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## G = C22×D19order 152 = 23·19

### Direct product of C22 and D19

Aliases: C22×D19, C19⋊C23, C38⋊C22, (C2×C38)⋊3C2, SmallGroup(152,11)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C19 — C22×D19
 Chief series C1 — C19 — D19 — D38 — C22×D19
 Lower central C19 — C22×D19
 Upper central C1 — C22

Generators and relations for C22×D19
G = < a,b,c,d | a2=b2=c19=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Smallest permutation representation of C22×D19
On 76 points
Generators in S76
(1 75)(2 76)(3 58)(4 59)(5 60)(6 61)(7 62)(8 63)(9 64)(10 65)(11 66)(12 67)(13 68)(14 69)(15 70)(16 71)(17 72)(18 73)(19 74)(20 46)(21 47)(22 48)(23 49)(24 50)(25 51)(26 52)(27 53)(28 54)(29 55)(30 56)(31 57)(32 39)(33 40)(34 41)(35 42)(36 43)(37 44)(38 45)
(1 31)(2 32)(3 33)(4 34)(5 35)(6 36)(7 37)(8 38)(9 20)(10 21)(11 22)(12 23)(13 24)(14 25)(15 26)(16 27)(17 28)(18 29)(19 30)(39 76)(40 58)(41 59)(42 60)(43 61)(44 62)(45 63)(46 64)(47 65)(48 66)(49 67)(50 68)(51 69)(52 70)(53 71)(54 72)(55 73)(56 74)(57 75)
(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 74)(2 73)(3 72)(4 71)(5 70)(6 69)(7 68)(8 67)(9 66)(10 65)(11 64)(12 63)(13 62)(14 61)(15 60)(16 59)(17 58)(18 76)(19 75)(20 48)(21 47)(22 46)(23 45)(24 44)(25 43)(26 42)(27 41)(28 40)(29 39)(30 57)(31 56)(32 55)(33 54)(34 53)(35 52)(36 51)(37 50)(38 49)

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

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

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

C22×D19 is a maximal subgroup of   D38⋊C4  D19⋊A4
C22×D19 is a maximal quotient of   D765C2  D42D19  D76⋊C2

44 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 19A ··· 19I 38A ··· 38AA order 1 2 2 2 2 2 2 2 19 ··· 19 38 ··· 38 size 1 1 1 1 19 19 19 19 2 ··· 2 2 ··· 2

44 irreducible representations

 dim 1 1 1 2 2 type + + + + + image C1 C2 C2 D19 D38 kernel C22×D19 D38 C2×C38 C22 C2 # reps 1 6 1 9 27

Matrix representation of C22×D19 in GL3(𝔽191) generated by

 190 0 0 0 1 0 0 0 1
,
 1 0 0 0 190 0 0 0 190
,
 1 0 0 0 33 1 0 73 118
,
 1 0 0 0 118 190 0 171 73
G:=sub<GL(3,GF(191))| [190,0,0,0,1,0,0,0,1],[1,0,0,0,190,0,0,0,190],[1,0,0,0,33,73,0,1,118],[1,0,0,0,118,171,0,190,73] >;

C22×D19 in GAP, Magma, Sage, TeX

C_2^2\times D_{19}
% in TeX

G:=Group("C2^2xD19");
// GroupNames label

G:=SmallGroup(152,11);
// by ID

G=gap.SmallGroup(152,11);
# by ID

G:=PCGroup([4,-2,-2,-2,-19,2307]);
// Polycyclic

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

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