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G = C4×D19order 152 = 23·19

Direct product of C4 and D19

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

Aliases: C4×D19, C762C2, D38.C2, C2.1D38, Dic192C2, C38.2C22, C191(C2×C4), SmallGroup(152,4)

Series: Derived Chief Lower central Upper central

C1C19 — C4×D19
C1C19C38D38 — C4×D19
C19 — C4×D19
C1C4

Generators and relations for C4×D19
 G = < a,b,c | a4=b19=c2=1, ab=ba, ac=ca, cbc=b-1 >

19C2
19C2
19C22
19C4
19C2×C4

Smallest permutation representation of C4×D19
On 76 points
Generators in S76
(1 62 28 42)(2 63 29 43)(3 64 30 44)(4 65 31 45)(5 66 32 46)(6 67 33 47)(7 68 34 48)(8 69 35 49)(9 70 36 50)(10 71 37 51)(11 72 38 52)(12 73 20 53)(13 74 21 54)(14 75 22 55)(15 76 23 56)(16 58 24 57)(17 59 25 39)(18 60 26 40)(19 61 27 41)
(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 35)(21 34)(22 33)(23 32)(24 31)(25 30)(26 29)(27 28)(36 38)(39 44)(40 43)(41 42)(45 57)(46 56)(47 55)(48 54)(49 53)(50 52)(58 65)(59 64)(60 63)(61 62)(66 76)(67 75)(68 74)(69 73)(70 72)

G:=sub<Sym(76)| (1,62,28,42)(2,63,29,43)(3,64,30,44)(4,65,31,45)(5,66,32,46)(6,67,33,47)(7,68,34,48)(8,69,35,49)(9,70,36,50)(10,71,37,51)(11,72,38,52)(12,73,20,53)(13,74,21,54)(14,75,22,55)(15,76,23,56)(16,58,24,57)(17,59,25,39)(18,60,26,40)(19,61,27,41), (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)(20,35)(21,34)(22,33)(23,32)(24,31)(25,30)(26,29)(27,28)(36,38)(39,44)(40,43)(41,42)(45,57)(46,56)(47,55)(48,54)(49,53)(50,52)(58,65)(59,64)(60,63)(61,62)(66,76)(67,75)(68,74)(69,73)(70,72)>;

G:=Group( (1,62,28,42)(2,63,29,43)(3,64,30,44)(4,65,31,45)(5,66,32,46)(6,67,33,47)(7,68,34,48)(8,69,35,49)(9,70,36,50)(10,71,37,51)(11,72,38,52)(12,73,20,53)(13,74,21,54)(14,75,22,55)(15,76,23,56)(16,58,24,57)(17,59,25,39)(18,60,26,40)(19,61,27,41), (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)(20,35)(21,34)(22,33)(23,32)(24,31)(25,30)(26,29)(27,28)(36,38)(39,44)(40,43)(41,42)(45,57)(46,56)(47,55)(48,54)(49,53)(50,52)(58,65)(59,64)(60,63)(61,62)(66,76)(67,75)(68,74)(69,73)(70,72) );

G=PermutationGroup([(1,62,28,42),(2,63,29,43),(3,64,30,44),(4,65,31,45),(5,66,32,46),(6,67,33,47),(7,68,34,48),(8,69,35,49),(9,70,36,50),(10,71,37,51),(11,72,38,52),(12,73,20,53),(13,74,21,54),(14,75,22,55),(15,76,23,56),(16,58,24,57),(17,59,25,39),(18,60,26,40),(19,61,27,41)], [(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),(20,35),(21,34),(22,33),(23,32),(24,31),(25,30),(26,29),(27,28),(36,38),(39,44),(40,43),(41,42),(45,57),(46,56),(47,55),(48,54),(49,53),(50,52),(58,65),(59,64),(60,63),(61,62),(66,76),(67,75),(68,74),(69,73),(70,72)])

C4×D19 is a maximal subgroup of   C8⋊D19  D765C2  D42D19  D76⋊C2  D57⋊C4
C4×D19 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++++++
imageC1C2C2C2C4D19D38C4×D19
kernelC4×D19Dic19C76D38D19C4C2C1
# reps111149918

Matrix representation of C4×D19 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] >;

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

Export

Subgroup lattice of C4×D19 in TeX

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