metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C19⋊2D4, C22⋊D19, D38⋊2C2, Dic19⋊C2, C2.5D38, C38.5C22, (C2×C38)⋊2C2, SmallGroup(152,7)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C19⋊D4
G = < a,b,c | a19=b4=c2=1, bab-1=cac=a-1, cbc=b-1 >
Smallest permutation representation of C19⋊D4
►On 76 points
Generators in S76
(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 55 21 75)(2 54 22 74)(3 53 23 73)(4 52 24 72)(5 51 25 71)(6 50 26 70)(7 49 27 69)(8 48 28 68)(9 47 29 67)(10 46 30 66)(11 45 31 65)(12 44 32 64)(13 43 33 63)(14 42 34 62)(15 41 35 61)(16 40 36 60)(17 39 37 59)(18 57 38 58)(19 56 20 76)
(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 11)(20 22)(23 38)(24 37)(25 36)(26 35)(27 34)(28 33)(29 32)(30 31)(39 72)(40 71)(41 70)(42 69)(43 68)(44 67)(45 66)(46 65)(47 64)(48 63)(49 62)(50 61)(51 60)(52 59)(53 58)(54 76)(55 75)(56 74)(57 73)
G:=sub<Sym(76)| (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), (1,55,21,75)(2,54,22,74)(3,53,23,73)(4,52,24,72)(5,51,25,71)(6,50,26,70)(7,49,27,69)(8,48,28,68)(9,47,29,67)(10,46,30,66)(11,45,31,65)(12,44,32,64)(13,43,33,63)(14,42,34,62)(15,41,35,61)(16,40,36,60)(17,39,37,59)(18,57,38,58)(19,56,20,76), (2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(20,22)(23,38)(24,37)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)(39,72)(40,71)(41,70)(42,69)(43,68)(44,67)(45,66)(46,65)(47,64)(48,63)(49,62)(50,61)(51,60)(52,59)(53,58)(54,76)(55,75)(56,74)(57,73)>;
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)(39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57)(58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76), (1,55,21,75)(2,54,22,74)(3,53,23,73)(4,52,24,72)(5,51,25,71)(6,50,26,70)(7,49,27,69)(8,48,28,68)(9,47,29,67)(10,46,30,66)(11,45,31,65)(12,44,32,64)(13,43,33,63)(14,42,34,62)(15,41,35,61)(16,40,36,60)(17,39,37,59)(18,57,38,58)(19,56,20,76), (2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(20,22)(23,38)(24,37)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)(39,72)(40,71)(41,70)(42,69)(43,68)(44,67)(45,66)(46,65)(47,64)(48,63)(49,62)(50,61)(51,60)(52,59)(53,58)(54,76)(55,75)(56,74)(57,73) );
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),(39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57),(58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76)], [(1,55,21,75),(2,54,22,74),(3,53,23,73),(4,52,24,72),(5,51,25,71),(6,50,26,70),(7,49,27,69),(8,48,28,68),(9,47,29,67),(10,46,30,66),(11,45,31,65),(12,44,32,64),(13,43,33,63),(14,42,34,62),(15,41,35,61),(16,40,36,60),(17,39,37,59),(18,57,38,58),(19,56,20,76)], [(2,19),(3,18),(4,17),(5,16),(6,15),(7,14),(8,13),(9,12),(10,11),(20,22),(23,38),(24,37),(25,36),(26,35),(27,34),(28,33),(29,32),(30,31),(39,72),(40,71),(41,70),(42,69),(43,68),(44,67),(45,66),(46,65),(47,64),(48,63),(49,62),(50,61),(51,60),(52,59),(53,58),(54,76),(55,75),(56,74),(57,73)]])
C19⋊D4 is a maximal subgroup of
D76⋊5C2 D4×D19 D4⋊2D19 D38⋊C6 C57⋊D4 C19⋊D12 C57⋊7D4 C19⋊S4
C19⋊D4 is a maximal quotient of
Dic19⋊C4 D38⋊C4 D4⋊D19 D4.D19 Q8⋊D19 C19⋊Q16 C23.D19 C57⋊D4 C19⋊D12 C57⋊7D4
41 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4 | 19A | ··· | 19I | 38A | ··· | 38AA |
| order | 1 | 2 | 2 | 2 | 4 | 19 | ··· | 19 | 38 | ··· | 38 |
| size | 1 | 1 | 2 | 38 | 38 | 2 | ··· | 2 | 2 | ··· | 2 |
41 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | D4 | D19 | D38 | C19⋊D4 |
| kernel | C19⋊D4 | Dic19 | D38 | C2×C38 | C19 | C22 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 9 | 9 | 18 |
Matrix representation of C19⋊D4 ►in GL2(𝔽229) generated by
| 0 | 1 |
| 228 | 18 |
| 39 | 72 |
| 87 | 190 |
| 1 | 0 |
| 18 | 228 |
G:=sub<GL(2,GF(229))| [0,228,1,18],[39,87,72,190],[1,18,0,228] >;
C19⋊D4 in GAP, Magma, Sage, TeX
C_{19}\rtimes D_4 % in TeX
G:=Group("C19:D4"); // GroupNames label
G:=SmallGroup(152,7);
// by ID
G=gap.SmallGroup(152,7);
# by ID
G:=PCGroup([4,-2,-2,-2,-19,49,2307]);
// Polycyclic
G:=Group<a,b,c|a^19=b^4=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations
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