metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C20.23D4, (C2×Q8)⋊4D5, (Q8×C10)⋊4C2, (C4×Dic5)⋊7C2, C10.58(C2×D4), (C2×C4).57D10, C5⋊4(C4.4D4), (C2×D20).10C2, C4.11(C5⋊D4), D10⋊C4⋊16C2, C10.37(C4○D4), (C2×C10).59C23, (C2×C20).40C22, C2.9(Q8⋊2D5), C22.65(C22×D5), (C2×Dic5).44C22, (C22×D5).13C22, C2.22(C2×C5⋊D4), SmallGroup(160,168)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C20.23D4
G = < a,b,c | a20=b4=c2=1, bab-1=a9, cac=a-1, cbc=a10b-1 >
Subgroups: 272 in 76 conjugacy classes, 33 normal (13 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C10, C42, C22⋊C4, C2×D4, C2×Q8, Dic5, C20, C20, D10, C2×C10, C4.4D4, D20, C2×Dic5, C2×C20, C2×C20, C5×Q8, C22×D5, C4×Dic5, D10⋊C4, C2×D20, Q8×C10, C20.23D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C4.4D4, C5⋊D4, C22×D5, Q8⋊2D5, C2×C5⋊D4, C20.23D4
►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 60 76 33)(2 49 77 22)(3 58 78 31)(4 47 79 40)(5 56 80 29)(6 45 61 38)(7 54 62 27)(8 43 63 36)(9 52 64 25)(10 41 65 34)(11 50 66 23)(12 59 67 32)(13 48 68 21)(14 57 69 30)(15 46 70 39)(16 55 71 28)(17 44 72 37)(18 53 73 26)(19 42 74 35)(20 51 75 24)
(2 20)(3 19)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)(10 12)(21 42)(22 41)(23 60)(24 59)(25 58)(26 57)(27 56)(28 55)(29 54)(30 53)(31 52)(32 51)(33 50)(34 49)(35 48)(36 47)(37 46)(38 45)(39 44)(40 43)(61 71)(62 70)(63 69)(64 68)(65 67)(72 80)(73 79)(74 78)(75 77)
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), (1,60,76,33)(2,49,77,22)(3,58,78,31)(4,47,79,40)(5,56,80,29)(6,45,61,38)(7,54,62,27)(8,43,63,36)(9,52,64,25)(10,41,65,34)(11,50,66,23)(12,59,67,32)(13,48,68,21)(14,57,69,30)(15,46,70,39)(16,55,71,28)(17,44,72,37)(18,53,73,26)(19,42,74,35)(20,51,75,24), (2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(21,42)(22,41)(23,60)(24,59)(25,58)(26,57)(27,56)(28,55)(29,54)(30,53)(31,52)(32,51)(33,50)(34,49)(35,48)(36,47)(37,46)(38,45)(39,44)(40,43)(61,71)(62,70)(63,69)(64,68)(65,67)(72,80)(73,79)(74,78)(75,77)>;
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), (1,60,76,33)(2,49,77,22)(3,58,78,31)(4,47,79,40)(5,56,80,29)(6,45,61,38)(7,54,62,27)(8,43,63,36)(9,52,64,25)(10,41,65,34)(11,50,66,23)(12,59,67,32)(13,48,68,21)(14,57,69,30)(15,46,70,39)(16,55,71,28)(17,44,72,37)(18,53,73,26)(19,42,74,35)(20,51,75,24), (2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(21,42)(22,41)(23,60)(24,59)(25,58)(26,57)(27,56)(28,55)(29,54)(30,53)(31,52)(32,51)(33,50)(34,49)(35,48)(36,47)(37,46)(38,45)(39,44)(40,43)(61,71)(62,70)(63,69)(64,68)(65,67)(72,80)(73,79)(74,78)(75,77) );
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)], [(1,60,76,33),(2,49,77,22),(3,58,78,31),(4,47,79,40),(5,56,80,29),(6,45,61,38),(7,54,62,27),(8,43,63,36),(9,52,64,25),(10,41,65,34),(11,50,66,23),(12,59,67,32),(13,48,68,21),(14,57,69,30),(15,46,70,39),(16,55,71,28),(17,44,72,37),(18,53,73,26),(19,42,74,35),(20,51,75,24)], [(2,20),(3,19),(4,18),(5,17),(6,16),(7,15),(8,14),(9,13),(10,12),(21,42),(22,41),(23,60),(24,59),(25,58),(26,57),(27,56),(28,55),(29,54),(30,53),(31,52),(32,51),(33,50),(34,49),(35,48),(36,47),(37,46),(38,45),(39,44),(40,43),(61,71),(62,70),(63,69),(64,68),(65,67),(72,80),(73,79),(74,78),(75,77)]])
C20.23D4 is a maximal subgroup of
(C2×C4).D20 (C2×Q8).F5 (C2×Q8)⋊F5 D20.5D4 Q8⋊C4⋊D5 Dic10.11D4 Q8⋊Dic5⋊C2 D20.12D4 (C5×D4).D4 C40.43D4 C40⋊9D4 (C2×Q16)⋊D5 C40.37D4 C40.28D4 D20.39D4 2- 1+4⋊2D5 C42.122D10 C42.131D10 C42.133D10 C42.136D10 C22⋊Q8⋊25D5 D20⋊21D4 Dic10⋊22D4 C10.532+ 1+4 C10.222- 1+4 C10.242- 1+4 C10.562+ 1+4 C10.262- 1+4 C42.138D10 D5×C4.4D4 C42⋊18D10 C42⋊20D10 C42.143D10 C42⋊22D10 C42.171D10 C42.240D10 C42.177D10 C42.178D10 C42.179D10 C42.180D10 C10.442- 1+4 C10.452- 1+4 C10.1452+ 1+4 C10.1462+ 1+4 (C2×C20)⋊17D4 C60.88D4 C60.70D4 (C2×Dic6)⋊D5 C60.23D4
C20.23D4 is a maximal quotient of
C20.48(C4⋊C4) (C2×C20).55D4 (C2×D20)⋊22C4 C10.90(C4×D4) (C2×C4)⋊3D20 (C2×C20).290D4 C42.70D10 C42.216D10 C42.71D10 C20.D8 C42.82D10 C20.11Q16 (Q8×C10)⋊17C4 (C22×D5)⋊Q8 C60.88D4 C60.70D4 (C2×Dic6)⋊D5 C60.23D4
34 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 10A | ··· | 10F | 20A | ··· | 20L |
| 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 | 20 | 20 | 2 | 2 | 4 | 4 | 10 | 10 | 10 | 10 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
34 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | D4 | D5 | C4○D4 | D10 | C5⋊D4 | Q8⋊2D5 |
| kernel | C20.23D4 | C4×Dic5 | D10⋊C4 | C2×D20 | Q8×C10 | C20 | C2×Q8 | C10 | C2×C4 | C4 | C2 |
| # reps | 1 | 1 | 4 | 1 | 1 | 2 | 2 | 4 | 6 | 8 | 4 |
Matrix representation of C20.23D4 ►in GL4(𝔽41) generated by
| 0 | 1 | 0 | 0 |
| 40 | 0 | 0 | 0 |
| 0 | 0 | 35 | 40 |
| 0 | 0 | 36 | 40 |
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 21 | 17 |
| 0 | 0 | 15 | 20 |
| 1 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 6 | 1 |
| 0 | 0 | 6 | 35 |
G:=sub<GL(4,GF(41))| [0,40,0,0,1,0,0,0,0,0,35,36,0,0,40,40],[9,0,0,0,0,9,0,0,0,0,21,15,0,0,17,20],[1,0,0,0,0,40,0,0,0,0,6,6,0,0,1,35] >;
C20.23D4 in GAP, Magma, Sage, TeX
C_{20}._{23}D_4 % in TeX
G:=Group("C20.23D4"); // GroupNames label
G:=SmallGroup(160,168);
// by ID
G=gap.SmallGroup(160,168);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,217,103,218,188,86,4613]);
// Polycyclic
G:=Group<a,b,c|a^20=b^4=c^2=1,b*a*b^-1=a^9,c*a*c=a^-1,c*b*c=a^10*b^-1>;
// generators/relations