direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4×D19, C4⋊1D38, C76⋊C22, D76⋊3C2, C22⋊1D38, D38⋊2C22, C38.5C23, Dic19⋊1C22, C19⋊2(C2×D4), (C2×C38)⋊C22, (D4×C19)⋊2C2, (C4×D19)⋊1C2, C19⋊D4⋊1C2, (C22×D19)⋊2C2, C2.6(C22×D19), SmallGroup(304,31)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4×D19
G = < a,b,c,d | a4=b2=c19=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 520 in 54 conjugacy classes, 25 normal (13 characteristic)
C1, C2, C2, C4, C4, C22, C22, C2×C4, D4, D4, C23, C2×D4, C19, D19, D19, C38, C38, Dic19, C76, D38, D38, D38, C2×C38, C4×D19, D76, C19⋊D4, D4×C19, C22×D19, D4×D19
Quotients: C1, C2, C22, D4, C23, C2×D4, D19, D38, C22×D19, D4×D19
►On 76 points
Generators in S76
(1 42 34 71)(2 43 35 72)(3 44 36 73)(4 45 37 74)(5 46 38 75)(6 47 20 76)(7 48 21 58)(8 49 22 59)(9 50 23 60)(10 51 24 61)(11 52 25 62)(12 53 26 63)(13 54 27 64)(14 55 28 65)(15 56 29 66)(16 57 30 67)(17 39 31 68)(18 40 32 69)(19 41 33 70)
(39 68)(40 69)(41 70)(42 71)(43 72)(44 73)(45 74)(46 75)(47 76)(48 58)(49 59)(50 60)(51 61)(52 62)(53 63)(54 64)(55 65)(56 66)(57 67)
(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)
(1 33)(2 32)(3 31)(4 30)(5 29)(6 28)(7 27)(8 26)(9 25)(10 24)(11 23)(12 22)(13 21)(14 20)(15 38)(16 37)(17 36)(18 35)(19 34)(39 73)(40 72)(41 71)(42 70)(43 69)(44 68)(45 67)(46 66)(47 65)(48 64)(49 63)(50 62)(51 61)(52 60)(53 59)(54 58)(55 76)(56 75)(57 74)
G:=sub<Sym(76)| (1,42,34,71)(2,43,35,72)(3,44,36,73)(4,45,37,74)(5,46,38,75)(6,47,20,76)(7,48,21,58)(8,49,22,59)(9,50,23,60)(10,51,24,61)(11,52,25,62)(12,53,26,63)(13,54,27,64)(14,55,28,65)(15,56,29,66)(16,57,30,67)(17,39,31,68)(18,40,32,69)(19,41,33,70), (39,68)(40,69)(41,70)(42,71)(43,72)(44,73)(45,74)(46,75)(47,76)(48,58)(49,59)(50,60)(51,61)(52,62)(53,63)(54,64)(55,65)(56,66)(57,67), (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), (1,33)(2,32)(3,31)(4,30)(5,29)(6,28)(7,27)(8,26)(9,25)(10,24)(11,23)(12,22)(13,21)(14,20)(15,38)(16,37)(17,36)(18,35)(19,34)(39,73)(40,72)(41,71)(42,70)(43,69)(44,68)(45,67)(46,66)(47,65)(48,64)(49,63)(50,62)(51,61)(52,60)(53,59)(54,58)(55,76)(56,75)(57,74)>;
G:=Group( (1,42,34,71)(2,43,35,72)(3,44,36,73)(4,45,37,74)(5,46,38,75)(6,47,20,76)(7,48,21,58)(8,49,22,59)(9,50,23,60)(10,51,24,61)(11,52,25,62)(12,53,26,63)(13,54,27,64)(14,55,28,65)(15,56,29,66)(16,57,30,67)(17,39,31,68)(18,40,32,69)(19,41,33,70), (39,68)(40,69)(41,70)(42,71)(43,72)(44,73)(45,74)(46,75)(47,76)(48,58)(49,59)(50,60)(51,61)(52,62)(53,63)(54,64)(55,65)(56,66)(57,67), (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), (1,33)(2,32)(3,31)(4,30)(5,29)(6,28)(7,27)(8,26)(9,25)(10,24)(11,23)(12,22)(13,21)(14,20)(15,38)(16,37)(17,36)(18,35)(19,34)(39,73)(40,72)(41,71)(42,70)(43,69)(44,68)(45,67)(46,66)(47,65)(48,64)(49,63)(50,62)(51,61)(52,60)(53,59)(54,58)(55,76)(56,75)(57,74) );
G=PermutationGroup([[(1,42,34,71),(2,43,35,72),(3,44,36,73),(4,45,37,74),(5,46,38,75),(6,47,20,76),(7,48,21,58),(8,49,22,59),(9,50,23,60),(10,51,24,61),(11,52,25,62),(12,53,26,63),(13,54,27,64),(14,55,28,65),(15,56,29,66),(16,57,30,67),(17,39,31,68),(18,40,32,69),(19,41,33,70)], [(39,68),(40,69),(41,70),(42,71),(43,72),(44,73),(45,74),(46,75),(47,76),(48,58),(49,59),(50,60),(51,61),(52,62),(53,63),(54,64),(55,65),(56,66),(57,67)], [(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)], [(1,33),(2,32),(3,31),(4,30),(5,29),(6,28),(7,27),(8,26),(9,25),(10,24),(11,23),(12,22),(13,21),(14,20),(15,38),(16,37),(17,36),(18,35),(19,34),(39,73),(40,72),(41,71),(42,70),(43,69),(44,68),(45,67),(46,66),(47,65),(48,64),(49,63),(50,62),(51,61),(52,60),(53,59),(54,58),(55,76),(56,75),(57,74)]])
55 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 19A | ··· | 19I | 38A | ··· | 38I | 38J | ··· | 38AA | 76A | ··· | 76I |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 19 | ··· | 19 | 38 | ··· | 38 | 38 | ··· | 38 | 76 | ··· | 76 |
| size | 1 | 1 | 2 | 2 | 19 | 19 | 38 | 38 | 2 | 38 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
55 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D19 | D38 | D38 | D4×D19 |
| kernel | D4×D19 | C4×D19 | D76 | C19⋊D4 | D4×C19 | C22×D19 | D19 | D4 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 9 | 9 | 18 | 9 |
Matrix representation of D4×D19 ►in GL4(𝔽229) generated by
| 228 | 0 | 0 | 0 |
| 0 | 228 | 0 | 0 |
| 0 | 0 | 162 | 213 |
| 0 | 0 | 23 | 67 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 192 | 228 |
| 132 | 1 | 0 | 0 |
| 161 | 199 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 164 | 131 | 0 | 0 |
| 202 | 65 | 0 | 0 |
| 0 | 0 | 228 | 0 |
| 0 | 0 | 0 | 228 |
G:=sub<GL(4,GF(229))| [228,0,0,0,0,228,0,0,0,0,162,23,0,0,213,67],[1,0,0,0,0,1,0,0,0,0,1,192,0,0,0,228],[132,161,0,0,1,199,0,0,0,0,1,0,0,0,0,1],[164,202,0,0,131,65,0,0,0,0,228,0,0,0,0,228] >;
D4×D19 in GAP, Magma, Sage, TeX
D_4\times D_{19} % in TeX
G:=Group("D4xD19"); // GroupNames label
G:=SmallGroup(304,31);
// by ID
G=gap.SmallGroup(304,31);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-19,97,7204]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^19=d^2=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=c^-1>;
// generators/relations