metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C20.17D4, C23.7D10, (C2×D4).6D5, (C4×Dic5)⋊5C2, (D4×C10).5C2, (C2×C4).50D10, C10.48(C2×D4), C4.7(C5⋊D4), C5⋊3(C4.4D4), C23.D5⋊9C2, (C2×Dic10)⋊10C2, C10.30(C4○D4), (C2×C20).33C22, (C2×C10).51C23, C2.16(D4⋊2D5), C22.58(C22×D5), (C22×C10).19C22, (C2×Dic5).18C22, C2.12(C2×C5⋊D4), SmallGroup(160,157)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C20.17D4
G = < a,b,c | a20=b4=1, c2=a10, bab-1=a9, cac-1=a-1, cbc-1=a10b-1 >
Subgroups: 208 in 76 conjugacy classes, 33 normal (13 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, D4, Q8, C23, C10, C10, C10, C42, C22⋊C4, C2×D4, C2×Q8, Dic5, C20, C2×C10, C2×C10, C4.4D4, Dic10, C2×Dic5, C2×C20, C5×D4, C22×C10, C4×Dic5, C23.D5, C2×Dic10, D4×C10, C20.17D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C4.4D4, C5⋊D4, C22×D5, D4⋊2D5, C2×C5⋊D4, C20.17D4
►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 78 26)(2 59 79 35)(3 48 80 24)(4 57 61 33)(5 46 62 22)(6 55 63 31)(7 44 64 40)(8 53 65 29)(9 42 66 38)(10 51 67 27)(11 60 68 36)(12 49 69 25)(13 58 70 34)(14 47 71 23)(15 56 72 32)(16 45 73 21)(17 54 74 30)(18 43 75 39)(19 52 76 28)(20 41 77 37)
(1 45 11 55)(2 44 12 54)(3 43 13 53)(4 42 14 52)(5 41 15 51)(6 60 16 50)(7 59 17 49)(8 58 18 48)(9 57 19 47)(10 56 20 46)(21 68 31 78)(22 67 32 77)(23 66 33 76)(24 65 34 75)(25 64 35 74)(26 63 36 73)(27 62 37 72)(28 61 38 71)(29 80 39 70)(30 79 40 69)
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), (1,50,78,26)(2,59,79,35)(3,48,80,24)(4,57,61,33)(5,46,62,22)(6,55,63,31)(7,44,64,40)(8,53,65,29)(9,42,66,38)(10,51,67,27)(11,60,68,36)(12,49,69,25)(13,58,70,34)(14,47,71,23)(15,56,72,32)(16,45,73,21)(17,54,74,30)(18,43,75,39)(19,52,76,28)(20,41,77,37), (1,45,11,55)(2,44,12,54)(3,43,13,53)(4,42,14,52)(5,41,15,51)(6,60,16,50)(7,59,17,49)(8,58,18,48)(9,57,19,47)(10,56,20,46)(21,68,31,78)(22,67,32,77)(23,66,33,76)(24,65,34,75)(25,64,35,74)(26,63,36,73)(27,62,37,72)(28,61,38,71)(29,80,39,70)(30,79,40,69)>;
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), (1,50,78,26)(2,59,79,35)(3,48,80,24)(4,57,61,33)(5,46,62,22)(6,55,63,31)(7,44,64,40)(8,53,65,29)(9,42,66,38)(10,51,67,27)(11,60,68,36)(12,49,69,25)(13,58,70,34)(14,47,71,23)(15,56,72,32)(16,45,73,21)(17,54,74,30)(18,43,75,39)(19,52,76,28)(20,41,77,37), (1,45,11,55)(2,44,12,54)(3,43,13,53)(4,42,14,52)(5,41,15,51)(6,60,16,50)(7,59,17,49)(8,58,18,48)(9,57,19,47)(10,56,20,46)(21,68,31,78)(22,67,32,77)(23,66,33,76)(24,65,34,75)(25,64,35,74)(26,63,36,73)(27,62,37,72)(28,61,38,71)(29,80,39,70)(30,79,40,69) );
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)], [(1,50,78,26),(2,59,79,35),(3,48,80,24),(4,57,61,33),(5,46,62,22),(6,55,63,31),(7,44,64,40),(8,53,65,29),(9,42,66,38),(10,51,67,27),(11,60,68,36),(12,49,69,25),(13,58,70,34),(14,47,71,23),(15,56,72,32),(16,45,73,21),(17,54,74,30),(18,43,75,39),(19,52,76,28),(20,41,77,37)], [(1,45,11,55),(2,44,12,54),(3,43,13,53),(4,42,14,52),(5,41,15,51),(6,60,16,50),(7,59,17,49),(8,58,18,48),(9,57,19,47),(10,56,20,46),(21,68,31,78),(22,67,32,77),(23,66,33,76),(24,65,34,75),(25,64,35,74),(26,63,36,73),(27,62,37,72),(28,61,38,71),(29,80,39,70),(30,79,40,69)]])
C20.17D4 is a maximal subgroup of
C23.D20 (C2×D4).F5 (D4×C10).C4 D20.1D4 C20⋊Q8⋊C2 Dic10.D4 (C8×Dic5)⋊C2 D20.D4 (C2×D8).D5 C40⋊11D4 C40.22D4 (C5×Q8).D4 C40.31D4 C40.43D4 D20.38D4 2+ 1+4.D5 C42.106D10 C42.229D10 C42.114D10 C42.115D10 C24.32D10 C24.35D10 C24⋊5D10 Dic10⋊19D4 C4⋊C4.178D10 C10.362+ 1+4 D20⋊20D4 C10.422+ 1+4 C10.452+ 1+4 C10.742- 1+4 C10.812- 1+4 C10.622+ 1+4 C10.842- 1+4 C42.139D10 D5×C4.4D4 C42.141D10 C42.166D10 C42⋊26D10 C42.238D10 C24.41D10 C24.42D10 C10.1052- 1+4 C10.1072- 1+4 (C2×C20)⋊17D4 C60.89D4 C60.69D4 C23.D5⋊S3 C60.17D4
C20.17D4 is a maximal quotient of
C24.4D10 C23⋊Dic10 C23.14D20 (C2×Dic5)⋊6Q8 C20.48(C4⋊C4) (C2×C20).288D4 C42.62D10 C42.213D10 C20.16D8 C42.72D10 C20.Q16 C42.77D10 C24.19D10 C24.20D10 C60.89D4 C60.69D4 C23.D5⋊S3 C60.17D4
34 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 10A | ··· | 10F | 10G | ··· | 10N | 20A | 20B | 20C | 20D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | 20 | 20 | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 10 | 10 | 10 | 10 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 |
34 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | D4 | D5 | C4○D4 | D10 | D10 | C5⋊D4 | D4⋊2D5 |
| kernel | C20.17D4 | C4×Dic5 | C23.D5 | C2×Dic10 | D4×C10 | C20 | C2×D4 | C10 | C2×C4 | C23 | C4 | C2 |
| # reps | 1 | 1 | 4 | 1 | 1 | 2 | 2 | 4 | 2 | 4 | 8 | 4 |
Matrix representation of C20.17D4 ►in GL4(𝔽41) generated by
| 37 | 0 | 0 | 0 |
| 0 | 10 | 0 | 0 |
| 0 | 0 | 40 | 18 |
| 0 | 0 | 9 | 1 |
| 0 | 1 | 0 | 0 |
| 40 | 0 | 0 | 0 |
| 0 | 0 | 9 | 2 |
| 0 | 0 | 1 | 32 |
| 0 | 40 | 0 | 0 |
| 40 | 0 | 0 | 0 |
| 0 | 0 | 32 | 0 |
| 0 | 0 | 40 | 9 |
G:=sub<GL(4,GF(41))| [37,0,0,0,0,10,0,0,0,0,40,9,0,0,18,1],[0,40,0,0,1,0,0,0,0,0,9,1,0,0,2,32],[0,40,0,0,40,0,0,0,0,0,32,40,0,0,0,9] >;
C20.17D4 in GAP, Magma, Sage, TeX
C_{20}._{17}D_4 % in TeX
G:=Group("C20.17D4"); // GroupNames label
G:=SmallGroup(160,157);
// by ID
G=gap.SmallGroup(160,157);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,96,55,506,116,4613]);
// Polycyclic
G:=Group<a,b,c|a^20=b^4=1,c^2=a^10,b*a*b^-1=a^9,c*a*c^-1=a^-1,c*b*c^-1=a^10*b^-1>;
// generators/relations