direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C4×Dic5, C20⋊4C4, C5⋊2C42, C22.3D10, (C2×C4).6D5, C2.2(C4×D5), (C2×C20).7C2, C10.10(C2×C4), C2.2(C2×Dic5), (C2×C10).3C22, (C2×Dic5).6C2, SmallGroup(80,11)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — C4×Dic5 |
Generators and relations for C4×Dic5
G = < a,b,c | a4=b10=1, c2=b5, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C4×Dic5
►Regular action on 80 points
Generators in S80
(1 45 28 38)(2 46 29 39)(3 47 30 40)(4 48 21 31)(5 49 22 32)(6 50 23 33)(7 41 24 34)(8 42 25 35)(9 43 26 36)(10 44 27 37)(11 62 71 51)(12 63 72 52)(13 64 73 53)(14 65 74 54)(15 66 75 55)(16 67 76 56)(17 68 77 57)(18 69 78 58)(19 70 79 59)(20 61 80 60)
(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 62 6 67)(2 61 7 66)(3 70 8 65)(4 69 9 64)(5 68 10 63)(11 33 16 38)(12 32 17 37)(13 31 18 36)(14 40 19 35)(15 39 20 34)(21 58 26 53)(22 57 27 52)(23 56 28 51)(24 55 29 60)(25 54 30 59)(41 75 46 80)(42 74 47 79)(43 73 48 78)(44 72 49 77)(45 71 50 76)
G:=sub<Sym(80)| (1,45,28,38)(2,46,29,39)(3,47,30,40)(4,48,21,31)(5,49,22,32)(6,50,23,33)(7,41,24,34)(8,42,25,35)(9,43,26,36)(10,44,27,37)(11,62,71,51)(12,63,72,52)(13,64,73,53)(14,65,74,54)(15,66,75,55)(16,67,76,56)(17,68,77,57)(18,69,78,58)(19,70,79,59)(20,61,80,60), (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), (1,62,6,67)(2,61,7,66)(3,70,8,65)(4,69,9,64)(5,68,10,63)(11,33,16,38)(12,32,17,37)(13,31,18,36)(14,40,19,35)(15,39,20,34)(21,58,26,53)(22,57,27,52)(23,56,28,51)(24,55,29,60)(25,54,30,59)(41,75,46,80)(42,74,47,79)(43,73,48,78)(44,72,49,77)(45,71,50,76)>;
G:=Group( (1,45,28,38)(2,46,29,39)(3,47,30,40)(4,48,21,31)(5,49,22,32)(6,50,23,33)(7,41,24,34)(8,42,25,35)(9,43,26,36)(10,44,27,37)(11,62,71,51)(12,63,72,52)(13,64,73,53)(14,65,74,54)(15,66,75,55)(16,67,76,56)(17,68,77,57)(18,69,78,58)(19,70,79,59)(20,61,80,60), (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), (1,62,6,67)(2,61,7,66)(3,70,8,65)(4,69,9,64)(5,68,10,63)(11,33,16,38)(12,32,17,37)(13,31,18,36)(14,40,19,35)(15,39,20,34)(21,58,26,53)(22,57,27,52)(23,56,28,51)(24,55,29,60)(25,54,30,59)(41,75,46,80)(42,74,47,79)(43,73,48,78)(44,72,49,77)(45,71,50,76) );
G=PermutationGroup([[(1,45,28,38),(2,46,29,39),(3,47,30,40),(4,48,21,31),(5,49,22,32),(6,50,23,33),(7,41,24,34),(8,42,25,35),(9,43,26,36),(10,44,27,37),(11,62,71,51),(12,63,72,52),(13,64,73,53),(14,65,74,54),(15,66,75,55),(16,67,76,56),(17,68,77,57),(18,69,78,58),(19,70,79,59),(20,61,80,60)], [(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)], [(1,62,6,67),(2,61,7,66),(3,70,8,65),(4,69,9,64),(5,68,10,63),(11,33,16,38),(12,32,17,37),(13,31,18,36),(14,40,19,35),(15,39,20,34),(21,58,26,53),(22,57,27,52),(23,56,28,51),(24,55,29,60),(25,54,30,59),(41,75,46,80),(42,74,47,79),(43,73,48,78),(44,72,49,77),(45,71,50,76)]])
C4×Dic5 is a maximal subgroup of
C20.8Q8 C40⋊8C4 D20⋊7C4 D4⋊2Dic5 C20⋊C8 C10.C42 Dic5⋊C8 D5×C42 C42⋊D5 C23.11D10 C23.D10 Dic5⋊4D4 Dic5.5D4 Dic5⋊3Q8 C20⋊Q8 Dic5.Q8 C4.Dic10 C4⋊C4⋊7D5 D20⋊8C4 C4⋊C4⋊D5 C23.21D10 C20.17D4 C20⋊D4 Dic5⋊Q8 C20.23D4
C4×Dic5 is a maximal quotient of
C42.D5 C40⋊8C4 C10.10C42
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 5A | 5B | 10A | ··· | 10F | 20A | ··· | 20H |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 5 | ··· | 5 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | |||
| image | C1 | C2 | C2 | C4 | C4 | D5 | Dic5 | D10 | C4×D5 |
| kernel | C4×Dic5 | C2×Dic5 | C2×C20 | Dic5 | C20 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 8 | 4 | 2 | 4 | 2 | 8 |
Matrix representation of C4×Dic5 ►in GL4(𝔽41) generated by
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 40 | 34 |
| 1 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 26 | 21 |
| 0 | 0 | 3 | 15 |
G:=sub<GL(4,GF(41))| [9,0,0,0,0,9,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,40,0,0,0,0,0,40,0,0,1,34],[1,0,0,0,0,9,0,0,0,0,26,3,0,0,21,15] >;
C4×Dic5 in GAP, Magma, Sage, TeX
C_4\times {\rm Dic}_5 % in TeX
G:=Group("C4xDic5"); // GroupNames label
G:=SmallGroup(80,11);
// by ID
G=gap.SmallGroup(80,11);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-5,20,46,1604]);
// Polycyclic
G:=Group<a,b,c|a^4=b^10=1,c^2=b^5,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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