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G = C3×C41⋊C4order 492 = 22·3·41

Direct product of C3 and C41⋊C4

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

Aliases: C3×C41⋊C4, C41⋊C12, C1232C4, D41.C6, (C3×D41).2C2, SmallGroup(492,5)

Series: Derived Chief Lower central Upper central

C1C41 — C3×C41⋊C4
C1C41D41C3×D41 — C3×C41⋊C4
C41 — C3×C41⋊C4
C1C3

Generators and relations for C3×C41⋊C4
 G = < a,b,c | a3=b41=c4=1, ab=ba, ac=ca, cbc-1=b9 >

41C2
41C4
41C6
41C12

Smallest permutation representation of C3×C41⋊C4
On 123 points
Generators in S123
(1 83 42)(2 84 43)(3 85 44)(4 86 45)(5 87 46)(6 88 47)(7 89 48)(8 90 49)(9 91 50)(10 92 51)(11 93 52)(12 94 53)(13 95 54)(14 96 55)(15 97 56)(16 98 57)(17 99 58)(18 100 59)(19 101 60)(20 102 61)(21 103 62)(22 104 63)(23 105 64)(24 106 65)(25 107 66)(26 108 67)(27 109 68)(28 110 69)(29 111 70)(30 112 71)(31 113 72)(32 114 73)(33 115 74)(34 116 75)(35 117 76)(36 118 77)(37 119 78)(38 120 79)(39 121 80)(40 122 81)(41 123 82)
(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)
(2 33 41 10)(3 24 40 19)(4 15 39 28)(5 6 38 37)(7 29 36 14)(8 20 35 23)(9 11 34 32)(12 25 31 18)(13 16 30 27)(17 21 26 22)(43 74 82 51)(44 65 81 60)(45 56 80 69)(46 47 79 78)(48 70 77 55)(49 61 76 64)(50 52 75 73)(53 66 72 59)(54 57 71 68)(58 62 67 63)(84 115 123 92)(85 106 122 101)(86 97 121 110)(87 88 120 119)(89 111 118 96)(90 102 117 105)(91 93 116 114)(94 107 113 100)(95 98 112 109)(99 103 108 104)

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

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

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

42 conjugacy classes

class 1  2 3A3B4A4B6A6B12A12B12C12D41A···41J123A···123T
order123344661212121241···41123···123
size1411141414141414141414···44···4

42 irreducible representations

dim11111144
type+++
imageC1C2C3C4C6C12C41⋊C4C3×C41⋊C4
kernelC3×C41⋊C4C3×D41C41⋊C4C123D41C41C3C1
# reps1122241020

Matrix representation of C3×C41⋊C4 in GL4(𝔽2953) generated by

800000
080000
008000
000800
,
0002952
1002095
0102046
0012095
,
1171823122538
091228742478
049422981354
027212242695
G:=sub<GL(4,GF(2953))| [800,0,0,0,0,800,0,0,0,0,800,0,0,0,0,800],[0,1,0,0,0,0,1,0,0,0,0,1,2952,2095,2046,2095],[1,0,0,0,1718,912,494,272,2312,2874,2298,1224,2538,2478,1354,2695] >;

C3×C41⋊C4 in GAP, Magma, Sage, TeX

C_3\times C_{41}\rtimes C_4
% in TeX

G:=Group("C3xC41:C4");
// GroupNames label

G:=SmallGroup(492,5);
// by ID

G=gap.SmallGroup(492,5);
# by ID

G:=PCGroup([4,-2,-3,-2,-41,24,6147,1291]);
// Polycyclic

G:=Group<a,b,c|a^3=b^41=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^9>;
// generators/relations

Export

Subgroup lattice of C3×C41⋊C4 in TeX

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