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## G = C3×C40⋊C4order 480 = 25·3·5

### Direct product of C3 and C40⋊C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C3×C40⋊C4
 Chief series C1 — C5 — C10 — C20 — C4×D5 — D5×C12 — C3×C4⋊F5 — C3×C40⋊C4
 Lower central C5 — C10 — C20 — C3×C40⋊C4
 Upper central C1 — C6 — C12 — C24

Generators and relations for C3×C40⋊C4
G = < a,b,c | a3=b40=c4=1, ab=ba, ac=ca, cbc-1=b3 >

Subgroups: 280 in 72 conjugacy classes, 36 normal (28 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C8, C8, C2×C4, D5, C10, C12, C12, C2×C6, C15, C4⋊C4, C2×C8, Dic5, C20, F5, D10, C24, C24, C2×C12, C3×D5, C30, C4.Q8, C52C8, C40, C4×D5, C2×F5, C3×C4⋊C4, C2×C24, C3×Dic5, C60, C3×F5, C6×D5, C8×D5, C4⋊F5, C3×C4.Q8, C3×C52C8, C120, D5×C12, C6×F5, C40⋊C4, D5×C24, C3×C4⋊F5, C3×C40⋊C4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, Q8, C12, C2×C6, C4⋊C4, SD16, F5, C2×C12, C3×D4, C3×Q8, C4.Q8, C2×F5, C3×C4⋊C4, C3×SD16, C3×F5, C4⋊F5, C3×C4.Q8, C6×F5, C40⋊C4, C3×C4⋊F5, C3×C40⋊C4

Smallest permutation representation of C3×C40⋊C4
On 120 points
Generators in S120
(1 109 52)(2 110 53)(3 111 54)(4 112 55)(5 113 56)(6 114 57)(7 115 58)(8 116 59)(9 117 60)(10 118 61)(11 119 62)(12 120 63)(13 81 64)(14 82 65)(15 83 66)(16 84 67)(17 85 68)(18 86 69)(19 87 70)(20 88 71)(21 89 72)(22 90 73)(23 91 74)(24 92 75)(25 93 76)(26 94 77)(27 95 78)(28 96 79)(29 97 80)(30 98 41)(31 99 42)(32 100 43)(33 101 44)(34 102 45)(35 103 46)(36 104 47)(37 105 48)(38 106 49)(39 107 50)(40 108 51)
(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)
(1 36 21 16)(2 23 30 19)(3 10 39 22)(4 37 8 25)(5 24 17 28)(6 11 26 31)(7 38 35 34)(9 12 13 40)(14 27 18 15)(20 29 32 33)(41 70 53 74)(42 57 62 77)(43 44 71 80)(45 58 49 46)(47 72 67 52)(48 59 76 55)(50 73 54 61)(51 60 63 64)(56 75 68 79)(65 78 69 66)(81 108 117 120)(82 95 86 83)(84 109 104 89)(85 96 113 92)(87 110 91 98)(88 97 100 101)(90 111 118 107)(93 112 105 116)(94 99 114 119)(102 115 106 103)

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

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

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

66 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4A 4B 4C 4D 4E 4F 5 6A 6B 6C 6D 6E 6F 8A 8B 8C 8D 10 12A 12B 12C 12D 12E ··· 12L 15A 15B 20A 20B 24A 24B 24C 24D 24E 24F 24G 24H 30A 30B 40A 40B 40C 40D 60A 60B 60C 60D 120A ··· 120H order 1 2 2 2 3 3 4 4 4 4 4 4 5 6 6 6 6 6 6 8 8 8 8 10 12 12 12 12 12 ··· 12 15 15 20 20 24 24 24 24 24 24 24 24 30 30 40 40 40 40 60 60 60 60 120 ··· 120 size 1 1 5 5 1 1 2 10 20 20 20 20 4 1 1 5 5 5 5 2 2 10 10 4 2 2 10 10 20 ··· 20 4 4 4 4 2 2 2 2 10 10 10 10 4 4 4 4 4 4 4 4 4 4 4 ··· 4

66 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 4 4 4 4 4 type + + + - + + + image C1 C2 C2 C3 C4 C4 C6 C6 C12 C12 Q8 D4 SD16 C3×Q8 C3×D4 C3×SD16 F5 C2×F5 C3×F5 C4⋊F5 C6×F5 C40⋊C4 C3×C4⋊F5 C3×C40⋊C4 kernel C3×C40⋊C4 D5×C24 C3×C4⋊F5 C40⋊C4 C3×C5⋊2C8 C120 C8×D5 C4⋊F5 C5⋊2C8 C40 C3×Dic5 C6×D5 C3×D5 Dic5 D10 D5 C24 C12 C8 C6 C4 C3 C2 C1 # reps 1 1 2 2 2 2 2 4 4 4 1 1 4 2 2 8 1 1 2 2 2 4 4 8

Matrix representation of C3×C40⋊C4 in GL6(𝔽241)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 15 0 0 0 0 0 0 15 0 0 0 0 0 0 15 0 0 0 0 0 0 15
,
 222 19 0 0 0 0 222 222 0 0 0 0 0 0 0 207 17 224 0 0 34 207 224 0 0 0 17 0 224 207 0 0 34 224 17 207
,
 11 230 0 0 0 0 230 230 0 0 0 0 0 0 207 34 17 0 0 0 224 34 0 207 0 0 207 0 34 224 0 0 0 17 34 207

G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,15,0,0,0,0,0,0,15,0,0,0,0,0,0,15,0,0,0,0,0,0,15],[222,222,0,0,0,0,19,222,0,0,0,0,0,0,0,34,17,34,0,0,207,207,0,224,0,0,17,224,224,17,0,0,224,0,207,207],[11,230,0,0,0,0,230,230,0,0,0,0,0,0,207,224,207,0,0,0,34,34,0,17,0,0,17,0,34,34,0,0,0,207,224,207] >;

C3×C40⋊C4 in GAP, Magma, Sage, TeX

C_3\times C_{40}\rtimes C_4
% in TeX

G:=Group("C3xC40:C4");
// GroupNames label

G:=SmallGroup(480,273);
// by ID

G=gap.SmallGroup(480,273);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-5,84,365,176,2524,102,9414,1595]);
// Polycyclic

G:=Group<a,b,c|a^3=b^40=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^3>;
// generators/relations

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