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G = C41⋊Dic3order 492 = 22·3·41

The semidirect product of C41 and Dic3 acting via Dic3/C3=C4

metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: C41⋊Dic3, C1231C4, D41.S3, C3⋊(C41⋊C4), (C3×D41).1C2, SmallGroup(492,6)

Series: Derived Chief Lower central Upper central

C1C123 — C41⋊Dic3
C1C41C123C3×D41 — C41⋊Dic3
C123 — C41⋊Dic3
C1

Generators and relations for C41⋊Dic3
 G = < a,b,c | a41=b6=1, c2=b3, bab-1=a-1, cac-1=a32, cbc-1=b-1 >

41C2
123C4
41C6
41Dic3
3C41⋊C4

Smallest permutation representation of C41⋊Dic3
On 123 points
Generators in S123
(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 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123)
(1 84 57)(2 83 58 41 85 56)(3 123 59 40 86 55)(4 122 60 39 87 54)(5 121 61 38 88 53)(6 120 62 37 89 52)(7 119 63 36 90 51)(8 118 64 35 91 50)(9 117 65 34 92 49)(10 116 66 33 93 48)(11 115 67 32 94 47)(12 114 68 31 95 46)(13 113 69 30 96 45)(14 112 70 29 97 44)(15 111 71 28 98 43)(16 110 72 27 99 42)(17 109 73 26 100 82)(18 108 74 25 101 81)(19 107 75 24 102 80)(20 106 76 23 103 79)(21 105 77 22 104 78)
(2 10 41 33)(3 19 40 24)(4 28 39 15)(5 37 38 6)(7 14 36 29)(8 23 35 20)(9 32 34 11)(12 18 31 25)(13 27 30 16)(17 22 26 21)(42 113 72 96)(43 122 71 87)(44 90 70 119)(45 99 69 110)(46 108 68 101)(47 117 67 92)(48 85 66 83)(49 94 65 115)(50 103 64 106)(51 112 63 97)(52 121 62 88)(53 89 61 120)(54 98 60 111)(55 107 59 102)(56 116 58 93)(57 84)(73 105 82 104)(74 114 81 95)(75 123 80 86)(76 91 79 118)(77 100 78 109)

G:=sub<Sym(123)| (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)(83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123), (1,84,57)(2,83,58,41,85,56)(3,123,59,40,86,55)(4,122,60,39,87,54)(5,121,61,38,88,53)(6,120,62,37,89,52)(7,119,63,36,90,51)(8,118,64,35,91,50)(9,117,65,34,92,49)(10,116,66,33,93,48)(11,115,67,32,94,47)(12,114,68,31,95,46)(13,113,69,30,96,45)(14,112,70,29,97,44)(15,111,71,28,98,43)(16,110,72,27,99,42)(17,109,73,26,100,82)(18,108,74,25,101,81)(19,107,75,24,102,80)(20,106,76,23,103,79)(21,105,77,22,104,78), (2,10,41,33)(3,19,40,24)(4,28,39,15)(5,37,38,6)(7,14,36,29)(8,23,35,20)(9,32,34,11)(12,18,31,25)(13,27,30,16)(17,22,26,21)(42,113,72,96)(43,122,71,87)(44,90,70,119)(45,99,69,110)(46,108,68,101)(47,117,67,92)(48,85,66,83)(49,94,65,115)(50,103,64,106)(51,112,63,97)(52,121,62,88)(53,89,61,120)(54,98,60,111)(55,107,59,102)(56,116,58,93)(57,84)(73,105,82,104)(74,114,81,95)(75,123,80,86)(76,91,79,118)(77,100,78,109)>;

G:=Group( (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)(83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123), (1,84,57)(2,83,58,41,85,56)(3,123,59,40,86,55)(4,122,60,39,87,54)(5,121,61,38,88,53)(6,120,62,37,89,52)(7,119,63,36,90,51)(8,118,64,35,91,50)(9,117,65,34,92,49)(10,116,66,33,93,48)(11,115,67,32,94,47)(12,114,68,31,95,46)(13,113,69,30,96,45)(14,112,70,29,97,44)(15,111,71,28,98,43)(16,110,72,27,99,42)(17,109,73,26,100,82)(18,108,74,25,101,81)(19,107,75,24,102,80)(20,106,76,23,103,79)(21,105,77,22,104,78), (2,10,41,33)(3,19,40,24)(4,28,39,15)(5,37,38,6)(7,14,36,29)(8,23,35,20)(9,32,34,11)(12,18,31,25)(13,27,30,16)(17,22,26,21)(42,113,72,96)(43,122,71,87)(44,90,70,119)(45,99,69,110)(46,108,68,101)(47,117,67,92)(48,85,66,83)(49,94,65,115)(50,103,64,106)(51,112,63,97)(52,121,62,88)(53,89,61,120)(54,98,60,111)(55,107,59,102)(56,116,58,93)(57,84)(73,105,82,104)(74,114,81,95)(75,123,80,86)(76,91,79,118)(77,100,78,109) );

G=PermutationGroup([(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),(83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123)], [(1,84,57),(2,83,58,41,85,56),(3,123,59,40,86,55),(4,122,60,39,87,54),(5,121,61,38,88,53),(6,120,62,37,89,52),(7,119,63,36,90,51),(8,118,64,35,91,50),(9,117,65,34,92,49),(10,116,66,33,93,48),(11,115,67,32,94,47),(12,114,68,31,95,46),(13,113,69,30,96,45),(14,112,70,29,97,44),(15,111,71,28,98,43),(16,110,72,27,99,42),(17,109,73,26,100,82),(18,108,74,25,101,81),(19,107,75,24,102,80),(20,106,76,23,103,79),(21,105,77,22,104,78)], [(2,10,41,33),(3,19,40,24),(4,28,39,15),(5,37,38,6),(7,14,36,29),(8,23,35,20),(9,32,34,11),(12,18,31,25),(13,27,30,16),(17,22,26,21),(42,113,72,96),(43,122,71,87),(44,90,70,119),(45,99,69,110),(46,108,68,101),(47,117,67,92),(48,85,66,83),(49,94,65,115),(50,103,64,106),(51,112,63,97),(52,121,62,88),(53,89,61,120),(54,98,60,111),(55,107,59,102),(56,116,58,93),(57,84),(73,105,82,104),(74,114,81,95),(75,123,80,86),(76,91,79,118),(77,100,78,109)])

36 conjugacy classes

class 1  2  3 4A4B 6 41A···41J123A···123T
order12344641···41123···123
size1412123123824···44···4

36 irreducible representations

dim1112244
type+++-+
imageC1C2C4S3Dic3C41⋊C4C41⋊Dic3
kernelC41⋊Dic3C3×D41C123D41C41C3C1
# reps112111020

Matrix representation of C41⋊Dic3 in GL4(𝔽2953) generated by

0100
0010
0001
295226445542644
,
971112619431392
2293243125591982
26141382165660
15343982056339
,
1000
8482292341268
10387802532614
193917972739191
G:=sub<GL(4,GF(2953))| [0,0,0,2952,1,0,0,2644,0,1,0,554,0,0,1,2644],[971,2293,2614,1534,1126,2431,138,398,1943,2559,2165,2056,1392,1982,660,339],[1,848,1038,1939,0,229,780,1797,0,234,2532,2739,0,1268,614,191] >;

C41⋊Dic3 in GAP, Magma, Sage, TeX

C_{41}\rtimes {\rm Dic}_3
% in TeX

G:=Group("C41:Dic3");
// GroupNames label

G:=SmallGroup(492,6);
// by ID

G=gap.SmallGroup(492,6);
# by ID

G:=PCGroup([4,-2,-2,-3,-41,8,98,1731,3847]);
// Polycyclic

G:=Group<a,b,c|a^41=b^6=1,c^2=b^3,b*a*b^-1=a^-1,c*a*c^-1=a^32,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of C41⋊Dic3 in TeX

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