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## G = C4×D19order 152 = 23·19

### Direct product of C4 and D19

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C19 — C4×D19
 Chief series C1 — C19 — C38 — D38 — C4×D19
 Lower central C19 — C4×D19
 Upper central C1 — C4

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

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 2A 2B 2C 4A 4B 4C 4D 19A ··· 19I 38A ··· 38I 76A ··· 76R order 1 2 2 2 4 4 4 4 19 ··· 19 38 ··· 38 76 ··· 76 size 1 1 19 19 1 1 19 19 2 ··· 2 2 ··· 2 2 ··· 2

44 irreducible representations

 dim 1 1 1 1 1 2 2 2 type + + + + + + image C1 C2 C2 C2 C4 D19 D38 C4×D19 kernel C4×D19 Dic19 C76 D38 D19 C4 C2 C1 # reps 1 1 1 1 4 9 9 18

Matrix representation of C4×D19 in GL2(𝔽37) generated by

 6 0 0 6
,
 36 29 29 9
,
 9 27 8 28
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

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