metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C22.4D20, C23.16D10, C4⋊Dic5⋊5C2, C22⋊C4⋊6D5, C10.6(C2×D4), (C2×C4).9D10, (C2×C10).4D4, C2.8(C2×D20), D10⋊C4⋊7C2, (C2×C20).3C22, C10.23(C4○D4), (C2×C10).27C23, (C22×Dic5)⋊2C2, C5⋊2(C22.D4), C2.10(D4⋊2D5), (C22×D5).5C22, C22.45(C22×D5), (C22×C10).16C22, (C2×Dic5).31C22, (C5×C22⋊C4)⋊4C2, (C2×C5⋊D4).5C2, SmallGroup(160,107)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C22.D20
G = < a,b,c,d | a2=b2=c20=1, d2=b, cac-1=ab=ba, ad=da, bc=cb, bd=db, dcd-1=bc-1 >
Subgroups: 256 in 78 conjugacy classes, 33 normal (15 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C5, C2×C4, C2×C4, D4, C23, C23, D5, C10, C10, C10, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic5, C20, D10, C2×C10, C2×C10, C2×C10, C22.D4, C2×Dic5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C22×D5, C22×C10, C4⋊Dic5, D10⋊C4, C5×C22⋊C4, C22×Dic5, C2×C5⋊D4, C22.D20
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C22.D4, D20, C22×D5, C2×D20, D4⋊2D5, C22.D20
►On 80 points
(1 31)(2 69)(3 33)(4 71)(5 35)(6 73)(7 37)(8 75)(9 39)(10 77)(11 21)(12 79)(13 23)(14 61)(15 25)(16 63)(17 27)(18 65)(19 29)(20 67)(22 57)(24 59)(26 41)(28 43)(30 45)(32 47)(34 49)(36 51)(38 53)(40 55)(42 64)(44 66)(46 68)(48 70)(50 72)(52 74)(54 76)(56 78)(58 80)(60 62)
(1 46)(2 47)(3 48)(4 49)(5 50)(6 51)(7 52)(8 53)(9 54)(10 55)(11 56)(12 57)(13 58)(14 59)(15 60)(16 41)(17 42)(18 43)(19 44)(20 45)(21 78)(22 79)(23 80)(24 61)(25 62)(26 63)(27 64)(28 65)(29 66)(30 67)(31 68)(32 69)(33 70)(34 71)(35 72)(36 73)(37 74)(38 75)(39 76)(40 77)
(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 45 46 20)(2 19 47 44)(3 43 48 18)(4 17 49 42)(5 41 50 16)(6 15 51 60)(7 59 52 14)(8 13 53 58)(9 57 54 12)(10 11 55 56)(21 40 78 77)(22 76 79 39)(23 38 80 75)(24 74 61 37)(25 36 62 73)(26 72 63 35)(27 34 64 71)(28 70 65 33)(29 32 66 69)(30 68 67 31)
G:=sub<Sym(80)| (1,31)(2,69)(3,33)(4,71)(5,35)(6,73)(7,37)(8,75)(9,39)(10,77)(11,21)(12,79)(13,23)(14,61)(15,25)(16,63)(17,27)(18,65)(19,29)(20,67)(22,57)(24,59)(26,41)(28,43)(30,45)(32,47)(34,49)(36,51)(38,53)(40,55)(42,64)(44,66)(46,68)(48,70)(50,72)(52,74)(54,76)(56,78)(58,80)(60,62), (1,46)(2,47)(3,48)(4,49)(5,50)(6,51)(7,52)(8,53)(9,54)(10,55)(11,56)(12,57)(13,58)(14,59)(15,60)(16,41)(17,42)(18,43)(19,44)(20,45)(21,78)(22,79)(23,80)(24,61)(25,62)(26,63)(27,64)(28,65)(29,66)(30,67)(31,68)(32,69)(33,70)(34,71)(35,72)(36,73)(37,74)(38,75)(39,76)(40,77), (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), (1,45,46,20)(2,19,47,44)(3,43,48,18)(4,17,49,42)(5,41,50,16)(6,15,51,60)(7,59,52,14)(8,13,53,58)(9,57,54,12)(10,11,55,56)(21,40,78,77)(22,76,79,39)(23,38,80,75)(24,74,61,37)(25,36,62,73)(26,72,63,35)(27,34,64,71)(28,70,65,33)(29,32,66,69)(30,68,67,31)>;
G:=Group( (1,31)(2,69)(3,33)(4,71)(5,35)(6,73)(7,37)(8,75)(9,39)(10,77)(11,21)(12,79)(13,23)(14,61)(15,25)(16,63)(17,27)(18,65)(19,29)(20,67)(22,57)(24,59)(26,41)(28,43)(30,45)(32,47)(34,49)(36,51)(38,53)(40,55)(42,64)(44,66)(46,68)(48,70)(50,72)(52,74)(54,76)(56,78)(58,80)(60,62), (1,46)(2,47)(3,48)(4,49)(5,50)(6,51)(7,52)(8,53)(9,54)(10,55)(11,56)(12,57)(13,58)(14,59)(15,60)(16,41)(17,42)(18,43)(19,44)(20,45)(21,78)(22,79)(23,80)(24,61)(25,62)(26,63)(27,64)(28,65)(29,66)(30,67)(31,68)(32,69)(33,70)(34,71)(35,72)(36,73)(37,74)(38,75)(39,76)(40,77), (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), (1,45,46,20)(2,19,47,44)(3,43,48,18)(4,17,49,42)(5,41,50,16)(6,15,51,60)(7,59,52,14)(8,13,53,58)(9,57,54,12)(10,11,55,56)(21,40,78,77)(22,76,79,39)(23,38,80,75)(24,74,61,37)(25,36,62,73)(26,72,63,35)(27,34,64,71)(28,70,65,33)(29,32,66,69)(30,68,67,31) );
G=PermutationGroup([[(1,31),(2,69),(3,33),(4,71),(5,35),(6,73),(7,37),(8,75),(9,39),(10,77),(11,21),(12,79),(13,23),(14,61),(15,25),(16,63),(17,27),(18,65),(19,29),(20,67),(22,57),(24,59),(26,41),(28,43),(30,45),(32,47),(34,49),(36,51),(38,53),(40,55),(42,64),(44,66),(46,68),(48,70),(50,72),(52,74),(54,76),(56,78),(58,80),(60,62)], [(1,46),(2,47),(3,48),(4,49),(5,50),(6,51),(7,52),(8,53),(9,54),(10,55),(11,56),(12,57),(13,58),(14,59),(15,60),(16,41),(17,42),(18,43),(19,44),(20,45),(21,78),(22,79),(23,80),(24,61),(25,62),(26,63),(27,64),(28,65),(29,66),(30,67),(31,68),(32,69),(33,70),(34,71),(35,72),(36,73),(37,74),(38,75),(39,76),(40,77)], [(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)], [(1,45,46,20),(2,19,47,44),(3,43,48,18),(4,17,49,42),(5,41,50,16),(6,15,51,60),(7,59,52,14),(8,13,53,58),(9,57,54,12),(10,11,55,56),(21,40,78,77),(22,76,79,39),(23,38,80,75),(24,74,61,37),(25,36,62,73),(26,72,63,35),(27,34,64,71),(28,70,65,33),(29,32,66,69),(30,68,67,31)]])
C22.D20 is a maximal subgroup of
C22⋊C4⋊F5 C23.5D20 C23⋊3D20 C24.31D10 C42⋊8D10 C42.92D10 C42.96D10 C42.102D10 D4⋊5D20 D4⋊6D20 C42.118D10 C24.56D10 C24⋊3D10 C24.33D10 C10.462+ 1+4 C10.1152+ 1+4 C10.472+ 1+4 C10.482+ 1+4 C22⋊Q8⋊25D5 C10.532+ 1+4 C10.772- 1+4 C10.572+ 1+4 C10.792- 1+4 D5×C22.D4 C10.822- 1+4 C10.1222+ 1+4 C10.662+ 1+4 C10.852- 1+4 C10.692+ 1+4 C42⋊20D10 C42.143D10 C42.144D10 C42.145D10 C42.161D10 C42.163D10 C42.164D10 C42.165D10 D6.D20 D6.9D20 C6.(C2×D20) C6.D4⋊D5 C22.D60
C22.D20 is a maximal quotient of
C2.(C4×D20) (C2×C20).28D4 C10.(C4⋊Q8) C10.55(C4×D4) (C2×C4).21D20 (C2×C20).33D4 C23.34D20 C23.35D20 C23.10D20 C23.38D20 C22.D40 C23.13D20 C23.42D20 C24.47D10 C23.14D20 C23.45D20 C24.16D10 D6.D20 D6.9D20 C6.(C2×D20) C6.D4⋊D5 C22.D60
34 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 20 | 4 | 4 | 10 | 10 | 10 | 10 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
34 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 | D5 | C4○D4 | D10 | D10 | D20 | D4⋊2D5 |
| kernel | C22.D20 | C4⋊Dic5 | D10⋊C4 | C5×C22⋊C4 | C22×Dic5 | C2×C5⋊D4 | C2×C10 | C22⋊C4 | C10 | C2×C4 | C23 | C22 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 2 | 8 | 4 |
Matrix representation of C22.D20 ►in GL4(𝔽41) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 21 | 1 |
| 0 | 0 | 11 | 20 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 16 | 30 | 0 | 0 |
| 27 | 2 | 0 | 0 |
| 0 | 0 | 25 | 9 |
| 0 | 0 | 40 | 16 |
| 39 | 30 | 0 | 0 |
| 4 | 2 | 0 | 0 |
| 0 | 0 | 25 | 9 |
| 0 | 0 | 17 | 16 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,21,11,0,0,1,20],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[16,27,0,0,30,2,0,0,0,0,25,40,0,0,9,16],[39,4,0,0,30,2,0,0,0,0,25,17,0,0,9,16] >;
C22.D20 in GAP, Magma, Sage, TeX
C_2^2.D_{20} % in TeX
G:=Group("C2^2.D20"); // GroupNames label
G:=SmallGroup(160,107);
// by ID
G=gap.SmallGroup(160,107);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,217,218,188,122,4613]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^20=1,d^2=b,c*a*c^-1=a*b=b*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=b*c^-1>;
// generators/relations