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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

 Derived series C1 — C2 — D4×C19
 Chief series C1 — C2 — C38 — C2×C38 — D4×C19
 Lower central C1 — C2 — D4×C19
 Upper central C1 — C38 — D4×C19

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

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

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

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

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

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

95 conjugacy classes

 class 1 2A 2B 2C 4 19A ··· 19R 38A ··· 38R 38S ··· 38BB 76A ··· 76R order 1 2 2 2 4 19 ··· 19 38 ··· 38 38 ··· 38 76 ··· 76 size 1 1 2 2 2 1 ··· 1 1 ··· 1 2 ··· 2 2 ··· 2

95 irreducible representations

 dim 1 1 1 1 1 1 2 2 type + + + + image C1 C2 C2 C19 C38 C38 D4 D4×C19 kernel D4×C19 C76 C2×C38 D4 C4 C22 C19 C1 # reps 1 1 2 18 18 36 1 18

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

 60 0 0 60
,
 15 113 227 214
,
 1 15 0 228
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

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