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G = C3×C31⋊C3order 279 = 32·31

Direct product of C3 and C31⋊C3

direct product, metacyclic, supersoluble, monomial, A-group, 3-hyperelementary

Aliases: C3×C31⋊C3, C93⋊C3, C31⋊C32, SmallGroup(279,3)

Series: Derived Chief Lower central Upper central

C1C31 — C3×C31⋊C3
C1C31C31⋊C3 — C3×C31⋊C3
C31 — C3×C31⋊C3
C1C3

Generators and relations for C3×C31⋊C3
 G = < a,b,c | a3=b31=c3=1, ab=ba, ac=ca, cbc-1=b5 >

31C3
31C3
31C3
31C32

Smallest permutation representation of C3×C31⋊C3
On 93 points
Generators in S93
(1 63 32)(2 64 33)(3 65 34)(4 66 35)(5 67 36)(6 68 37)(7 69 38)(8 70 39)(9 71 40)(10 72 41)(11 73 42)(12 74 43)(13 75 44)(14 76 45)(15 77 46)(16 78 47)(17 79 48)(18 80 49)(19 81 50)(20 82 51)(21 83 52)(22 84 53)(23 85 54)(24 86 55)(25 87 56)(26 88 57)(27 89 58)(28 90 59)(29 91 60)(30 92 61)(31 93 62)
(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 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93)
(2 26 6)(3 20 11)(4 14 16)(5 8 21)(7 27 31)(9 15 10)(12 28 25)(13 22 30)(17 29 19)(18 23 24)(33 57 37)(34 51 42)(35 45 47)(36 39 52)(38 58 62)(40 46 41)(43 59 56)(44 53 61)(48 60 50)(49 54 55)(64 88 68)(65 82 73)(66 76 78)(67 70 83)(69 89 93)(71 77 72)(74 90 87)(75 84 92)(79 91 81)(80 85 86)

G:=sub<Sym(93)| (1,63,32)(2,64,33)(3,65,34)(4,66,35)(5,67,36)(6,68,37)(7,69,38)(8,70,39)(9,71,40)(10,72,41)(11,73,42)(12,74,43)(13,75,44)(14,76,45)(15,77,46)(16,78,47)(17,79,48)(18,80,49)(19,81,50)(20,82,51)(21,83,52)(22,84,53)(23,85,54)(24,86,55)(25,87,56)(26,88,57)(27,89,58)(28,90,59)(29,91,60)(30,92,61)(31,93,62), (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)(63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93), (2,26,6)(3,20,11)(4,14,16)(5,8,21)(7,27,31)(9,15,10)(12,28,25)(13,22,30)(17,29,19)(18,23,24)(33,57,37)(34,51,42)(35,45,47)(36,39,52)(38,58,62)(40,46,41)(43,59,56)(44,53,61)(48,60,50)(49,54,55)(64,88,68)(65,82,73)(66,76,78)(67,70,83)(69,89,93)(71,77,72)(74,90,87)(75,84,92)(79,91,81)(80,85,86)>;

G:=Group( (1,63,32)(2,64,33)(3,65,34)(4,66,35)(5,67,36)(6,68,37)(7,69,38)(8,70,39)(9,71,40)(10,72,41)(11,73,42)(12,74,43)(13,75,44)(14,76,45)(15,77,46)(16,78,47)(17,79,48)(18,80,49)(19,81,50)(20,82,51)(21,83,52)(22,84,53)(23,85,54)(24,86,55)(25,87,56)(26,88,57)(27,89,58)(28,90,59)(29,91,60)(30,92,61)(31,93,62), (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)(63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93), (2,26,6)(3,20,11)(4,14,16)(5,8,21)(7,27,31)(9,15,10)(12,28,25)(13,22,30)(17,29,19)(18,23,24)(33,57,37)(34,51,42)(35,45,47)(36,39,52)(38,58,62)(40,46,41)(43,59,56)(44,53,61)(48,60,50)(49,54,55)(64,88,68)(65,82,73)(66,76,78)(67,70,83)(69,89,93)(71,77,72)(74,90,87)(75,84,92)(79,91,81)(80,85,86) );

G=PermutationGroup([(1,63,32),(2,64,33),(3,65,34),(4,66,35),(5,67,36),(6,68,37),(7,69,38),(8,70,39),(9,71,40),(10,72,41),(11,73,42),(12,74,43),(13,75,44),(14,76,45),(15,77,46),(16,78,47),(17,79,48),(18,80,49),(19,81,50),(20,82,51),(21,83,52),(22,84,53),(23,85,54),(24,86,55),(25,87,56),(26,88,57),(27,89,58),(28,90,59),(29,91,60),(30,92,61),(31,93,62)], [(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),(63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93)], [(2,26,6),(3,20,11),(4,14,16),(5,8,21),(7,27,31),(9,15,10),(12,28,25),(13,22,30),(17,29,19),(18,23,24),(33,57,37),(34,51,42),(35,45,47),(36,39,52),(38,58,62),(40,46,41),(43,59,56),(44,53,61),(48,60,50),(49,54,55),(64,88,68),(65,82,73),(66,76,78),(67,70,83),(69,89,93),(71,77,72),(74,90,87),(75,84,92),(79,91,81),(80,85,86)])

39 conjugacy classes

class 1 3A3B3C···3H31A···31J93A···93T
order1333···331···3193···93
size11131···313···33···3

39 irreducible representations

dim11133
type+
imageC1C3C3C31⋊C3C3×C31⋊C3
kernelC3×C31⋊C3C31⋊C3C93C3C1
# reps1621020

Matrix representation of C3×C31⋊C3 in GL3(𝔽373) generated by

28400
02840
00284
,
10541
100
010
,
100
2841631
220292356
G:=sub<GL(3,GF(373))| [284,0,0,0,284,0,0,0,284],[105,1,0,4,0,1,1,0,0],[1,284,220,0,16,292,0,31,356] >;

C3×C31⋊C3 in GAP, Magma, Sage, TeX

C_3\times C_{31}\rtimes C_3
% in TeX

G:=Group("C3xC31:C3");
// GroupNames label

G:=SmallGroup(279,3);
// by ID

G=gap.SmallGroup(279,3);
# by ID

G:=PCGroup([3,-3,-3,-31,2027]);
// Polycyclic

G:=Group<a,b,c|a^3=b^31=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^5>;
// generators/relations

Export

Subgroup lattice of C3×C31⋊C3 in TeX

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