direct product, metacyclic, supersoluble, monomial
Aliases: D4×C13⋊C3, C52⋊3C6, (D4×C13)⋊C3, C13⋊3(C3×D4), (C2×C26)⋊5C6, C26.7(C2×C6), C4⋊(C2×C13⋊C3), (C4×C13⋊C3)⋊3C2, C22⋊2(C2×C13⋊C3), (C22×C13⋊C3)⋊3C2, C2.2(C22×C13⋊C3), (C2×C13⋊C3).7C22, SmallGroup(312,23)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C13 — C26 — C2×C13⋊C3 — C22×C13⋊C3 — D4×C13⋊C3 |
Generators and relations for D4×C13⋊C3
G = < a,b,c,d | a4=b2=c13=d3=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c9 >
Smallest permutation representation of D4×C13⋊C3
►On 52 points
Generators in S52
(1 40 14 27)(2 41 15 28)(3 42 16 29)(4 43 17 30)(5 44 18 31)(6 45 19 32)(7 46 20 33)(8 47 21 34)(9 48 22 35)(10 49 23 36)(11 50 24 37)(12 51 25 38)(13 52 26 39)
(1 27)(2 28)(3 29)(4 30)(5 31)(6 32)(7 33)(8 34)(9 35)(10 36)(11 37)(12 38)(13 39)(14 40)(15 41)(16 42)(17 43)(18 44)(19 45)(20 46)(21 47)(22 48)(23 49)(24 50)(25 51)(26 52)
(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)
(2 4 10)(3 7 6)(5 13 11)(8 9 12)(15 17 23)(16 20 19)(18 26 24)(21 22 25)(28 30 36)(29 33 32)(31 39 37)(34 35 38)(41 43 49)(42 46 45)(44 52 50)(47 48 51)
G:=sub<Sym(52)| (1,40,14,27)(2,41,15,28)(3,42,16,29)(4,43,17,30)(5,44,18,31)(6,45,19,32)(7,46,20,33)(8,47,21,34)(9,48,22,35)(10,49,23,36)(11,50,24,37)(12,51,25,38)(13,52,26,39), (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,33)(8,34)(9,35)(10,36)(11,37)(12,38)(13,39)(14,40)(15,41)(16,42)(17,43)(18,44)(19,45)(20,46)(21,47)(22,48)(23,49)(24,50)(25,51)(26,52), (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), (2,4,10)(3,7,6)(5,13,11)(8,9,12)(15,17,23)(16,20,19)(18,26,24)(21,22,25)(28,30,36)(29,33,32)(31,39,37)(34,35,38)(41,43,49)(42,46,45)(44,52,50)(47,48,51)>;
G:=Group( (1,40,14,27)(2,41,15,28)(3,42,16,29)(4,43,17,30)(5,44,18,31)(6,45,19,32)(7,46,20,33)(8,47,21,34)(9,48,22,35)(10,49,23,36)(11,50,24,37)(12,51,25,38)(13,52,26,39), (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,33)(8,34)(9,35)(10,36)(11,37)(12,38)(13,39)(14,40)(15,41)(16,42)(17,43)(18,44)(19,45)(20,46)(21,47)(22,48)(23,49)(24,50)(25,51)(26,52), (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), (2,4,10)(3,7,6)(5,13,11)(8,9,12)(15,17,23)(16,20,19)(18,26,24)(21,22,25)(28,30,36)(29,33,32)(31,39,37)(34,35,38)(41,43,49)(42,46,45)(44,52,50)(47,48,51) );
G=PermutationGroup([[(1,40,14,27),(2,41,15,28),(3,42,16,29),(4,43,17,30),(5,44,18,31),(6,45,19,32),(7,46,20,33),(8,47,21,34),(9,48,22,35),(10,49,23,36),(11,50,24,37),(12,51,25,38),(13,52,26,39)], [(1,27),(2,28),(3,29),(4,30),(5,31),(6,32),(7,33),(8,34),(9,35),(10,36),(11,37),(12,38),(13,39),(14,40),(15,41),(16,42),(17,43),(18,44),(19,45),(20,46),(21,47),(22,48),(23,49),(24,50),(25,51),(26,52)], [(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)], [(2,4,10),(3,7,6),(5,13,11),(8,9,12),(15,17,23),(16,20,19),(18,26,24),(21,22,25),(28,30,36),(29,33,32),(31,39,37),(34,35,38),(41,43,49),(42,46,45),(44,52,50),(47,48,51)]])
35 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4 | 6A | 6B | 6C | 6D | 6E | 6F | 12A | 12B | 13A | 13B | 13C | 13D | 26A | 26B | 26C | 26D | 26E | ··· | 26L | 52A | 52B | 52C | 52D |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 13 | 13 | 13 | 13 | 26 | 26 | 26 | 26 | 26 | ··· | 26 | 52 | 52 | 52 | 52 |
| size | 1 | 1 | 2 | 2 | 13 | 13 | 2 | 13 | 13 | 26 | 26 | 26 | 26 | 26 | 26 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 6 | ··· | 6 | 6 | 6 | 6 | 6 |
35 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 3 | 3 | 3 | 6 |
| type | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | D4 | C3×D4 | C13⋊C3 | C2×C13⋊C3 | C2×C13⋊C3 | D4×C13⋊C3 |
| kernel | D4×C13⋊C3 | C4×C13⋊C3 | C22×C13⋊C3 | D4×C13 | C52 | C2×C26 | C13⋊C3 | C13 | D4 | C4 | C22 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 1 | 2 | 4 | 4 | 8 | 4 |
Matrix representation of D4×C13⋊C3 ►in GL5(𝔽157)
| 0 | 1 | 0 | 0 | 0 |
| 156 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 135 | 46 | 1 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 144 | 0 | 0 | 0 | 0 |
| 0 | 144 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 110 | 134 | 46 |
| 0 | 0 | 97 | 47 | 22 |
G:=sub<GL(5,GF(157))| [0,156,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,135,1,0,0,0,46,0,1,0,0,1,0,0],[144,0,0,0,0,0,144,0,0,0,0,0,1,110,97,0,0,0,134,47,0,0,0,46,22] >;
D4×C13⋊C3 in GAP, Magma, Sage, TeX
D_4\times C_{13}\rtimes C_3 % in TeX
G:=Group("D4xC13:C3"); // GroupNames label
G:=SmallGroup(312,23);
// by ID
G=gap.SmallGroup(312,23);
# by ID
G:=PCGroup([5,-2,-2,-3,-2,-13,141,464]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^13=d^3=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^9>;
// generators/relations
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