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G = C2×C38  order 76 = 22·19

Abelian group of type [2,38]

direct product, abelian, monomial, 2-elementary

Aliases: C2×C38, SmallGroup(76,4)

Series: Derived ►Chief ►Lower central ►Upper central

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

Generators and relations for C2×C38
 G = < a,b | a2=b38=1, ab=ba >

C1
C2
C2
C2
C19
C22
C38
C38
C38
C2×C38

Smallest permutation representation of C2×C38
►Regular action on 76 points
Generators in S76
(1 69)(2 70)(3 71)(4 72)(5 73)(6 74)(7 75)(8 76)(9 39)(10 40)(11 41)(12 42)(13 43)(14 44)(15 45)(16 46)(17 47)(18 48)(19 49)(20 50)(21 51)(22 52)(23 53)(24 54)(25 55)(26 56)(27 57)(28 58)(29 59)(30 60)(31 61)(32 62)(33 63)(34 64)(35 65)(36 66)(37 67)(38 68)

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

C2×C38 is a maximal subgroup of   C19⋊D4  C19⋊A4

76 conjugacy classes

class 1 2A2B2C19A···19R38A···38BB
order122219···1938···38
size11111···11···1

76 irreducible representations

dim1111
type++
imageC1C2C19C38
kernelC2×C38C38C22C2
# reps131854

Matrix representation of C2×C38 ►in GL2(𝔽191) generated by

1900
0190
,
1800
0122
G:=sub<GL(2,GF(191))| [190,0,0,190],[180,0,0,122] >;
 

C2×C38 in GAP, Magma, Sage, TeX 

C_2\times C_{38}
 
% in TeX
 
G:=Group("C2xC38");
 
// GroupNames label
 
G:=SmallGroup(76,4);
 
// by ID
 
G=gap.SmallGroup(76,4);
 
# by ID
 
G:=PCGroup([3,-2,-2,-19]);
 
// Polycyclic
 
G:=Group<a,b|a^2=b^38=1,a*b=b*a>;
 
// generators/relations
 

Export

Subgroup lattice of C2×C38 in TeX

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