direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C5×C4○D8, D8⋊3C10, Q16⋊3C10, C20.69D4, SD16⋊3C10, C20.47C23, C40.28C22, (C2×C8)⋊4C10, (C5×D8)⋊7C2, (C2×C40)⋊12C2, C4○D4⋊1C10, C8.6(C2×C10), (C5×Q16)⋊7C2, C4.20(C5×D4), (C5×SD16)⋊7C2, D4.2(C2×C10), C2.14(D4×C10), C10.77(C2×D4), (C2×C10).11D4, Q8.2(C2×C10), C22.1(C5×D4), C4.4(C22×C10), (C5×D4).12C22, (C5×Q8).13C22, (C2×C20).132C22, (C5×C4○D4)⋊6C2, (C2×C4).28(C2×C10), SmallGroup(160,196)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C4○D8
G = < a,b,c,d | a5=b4=d2=1, c4=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c3 >
Subgroups: 92 in 62 conjugacy classes, 40 normal (24 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, Q8, C10, C10, C2×C8, D8, SD16, Q16, C4○D4, C20, C20, C2×C10, C2×C10, C4○D8, C40, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C2×C40, C5×D8, C5×SD16, C5×Q16, C5×C4○D4, C5×C4○D8
Quotients: C1, C2, C22, C5, D4, C23, C10, C2×D4, C2×C10, C4○D8, C5×D4, C22×C10, D4×C10, C5×C4○D8
►On 80 points
(1 23 69 49 27)(2 24 70 50 28)(3 17 71 51 29)(4 18 72 52 30)(5 19 65 53 31)(6 20 66 54 32)(7 21 67 55 25)(8 22 68 56 26)(9 78 58 36 47)(10 79 59 37 48)(11 80 60 38 41)(12 73 61 39 42)(13 74 62 40 43)(14 75 63 33 44)(15 76 64 34 45)(16 77 57 35 46)
(1 39 5 35)(2 40 6 36)(3 33 7 37)(4 34 8 38)(9 70 13 66)(10 71 14 67)(11 72 15 68)(12 65 16 69)(17 44 21 48)(18 45 22 41)(19 46 23 42)(20 47 24 43)(25 59 29 63)(26 60 30 64)(27 61 31 57)(28 62 32 58)(49 73 53 77)(50 74 54 78)(51 75 55 79)(52 76 56 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)
(2 8)(3 7)(4 6)(9 15)(10 14)(11 13)(17 21)(18 20)(22 24)(25 29)(26 28)(30 32)(33 37)(34 36)(38 40)(41 43)(44 48)(45 47)(50 56)(51 55)(52 54)(58 64)(59 63)(60 62)(66 72)(67 71)(68 70)(74 80)(75 79)(76 78)
G:=sub<Sym(80)| (1,23,69,49,27)(2,24,70,50,28)(3,17,71,51,29)(4,18,72,52,30)(5,19,65,53,31)(6,20,66,54,32)(7,21,67,55,25)(8,22,68,56,26)(9,78,58,36,47)(10,79,59,37,48)(11,80,60,38,41)(12,73,61,39,42)(13,74,62,40,43)(14,75,63,33,44)(15,76,64,34,45)(16,77,57,35,46), (1,39,5,35)(2,40,6,36)(3,33,7,37)(4,34,8,38)(9,70,13,66)(10,71,14,67)(11,72,15,68)(12,65,16,69)(17,44,21,48)(18,45,22,41)(19,46,23,42)(20,47,24,43)(25,59,29,63)(26,60,30,64)(27,61,31,57)(28,62,32,58)(49,73,53,77)(50,74,54,78)(51,75,55,79)(52,76,56,80), (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), (2,8)(3,7)(4,6)(9,15)(10,14)(11,13)(17,21)(18,20)(22,24)(25,29)(26,28)(30,32)(33,37)(34,36)(38,40)(41,43)(44,48)(45,47)(50,56)(51,55)(52,54)(58,64)(59,63)(60,62)(66,72)(67,71)(68,70)(74,80)(75,79)(76,78)>;
G:=Group( (1,23,69,49,27)(2,24,70,50,28)(3,17,71,51,29)(4,18,72,52,30)(5,19,65,53,31)(6,20,66,54,32)(7,21,67,55,25)(8,22,68,56,26)(9,78,58,36,47)(10,79,59,37,48)(11,80,60,38,41)(12,73,61,39,42)(13,74,62,40,43)(14,75,63,33,44)(15,76,64,34,45)(16,77,57,35,46), (1,39,5,35)(2,40,6,36)(3,33,7,37)(4,34,8,38)(9,70,13,66)(10,71,14,67)(11,72,15,68)(12,65,16,69)(17,44,21,48)(18,45,22,41)(19,46,23,42)(20,47,24,43)(25,59,29,63)(26,60,30,64)(27,61,31,57)(28,62,32,58)(49,73,53,77)(50,74,54,78)(51,75,55,79)(52,76,56,80), (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), (2,8)(3,7)(4,6)(9,15)(10,14)(11,13)(17,21)(18,20)(22,24)(25,29)(26,28)(30,32)(33,37)(34,36)(38,40)(41,43)(44,48)(45,47)(50,56)(51,55)(52,54)(58,64)(59,63)(60,62)(66,72)(67,71)(68,70)(74,80)(75,79)(76,78) );
G=PermutationGroup([[(1,23,69,49,27),(2,24,70,50,28),(3,17,71,51,29),(4,18,72,52,30),(5,19,65,53,31),(6,20,66,54,32),(7,21,67,55,25),(8,22,68,56,26),(9,78,58,36,47),(10,79,59,37,48),(11,80,60,38,41),(12,73,61,39,42),(13,74,62,40,43),(14,75,63,33,44),(15,76,64,34,45),(16,77,57,35,46)], [(1,39,5,35),(2,40,6,36),(3,33,7,37),(4,34,8,38),(9,70,13,66),(10,71,14,67),(11,72,15,68),(12,65,16,69),(17,44,21,48),(18,45,22,41),(19,46,23,42),(20,47,24,43),(25,59,29,63),(26,60,30,64),(27,61,31,57),(28,62,32,58),(49,73,53,77),(50,74,54,78),(51,75,55,79),(52,76,56,80)], [(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)], [(2,8),(3,7),(4,6),(9,15),(10,14),(11,13),(17,21),(18,20),(22,24),(25,29),(26,28),(30,32),(33,37),(34,36),(38,40),(41,43),(44,48),(45,47),(50,56),(51,55),(52,54),(58,64),(59,63),(60,62),(66,72),(67,71),(68,70),(74,80),(75,79),(76,78)]])
C5×C4○D8 is a maximal subgroup of
D8⋊2Dic5 C20.58D8 D8⋊D10 C40.30C23 C40.31C23 D8⋊5Dic5 D8⋊4Dic5 Q16⋊D10 D8⋊15D10 D8⋊11D10 D20.47D4
C5×C4○D8 is a maximal quotient of
D8×C20 SD16×C20 Q16×C20
70 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 5C | 5D | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 10I | ··· | 10P | 20A | ··· | 20H | 20I | 20J | 20K | 20L | 20M | ··· | 20T | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 2 | 4 | 4 | 1 | 1 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 |
70 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | C10 | D4 | D4 | C4○D8 | C5×D4 | C5×D4 | C5×C4○D8 |
| kernel | C5×C4○D8 | C2×C40 | C5×D8 | C5×SD16 | C5×Q16 | C5×C4○D4 | C4○D8 | C2×C8 | D8 | SD16 | Q16 | C4○D4 | C20 | C2×C10 | C5 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 2 | 1 | 2 | 4 | 4 | 4 | 8 | 4 | 8 | 1 | 1 | 4 | 4 | 4 | 16 |
Matrix representation of C5×C4○D8 ►in GL2(𝔽41) generated by
| 10 | 0 |
| 0 | 10 |
| 9 | 0 |
| 0 | 9 |
| 12 | 29 |
| 12 | 12 |
| 1 | 0 |
| 0 | 40 |
G:=sub<GL(2,GF(41))| [10,0,0,10],[9,0,0,9],[12,12,29,12],[1,0,0,40] >;
C5×C4○D8 in GAP, Magma, Sage, TeX
C_5\times C_4\circ D_8
% in TeX
G:=Group("C5xC4oD8"); // GroupNames label
G:=SmallGroup(160,196);
// by ID
G=gap.SmallGroup(160,196);
# by ID
G:=PCGroup([6,-2,-2,-2,-5,-2,-2,505,374,3604,1810,88]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^4=d^2=1,c^4=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^2*c^3>;
// generators/relations