direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C4×D19, C76⋊2C2, D38.C2, C2.1D38, Dic19⋊2C2, C38.2C22, C19⋊1(C2×C4), SmallGroup(152,4)
Series: Derived ►Chief ►Lower central ►Upper central
| C19 — C4×D19 |
Generators and relations for C4×D19
G = < a,b,c | a4=b19=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C4×D19
►On 76 points
Generators in S76
(1 59 27 40)(2 60 28 41)(3 61 29 42)(4 62 30 43)(5 63 31 44)(6 64 32 45)(7 65 33 46)(8 66 34 47)(9 67 35 48)(10 68 36 49)(11 69 37 50)(12 70 38 51)(13 71 20 52)(14 72 21 53)(15 73 22 54)(16 74 23 55)(17 75 24 56)(18 76 25 57)(19 58 26 39)
(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 19)(2 18)(3 17)(4 16)(5 15)(6 14)(7 13)(8 12)(9 11)(20 33)(21 32)(22 31)(23 30)(24 29)(25 28)(26 27)(34 38)(35 37)(39 40)(41 57)(42 56)(43 55)(44 54)(45 53)(46 52)(47 51)(48 50)(58 59)(60 76)(61 75)(62 74)(63 73)(64 72)(65 71)(66 70)(67 69)
G:=sub<Sym(76)| (1,59,27,40)(2,60,28,41)(3,61,29,42)(4,62,30,43)(5,63,31,44)(6,64,32,45)(7,65,33,46)(8,66,34,47)(9,67,35,48)(10,68,36,49)(11,69,37,50)(12,70,38,51)(13,71,20,52)(14,72,21,53)(15,73,22,54)(16,74,23,55)(17,75,24,56)(18,76,25,57)(19,58,26,39), (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), (1,19)(2,18)(3,17)(4,16)(5,15)(6,14)(7,13)(8,12)(9,11)(20,33)(21,32)(22,31)(23,30)(24,29)(25,28)(26,27)(34,38)(35,37)(39,40)(41,57)(42,56)(43,55)(44,54)(45,53)(46,52)(47,51)(48,50)(58,59)(60,76)(61,75)(62,74)(63,73)(64,72)(65,71)(66,70)(67,69)>;
G:=Group( (1,59,27,40)(2,60,28,41)(3,61,29,42)(4,62,30,43)(5,63,31,44)(6,64,32,45)(7,65,33,46)(8,66,34,47)(9,67,35,48)(10,68,36,49)(11,69,37,50)(12,70,38,51)(13,71,20,52)(14,72,21,53)(15,73,22,54)(16,74,23,55)(17,75,24,56)(18,76,25,57)(19,58,26,39), (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), (1,19)(2,18)(3,17)(4,16)(5,15)(6,14)(7,13)(8,12)(9,11)(20,33)(21,32)(22,31)(23,30)(24,29)(25,28)(26,27)(34,38)(35,37)(39,40)(41,57)(42,56)(43,55)(44,54)(45,53)(46,52)(47,51)(48,50)(58,59)(60,76)(61,75)(62,74)(63,73)(64,72)(65,71)(66,70)(67,69) );
G=PermutationGroup([[(1,59,27,40),(2,60,28,41),(3,61,29,42),(4,62,30,43),(5,63,31,44),(6,64,32,45),(7,65,33,46),(8,66,34,47),(9,67,35,48),(10,68,36,49),(11,69,37,50),(12,70,38,51),(13,71,20,52),(14,72,21,53),(15,73,22,54),(16,74,23,55),(17,75,24,56),(18,76,25,57),(19,58,26,39)], [(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)], [(1,19),(2,18),(3,17),(4,16),(5,15),(6,14),(7,13),(8,12),(9,11),(20,33),(21,32),(22,31),(23,30),(24,29),(25,28),(26,27),(34,38),(35,37),(39,40),(41,57),(42,56),(43,55),(44,54),(45,53),(46,52),(47,51),(48,50),(58,59),(60,76),(61,75),(62,74),(63,73),(64,72),(65,71),(66,70),(67,69)]])
C4×D19 is a maximal subgroup of
C8⋊D19 D76⋊5C2 D4⋊2D19 D76⋊C2 D57⋊C4
C4×D19 is a maximal quotient of C8⋊D19 Dic19⋊C4 D38⋊C4 D57⋊C4
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 19A | ··· | 19I | 38A | ··· | 38I | 76A | ··· | 76R |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 19 | ··· | 19 | 38 | ··· | 38 | 76 | ··· | 76 |
| size | 1 | 1 | 19 | 19 | 1 | 1 | 19 | 19 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C4 | D19 | D38 | C4×D19 |
| kernel | C4×D19 | Dic19 | C76 | D38 | D19 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 9 | 9 | 18 |
Matrix representation of C4×D19 ►in GL2(𝔽37) generated by
| 6 | 0 |
| 0 | 6 |
| 36 | 29 |
| 29 | 9 |
| 9 | 27 |
| 8 | 28 |
G:=sub<GL(2,GF(37))| [6,0,0,6],[36,29,29,9],[9,8,27,28] >;
C4×D19 in GAP, Magma, Sage, TeX
C_4\times D_{19} % in TeX
G:=Group("C4xD19"); // GroupNames label
G:=SmallGroup(152,4);
// by ID
G=gap.SmallGroup(152,4);
# by ID
G:=PCGroup([4,-2,-2,-2,-19,21,2307]);
// Polycyclic
G:=Group<a,b,c|a^4=b^19=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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