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G = D4×C19order 152 = 23·19

Direct product of C19 and D4

direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary

Aliases: D4×C19, C4⋊C38, C763C2, C22⋊C38, C38.6C22, (C2×C38)⋊1C2, C2.1(C2×C38), SmallGroup(152,9)

Series: Derived Chief Lower central Upper central

C1C2 — D4×C19
C1C2C38C2×C38 — D4×C19
C1C2 — D4×C19
C1C38 — D4×C19

Generators and relations for D4×C19
 G = < a,b,c | a19=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >

2C2
2C2
2C38
2C38

Smallest permutation representation of D4×C19
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 48 28 71)(2 49 29 72)(3 50 30 73)(4 51 31 74)(5 52 32 75)(6 53 33 76)(7 54 34 58)(8 55 35 59)(9 56 36 60)(10 57 37 61)(11 39 38 62)(12 40 20 63)(13 41 21 64)(14 42 22 65)(15 43 23 66)(16 44 24 67)(17 45 25 68)(18 46 26 69)(19 47 27 70)
(39 62)(40 63)(41 64)(42 65)(43 66)(44 67)(45 68)(46 69)(47 70)(48 71)(49 72)(50 73)(51 74)(52 75)(53 76)(54 58)(55 59)(56 60)(57 61)

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

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

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

D4×C19 is a maximal subgroup of   D4⋊D19  D4.D19  D42D19

95 conjugacy classes

class 1 2A2B2C 4 19A···19R38A···38R38S···38BB76A···76R
order1222419···1938···3838···3876···76
size112221···11···12···22···2

95 irreducible representations

dim11111122
type++++
imageC1C2C2C19C38C38D4D4×C19
kernelD4×C19C76C2×C38D4C4C22C19C1
# reps112181836118

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

600
060
,
15113
227214
,
115
0228
G:=sub<GL(2,GF(229))| [60,0,0,60],[15,227,113,214],[1,0,15,228] >;

D4×C19 in GAP, Magma, Sage, TeX

D_4\times C_{19}
% in TeX

G:=Group("D4xC19");
// GroupNames label

G:=SmallGroup(152,9);
// by ID

G=gap.SmallGroup(152,9);
# by ID

G:=PCGroup([4,-2,-2,-19,-2,625]);
// Polycyclic

G:=Group<a,b,c|a^19=b^4=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of D4×C19 in TeX

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