(9,78,19,68)(10,77,20,67)(21,59,31,49)(22,58,32,48)(23,57,33,47)(24,56,34,46)(25,55,35,45)(26,54,36,44)(27,53,37,43)(28,52,38,42)(29,51,39,41)(30,50,40,60) );
G=PermutationGroup([[(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),(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)], [(1,50,40,76),(2,49,21,75),(3,48,22,74),(4,47,23,73),(5,46,24,72),(6,45,25,71),(7,44,26,70),(8,43,27,69),(9,42,28,68),(10,41,29,67),(11,60,30,66),(12,59,31,65),(13,58,32,64),(14,57,33,63),(15,56,34,62),(16,55,35,61),(17,54,36,80),(18,53,37,79),(19,52,38,78),(20,51,39,77)], [(1,66,11,76),(2,65,12,75),(3,64,13,74),(4,63,14,73),(5,62,15,72),(6,61,16,71),(7,80,17,70),(8,79,18,69),(9,78,19,68),(10,77,20,67),(21,59,31,49),(22,58,32,48),(23,57,33,47),(24,56,34,46),(25,55,35,45),(26,54,36,44),(27,53,37,43),(28,52,38,42),(29,51,39,41),(30,50,40,60)]])
C20.48D4 is a maximal subgroup of
C4⋊Dic5⋊C4 C23.30D20 C23.34D20 C23.35D20 C23.10D20 D20.31D4 D20.32D4 C22⋊Dic20 C4⋊C4.230D10 C4⋊C4.231D10 C4⋊C4.233D10 C5⋊2C8⋊23D4 C4.(D4×D5) (C2×C10)⋊Q16 C5⋊(C8.D4) C40⋊30D4 C40.82D4 C40⋊2D4 C40.4D4 (C5×D4).31D4 (C2×C10)⋊8Q16 (C5×D4).32D4 C42.274D10 C42.277D10 C23⋊2Dic10 C24.30D10 C24.31D10 C10.12- 1+4 C10.102+ 1+4 C10.52- 1+4 C10.62- 1+4 C42.89D10 C42.94D10 C42.98D10 C42.99D10 D4×Dic10 D4⋊5Dic10 C42.105D10 C42.106D10 D4⋊6Dic10 D20⋊23D4 D20⋊24D4 Dic10⋊23D4 C42⋊16D10 C42.115D10 C42.118D10 C4⋊C4.178D10 C10.352+ 1+4 C10.362+ 1+4 C4⋊C4⋊21D10 C10.392+ 1+4 C10.462+ 1+4 C10.742- 1+4 (Q8×Dic5)⋊C2 C10.502+ 1+4 C10.152- 1+4 D5×C22⋊Q8 C10.162- 1+4 C10.512+ 1+4 C10.582+ 1+4 C10.812- 1+4 C10.632+ 1+4 C10.842- 1+4 C10.692+ 1+4 C24.72D10 C24.42D10 Q8×C5⋊D4 C10.1042- 1+4 C10.1052- 1+4 C10.1072- 1+4 C10.1472+ 1+4 D6⋊Dic10 C60.45D4 C60.46D4 (C2×C10)⋊8Dic6 C60.205D4 C20.1S4
C20.48D4 is a maximal quotient of
C20⋊7(C4⋊C4) (C2×C20)⋊10Q8 C10.92(C4×D4) C23⋊Dic10 C24.6D10 C24.7D10 (C2×C4)⋊Dic10 (C2×C20).54D4 (C2×C20).55D4 C20.50D8 C20.38SD16 D4.3Dic10 C20.48SD16 C20.23Q16 Q8.3Dic10 C24.62D10 C24.64D10 D6⋊Dic10 C60.45D4 C60.46D4 (C2×C10)⋊8Dic6 C60.205D4
46 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 10A | ··· | 10N | 20A | ··· | 20P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
46 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | - | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | Q8 | D5 | C4○D4 | D10 | D10 | C5⋊D4 | Dic10 | C4○D20 |
| kernel | C20.48D4 | C10.D4 | C4⋊Dic5 | C23.D5 | C2×Dic10 | C22×C20 | C20 | C2×C10 | C22×C4 | C10 | C2×C4 | C23 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 2 | 8 | 8 | 8 |
Matrix representation of C20.48D4 ►in GL4(𝔽41) generated by
| 39 | 0 | 0 | 0 |
| 0 | 20 | 0 | 0 |
| 0 | 0 | 18 | 17 |
| 0 | 0 | 0 | 16 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 17 |
| 0 | 0 | 24 | 40 |
| 0 | 1 | 0 | 0 |
| 40 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 17 | 1 |
G:=sub<GL(4,GF(41))| [39,0,0,0,0,20,0,0,0,0,18,0,0,0,17,16],[0,1,0,0,1,0,0,0,0,0,1,24,0,0,17,40],[0,40,0,0,1,0,0,0,0,0,40,17,0,0,0,1] >;
C20.48D4 in GAP, Magma, Sage, TeX
C_{20}._{48}D_4 % in TeX
G:=Group("C20.48D4"); // GroupNames label
G:=SmallGroup(160,145);
// by ID
G=gap.SmallGroup(160,145);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,96,217,103,218,4613]);
// Polycyclic
G:=Group<a,b,c|a^20=b^4=1,c^2=a^10,b*a*b^-1=c*a*c^-1=a^-1,c*b*c^-1=a^10*b^-1>;
// generators/relations