direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C5×C4○D4, D4○C20, Q8○C20, D4⋊2C10, Q8⋊2C10, C20.21C22, C10.13C23, C4○(C5×D4), C4○(C5×Q8), C20○(C5×D4), C20○(C5×Q8), (C2×C20)⋊7C2, (C2×C4)⋊3C10, (C5×D4)⋊5C2, (C5×Q8)⋊5C2, C4.5(C2×C10), C22.(C2×C10), C2.3(C22×C10), (C2×C10).2C22, SmallGroup(80,48)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C4○D4
G = < a,b,c,d | a5=b4=d2=1, c2=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c >
Smallest permutation representation of C5×C4○D4
►On 40 points
(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)
(1 23 13 17)(2 24 14 18)(3 25 15 19)(4 21 11 20)(5 22 12 16)(6 31 40 30)(7 32 36 26)(8 33 37 27)(9 34 38 28)(10 35 39 29)
(1 17 13 23)(2 18 14 24)(3 19 15 25)(4 20 11 21)(5 16 12 22)(6 31 40 30)(7 32 36 26)(8 33 37 27)(9 34 38 28)(10 35 39 29)
(1 37)(2 38)(3 39)(4 40)(5 36)(6 11)(7 12)(8 13)(9 14)(10 15)(16 32)(17 33)(18 34)(19 35)(20 31)(21 30)(22 26)(23 27)(24 28)(25 29)
G:=sub<Sym(40)| (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), (1,23,13,17)(2,24,14,18)(3,25,15,19)(4,21,11,20)(5,22,12,16)(6,31,40,30)(7,32,36,26)(8,33,37,27)(9,34,38,28)(10,35,39,29), (1,17,13,23)(2,18,14,24)(3,19,15,25)(4,20,11,21)(5,16,12,22)(6,31,40,30)(7,32,36,26)(8,33,37,27)(9,34,38,28)(10,35,39,29), (1,37)(2,38)(3,39)(4,40)(5,36)(6,11)(7,12)(8,13)(9,14)(10,15)(16,32)(17,33)(18,34)(19,35)(20,31)(21,30)(22,26)(23,27)(24,28)(25,29)>;
G:=Group( (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), (1,23,13,17)(2,24,14,18)(3,25,15,19)(4,21,11,20)(5,22,12,16)(6,31,40,30)(7,32,36,26)(8,33,37,27)(9,34,38,28)(10,35,39,29), (1,17,13,23)(2,18,14,24)(3,19,15,25)(4,20,11,21)(5,16,12,22)(6,31,40,30)(7,32,36,26)(8,33,37,27)(9,34,38,28)(10,35,39,29), (1,37)(2,38)(3,39)(4,40)(5,36)(6,11)(7,12)(8,13)(9,14)(10,15)(16,32)(17,33)(18,34)(19,35)(20,31)(21,30)(22,26)(23,27)(24,28)(25,29) );
G=PermutationGroup([[(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)], [(1,23,13,17),(2,24,14,18),(3,25,15,19),(4,21,11,20),(5,22,12,16),(6,31,40,30),(7,32,36,26),(8,33,37,27),(9,34,38,28),(10,35,39,29)], [(1,17,13,23),(2,18,14,24),(3,19,15,25),(4,20,11,21),(5,16,12,22),(6,31,40,30),(7,32,36,26),(8,33,37,27),(9,34,38,28),(10,35,39,29)], [(1,37),(2,38),(3,39),(4,40),(5,36),(6,11),(7,12),(8,13),(9,14),(10,15),(16,32),(17,33),(18,34),(19,35),(20,31),(21,30),(22,26),(23,27),(24,28),(25,29)]])
C5×C4○D4 is a maximal subgroup of
D4⋊2Dic5 D4.Dic5 D4⋊D10 D4.8D10 D4.9D10 D4⋊8D10 D4.10D10
C5×C4○D4 is a maximal quotient of D4×C20 Q8×C20
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 5C | 5D | 10A | 10B | 10C | 10D | 10E | ··· | 10P | 20A | ··· | 20H | 20I | ··· | 20T |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C4○D4 | C5×C4○D4 |
| kernel | C5×C4○D4 | C2×C20 | C5×D4 | C5×Q8 | C4○D4 | C2×C4 | D4 | Q8 | C5 | C1 |
| # reps | 1 | 3 | 3 | 1 | 4 | 12 | 12 | 4 | 2 | 8 |
Matrix representation of C5×C4○D4 ►in GL2(𝔽41) generated by
| 16 | 0 |
| 0 | 16 |
| 9 | 0 |
| 0 | 9 |
| 32 | 13 |
| 0 | 9 |
| 13 | 18 |
| 18 | 28 |
G:=sub<GL(2,GF(41))| [16,0,0,16],[9,0,0,9],[32,0,13,9],[13,18,18,28] >;
C5×C4○D4 in GAP, Magma, Sage, TeX
C_5\times C_4\circ D_4
% in TeX
G:=Group("C5xC4oD4"); // GroupNames label
G:=SmallGroup(80,48);
// by ID
G=gap.SmallGroup(80,48);
# by ID
G:=PCGroup([5,-2,-2,-2,-5,-2,421,162]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^4=d^2=1,c^2=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>;
// generators/relations
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