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## G = C3×C41⋊C4order 492 = 22·3·41

### Direct product of C3 and C41⋊C4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C41 — C3×C41⋊C4
 Chief series C1 — C41 — D41 — C3×D41 — C3×C41⋊C4
 Lower central C41 — C3×C41⋊C4
 Upper central C1 — C3

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

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 3A 3B 4A 4B 6A 6B 12A 12B 12C 12D 41A ··· 41J 123A ··· 123T order 1 2 3 3 4 4 6 6 12 12 12 12 41 ··· 41 123 ··· 123 size 1 41 1 1 41 41 41 41 41 41 41 41 4 ··· 4 4 ··· 4

42 irreducible representations

 dim 1 1 1 1 1 1 4 4 type + + + image C1 C2 C3 C4 C6 C12 C41⋊C4 C3×C41⋊C4 kernel C3×C41⋊C4 C3×D41 C41⋊C4 C123 D41 C41 C3 C1 # reps 1 1 2 2 2 4 10 20

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

 800 0 0 0 0 800 0 0 0 0 800 0 0 0 0 800
,
 0 0 0 2952 1 0 0 2095 0 1 0 2046 0 0 1 2095
,
 1 1718 2312 2538 0 912 2874 2478 0 494 2298 1354 0 272 1224 2695
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

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