metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2×C20).2D4, (C2×C4).2D20, (C2×D4).2D10, C23⋊C4.1D5, C23.D5⋊2C4, C23.2(C4×D5), (C22×C10).9D4, C20.D4.1C2, (C22×Dic5)⋊1C4, (D4×C10).2C22, C23.2(C5⋊D4), C5⋊3(C23.D4), C10.29(C23⋊C4), C22.9(D10⋊C4), C23.18D10.1C2, C2.9(C23.1D10), (C5×C23⋊C4).1C2, (C22×C10).2(C2×C4), (C2×C10).66(C22⋊C4), SmallGroup(320,30)
Series: Derived ►Chief ►Lower central ►Upper central
C1 — C2 — C22 — C2×D4 — C23⋊C4 |
Generators and relations for (C2×C20).D4
G = < a,b,c,d | a2=b4=c20=1, d2=b-1, ab=ba, cac-1=dad-1=ab2, cbc-1=ab-1, bd=db, dcd-1=bc-1 >
Subgroups: 286 in 68 conjugacy classes, 21 normal (all characteristic)
C1, C2, C2 [×3], C4 [×4], C22, C22 [×4], C5, C8, C2×C4, C2×C4 [×4], D4, C23 [×2], C10, C10 [×3], C22⋊C4 [×3], C4⋊C4, M4(2), C22×C4, C2×D4, Dic5 [×2], C20 [×2], C2×C10, C2×C10 [×4], C23⋊C4, C4.D4, C22.D4, C5⋊2C8, C2×Dic5 [×3], C2×C20, C2×C20, C5×D4, C22×C10 [×2], C23.D4, C4.Dic5, C10.D4, C23.D5, C23.D5, C5×C22⋊C4, C22×Dic5, D4×C10, C20.D4, C5×C23⋊C4, C23.18D10, (C2×C20).D4
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], D5, C22⋊C4, D10, C23⋊C4, C4×D5, D20, C5⋊D4, C23.D4, D10⋊C4, C23.1D10, (C2×C20).D4
(1 22)(3 24)(5 26)(7 28)(9 30)(11 32)(13 34)(15 36)(17 38)(19 40)(41 74)(43 76)(45 78)(47 80)(49 62)(51 64)(53 66)(55 68)(57 70)(59 72)
(1 66 22 53)(2 67 23 54)(3 55 24 68)(4 56 25 69)(5 70 26 57)(6 71 27 58)(7 59 28 72)(8 60 29 73)(9 74 30 41)(10 75 31 42)(11 43 32 76)(12 44 33 77)(13 78 34 45)(14 79 35 46)(15 47 36 80)(16 48 37 61)(17 62 38 49)(18 63 39 50)(19 51 40 64)(20 52 21 65)
(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 75 53 10 22 42 66 31)(2 30 54 74 23 9 67 41)(3 73 68 29 24 60 55 8)(4 7 69 72 25 28 56 59)(5 71 57 6 26 58 70 27)(11 65 76 21 32 52 43 20)(12 19 77 64 33 40 44 51)(13 63 45 18 34 50 78 39)(14 38 46 62 35 17 79 49)(15 61 80 37 36 48 47 16)
G:=sub<Sym(80)| (1,22)(3,24)(5,26)(7,28)(9,30)(11,32)(13,34)(15,36)(17,38)(19,40)(41,74)(43,76)(45,78)(47,80)(49,62)(51,64)(53,66)(55,68)(57,70)(59,72), (1,66,22,53)(2,67,23,54)(3,55,24,68)(4,56,25,69)(5,70,26,57)(6,71,27,58)(7,59,28,72)(8,60,29,73)(9,74,30,41)(10,75,31,42)(11,43,32,76)(12,44,33,77)(13,78,34,45)(14,79,35,46)(15,47,36,80)(16,48,37,61)(17,62,38,49)(18,63,39,50)(19,51,40,64)(20,52,21,65), (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), (1,75,53,10,22,42,66,31)(2,30,54,74,23,9,67,41)(3,73,68,29,24,60,55,8)(4,7,69,72,25,28,56,59)(5,71,57,6,26,58,70,27)(11,65,76,21,32,52,43,20)(12,19,77,64,33,40,44,51)(13,63,45,18,34,50,78,39)(14,38,46,62,35,17,79,49)(15,61,80,37,36,48,47,16)>;
G:=Group( (1,22)(3,24)(5,26)(7,28)(9,30)(11,32)(13,34)(15,36)(17,38)(19,40)(41,74)(43,76)(45,78)(47,80)(49,62)(51,64)(53,66)(55,68)(57,70)(59,72), (1,66,22,53)(2,67,23,54)(3,55,24,68)(4,56,25,69)(5,70,26,57)(6,71,27,58)(7,59,28,72)(8,60,29,73)(9,74,30,41)(10,75,31,42)(11,43,32,76)(12,44,33,77)(13,78,34,45)(14,79,35,46)(15,47,36,80)(16,48,37,61)(17,62,38,49)(18,63,39,50)(19,51,40,64)(20,52,21,65), (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), (1,75,53,10,22,42,66,31)(2,30,54,74,23,9,67,41)(3,73,68,29,24,60,55,8)(4,7,69,72,25,28,56,59)(5,71,57,6,26,58,70,27)(11,65,76,21,32,52,43,20)(12,19,77,64,33,40,44,51)(13,63,45,18,34,50,78,39)(14,38,46,62,35,17,79,49)(15,61,80,37,36,48,47,16) );
G=PermutationGroup([(1,22),(3,24),(5,26),(7,28),(9,30),(11,32),(13,34),(15,36),(17,38),(19,40),(41,74),(43,76),(45,78),(47,80),(49,62),(51,64),(53,66),(55,68),(57,70),(59,72)], [(1,66,22,53),(2,67,23,54),(3,55,24,68),(4,56,25,69),(5,70,26,57),(6,71,27,58),(7,59,28,72),(8,60,29,73),(9,74,30,41),(10,75,31,42),(11,43,32,76),(12,44,33,77),(13,78,34,45),(14,79,35,46),(15,47,36,80),(16,48,37,61),(17,62,38,49),(18,63,39,50),(19,51,40,64),(20,52,21,65)], [(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)], [(1,75,53,10,22,42,66,31),(2,30,54,74,23,9,67,41),(3,73,68,29,24,60,55,8),(4,7,69,72,25,28,56,59),(5,71,57,6,26,58,70,27),(11,65,76,21,32,52,43,20),(12,19,77,64,33,40,44,51),(13,63,45,18,34,50,78,39),(14,38,46,62,35,17,79,49),(15,61,80,37,36,48,47,16)])
35 conjugacy classes
class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 8A | 8B | 10A | 10B | 10C | ··· | 10H | 10I | 10J | 20A | ··· | 20J |
order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 10 | 10 | 10 | ··· | 10 | 10 | 10 | 20 | ··· | 20 |
size | 1 | 1 | 2 | 4 | 4 | 4 | 8 | 8 | 20 | 20 | 40 | 2 | 2 | 40 | 40 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | ··· | 8 |
35 irreducible representations
dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 |
type | + | + | + | + | + | + | + | + | + | + | - | ||||||
image | C1 | C2 | C2 | C2 | C4 | C4 | D4 | D4 | D5 | D10 | D20 | C4×D5 | C5⋊D4 | C23⋊C4 | C23.D4 | C23.1D10 | (C2×C20).D4 |
kernel | (C2×C20).D4 | C20.D4 | C5×C23⋊C4 | C23.18D10 | C23.D5 | C22×Dic5 | C2×C20 | C22×C10 | C23⋊C4 | C2×D4 | C2×C4 | C23 | C23 | C10 | C5 | C2 | C1 |
# reps | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 1 | 2 | 4 | 2 |
Matrix representation of (C2×C20).D4 ►in GL6(𝔽41)
1 | 0 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 40 | 0 | 0 | 0 |
0 | 0 | 0 | 40 | 0 | 0 |
0 | 0 | 20 | 20 | 1 | 0 |
0 | 0 | 19 | 19 | 0 | 1 |
1 | 0 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 9 | 0 | 0 | 0 |
0 | 0 | 0 | 32 | 0 | 0 |
0 | 0 | 25 | 0 | 32 | 0 |
0 | 0 | 0 | 7 | 0 | 9 |
0 | 40 | 0 | 0 | 0 | 0 |
1 | 7 | 0 | 0 | 0 | 0 |
0 | 0 | 8 | 8 | 9 | 0 |
0 | 0 | 24 | 24 | 0 | 9 |
0 | 0 | 8 | 7 | 33 | 33 |
0 | 0 | 25 | 24 | 17 | 17 |
0 | 40 | 0 | 0 | 0 | 0 |
40 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 11 | 11 | 0 | 40 |
0 | 0 | 31 | 31 | 40 | 0 |
0 | 0 | 31 | 22 | 10 | 10 |
0 | 0 | 20 | 11 | 30 | 30 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,20,19,0,0,0,40,20,19,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,0,25,0,0,0,0,32,0,7,0,0,0,0,32,0,0,0,0,0,0,9],[0,1,0,0,0,0,40,7,0,0,0,0,0,0,8,24,8,25,0,0,8,24,7,24,0,0,9,0,33,17,0,0,0,9,33,17],[0,40,0,0,0,0,40,0,0,0,0,0,0,0,11,31,31,20,0,0,11,31,22,11,0,0,0,40,10,30,0,0,40,0,10,30] >;
(C2×C20).D4 in GAP, Magma, Sage, TeX
(C_2\times C_{20}).D_4
% in TeX
G:=Group("(C2xC20).D4");
// GroupNames label
G:=SmallGroup(320,30);
// by ID
G=gap.SmallGroup(320,30);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,141,36,422,346,297,851,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^20=1,d^2=b^-1,a*b=b*a,c*a*c^-1=d*a*d^-1=a*b^2,c*b*c^-1=a*b^-1,b*d=d*b,d*c*d^-1=b*c^-1>;
// generators/relations