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G = C4xD19order 152 = 23·19

Direct product of C4 and D19

direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: C4xD19, C76:2C2, D38.C2, C2.1D38, Dic19:2C2, C38.2C22, C19:1(C2xC4), SmallGroup(152,4)

Series: Derived Chief Lower central Upper central

C1C19 — C4xD19
C1C19C38D38 — C4xD19
C19 — C4xD19
C1C4

Generators and relations for C4xD19
 G = < a,b,c | a4=b19=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 106 in 16 conjugacy classes, 11 normal (9 characteristic)
Quotients: C1, C2, C4, C22, C2xC4, D19, D38, C4xD19
19C2
19C2
19C22
19C4
19C2xC4

Smallest permutation representation of C4xD19
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)]])

C4xD19 is a maximal subgroup of   C8:D19  D76:5C2  D4:2D19  D76:C2  D57:C4
C4xD19 is a maximal quotient of   C8:D19  Dic19:C4  D38:C4  D57:C4

44 conjugacy classes

class 1 2A2B2C4A4B4C4D19A···19I38A···38I76A···76R
order1222444419···1938···3876···76
size1119191119192···22···22···2

44 irreducible representations

dim11111222
type++++++
imageC1C2C2C2C4D19D38C4xD19
kernelC4xD19Dic19C76D38D19C4C2C1
# reps111149918

Matrix representation of C4xD19 in GL2(F37) generated by

60
06
,
3629
299
,
927
828
G:=sub<GL(2,GF(37))| [6,0,0,6],[36,29,29,9],[9,8,27,28] >;

C4xD19 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

Export

Subgroup lattice of C4xD19 in TeX

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