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G = C2×C38order 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

C1 — C2×C38
C1C19C38 — C2×C38
C1 — C2×C38
C1 — C2×C38

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


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