metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C20⋊4D4, C4⋊1D20, C42⋊5D5, (C4×C20)⋊4C2, (C2×D20)⋊1C2, C5⋊1(C4⋊1D4), C2.5(C2×D20), C10.3(C2×D4), (C2×C4).76D10, (C2×C10).15C23, (C2×C20).87C22, (C22×D5).1C22, C22.36(C22×D5), SmallGroup(160,95)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C20⋊4D4
G = < a,b,c | a4=b20=c2=1, ab=ba, cac=a-1, cbc=b-1 >
Subgroups: 480 in 108 conjugacy classes, 41 normal (7 characteristic)
C1, C2, C2, C4, C22, C22, C5, C2×C4, D4, C23, D5, C10, C42, C2×D4, C20, D10, C2×C10, C4⋊1D4, D20, C2×C20, C22×D5, C4×C20, C2×D20, C20⋊4D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, D10, C4⋊1D4, D20, C22×D5, C2×D20, C20⋊4D4
►On 80 points
(1 34 41 80)(2 35 42 61)(3 36 43 62)(4 37 44 63)(5 38 45 64)(6 39 46 65)(7 40 47 66)(8 21 48 67)(9 22 49 68)(10 23 50 69)(11 24 51 70)(12 25 52 71)(13 26 53 72)(14 27 54 73)(15 28 55 74)(16 29 56 75)(17 30 57 76)(18 31 58 77)(19 32 59 78)(20 33 60 79)
(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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 20)(17 19)(21 67)(22 66)(23 65)(24 64)(25 63)(26 62)(27 61)(28 80)(29 79)(30 78)(31 77)(32 76)(33 75)(34 74)(35 73)(36 72)(37 71)(38 70)(39 69)(40 68)(41 55)(42 54)(43 53)(44 52)(45 51)(46 50)(47 49)(56 60)(57 59)
G:=sub<Sym(80)| (1,34,41,80)(2,35,42,61)(3,36,43,62)(4,37,44,63)(5,38,45,64)(6,39,46,65)(7,40,47,66)(8,21,48,67)(9,22,49,68)(10,23,50,69)(11,24,51,70)(12,25,52,71)(13,26,53,72)(14,27,54,73)(15,28,55,74)(16,29,56,75)(17,30,57,76)(18,31,58,77)(19,32,59,78)(20,33,60,79), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,20)(17,19)(21,67)(22,66)(23,65)(24,64)(25,63)(26,62)(27,61)(28,80)(29,79)(30,78)(31,77)(32,76)(33,75)(34,74)(35,73)(36,72)(37,71)(38,70)(39,69)(40,68)(41,55)(42,54)(43,53)(44,52)(45,51)(46,50)(47,49)(56,60)(57,59)>;
G:=Group( (1,34,41,80)(2,35,42,61)(3,36,43,62)(4,37,44,63)(5,38,45,64)(6,39,46,65)(7,40,47,66)(8,21,48,67)(9,22,49,68)(10,23,50,69)(11,24,51,70)(12,25,52,71)(13,26,53,72)(14,27,54,73)(15,28,55,74)(16,29,56,75)(17,30,57,76)(18,31,58,77)(19,32,59,78)(20,33,60,79), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,20)(17,19)(21,67)(22,66)(23,65)(24,64)(25,63)(26,62)(27,61)(28,80)(29,79)(30,78)(31,77)(32,76)(33,75)(34,74)(35,73)(36,72)(37,71)(38,70)(39,69)(40,68)(41,55)(42,54)(43,53)(44,52)(45,51)(46,50)(47,49)(56,60)(57,59) );
G=PermutationGroup([[(1,34,41,80),(2,35,42,61),(3,36,43,62),(4,37,44,63),(5,38,45,64),(6,39,46,65),(7,40,47,66),(8,21,48,67),(9,22,49,68),(10,23,50,69),(11,24,51,70),(12,25,52,71),(13,26,53,72),(14,27,54,73),(15,28,55,74),(16,29,56,75),(17,30,57,76),(18,31,58,77),(19,32,59,78),(20,33,60,79)], [(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)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(16,20),(17,19),(21,67),(22,66),(23,65),(24,64),(25,63),(26,62),(27,61),(28,80),(29,79),(30,78),(31,77),(32,76),(33,75),(34,74),(35,73),(36,72),(37,71),(38,70),(39,69),(40,68),(41,55),(42,54),(43,53),(44,52),(45,51),(46,50),(47,49),(56,60),(57,59)]])
C20⋊4D4 is a maximal subgroup of
C4.D40 C42⋊F5 C8⋊5D20 C4.5D40 C20⋊4D8 C8⋊D20 C42.19D10 D4⋊4D20 C4⋊D40 Dic10⋊8D4 C20⋊7D8 Q8⋊D20 C42.64D10 C42.70D10 C20⋊D8 C20⋊6SD16 C20.D8 C42.276D10 C42⋊9D10 C42.100D10 D4×D20 Dic10⋊24D4 Q8⋊6D20 C42.136D10 C42⋊18D10 C42.156D10 C42⋊25D10 D5×C4⋊1D4 C42.240D10 C20⋊4D4⋊C3 C12⋊D20 C42⋊6D15
C20⋊4D4 is a maximal quotient of
(C2×C20)⋊5D4 (C2×C20).33D4 C8⋊5D20 C20⋊4D8 C8.8D20 C20⋊4Q16 C8⋊D20 C8.D20 C42⋊8Dic5 (C2×C4)⋊6D20 C12⋊D20 C42⋊6D15
46 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4F | 5A | 5B | 10A | ··· | 10F | 20A | ··· | 20X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
46 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | D4 | D5 | D10 | D20 |
| kernel | C20⋊4D4 | C4×C20 | C2×D20 | C20 | C42 | C2×C4 | C4 |
| # reps | 1 | 1 | 6 | 6 | 2 | 6 | 24 |
Matrix representation of C20⋊4D4 ►in GL4(𝔽41) generated by
| 40 | 2 | 0 | 0 |
| 40 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 32 | 30 |
| 0 | 0 | 11 | 27 |
| 1 | 0 | 0 | 0 |
| 1 | 40 | 0 | 0 |
| 0 | 0 | 0 | 40 |
| 0 | 0 | 40 | 0 |
G:=sub<GL(4,GF(41))| [40,40,0,0,2,1,0,0,0,0,40,0,0,0,0,40],[1,0,0,0,0,1,0,0,0,0,32,11,0,0,30,27],[1,1,0,0,0,40,0,0,0,0,0,40,0,0,40,0] >;
C20⋊4D4 in GAP, Magma, Sage, TeX
C_{20}\rtimes_4D_4 % in TeX
G:=Group("C20:4D4"); // GroupNames label
G:=SmallGroup(160,95);
// by ID
G=gap.SmallGroup(160,95);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,217,103,218,50,4613]);
// Polycyclic
G:=Group<a,b,c|a^4=b^20=c^2=1,a*b=b*a,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations