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G = C19⋊D4order 152 = 23·19

The semidirect product of C19 and D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C192D4, C22⋊D19, D382C2, Dic19⋊C2, C2.5D38, C38.5C22, (C2×C38)⋊2C2, SmallGroup(152,7)

Series: Derived Chief Lower central Upper central

C1C38 — C19⋊D4
C1C19C38D38 — C19⋊D4
C19C38 — C19⋊D4
C1C2C22

Generators and relations for C19⋊D4
 G = < a,b,c | a19=b4=c2=1, bab-1=cac=a-1, cbc=b-1 >

2C2
38C2
19C4
19C22
2D19
2C38
19D4

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

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

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

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

C19⋊D4 is a maximal subgroup of
D765C2  D4×D19  D42D19  D38⋊C6  C57⋊D4  C19⋊D12  C577D4  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  C577D4

41 conjugacy classes

class 1 2A2B2C 4 19A···19I38A···38AA
order1222419···1938···38
size11238382···22···2

41 irreducible representations

dim11112222
type+++++++
imageC1C2C2C2D4D19D38C19⋊D4
kernelC19⋊D4Dic19D38C2×C38C19C22C2C1
# reps111119918

Matrix representation of C19⋊D4 in GL2(𝔽229) generated by

01
22818
,
3972
87190
,
10
18228
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

Export

Subgroup lattice of C19⋊D4 in TeX

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