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## G = C3×D40⋊C2order 480 = 25·3·5

### Direct product of C3 and D40⋊C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C3×D40⋊C2
 Chief series C1 — C5 — C10 — C20 — C60 — D5×C12 — C3×D4×D5 — C3×D40⋊C2
 Lower central C5 — C10 — C20 — C3×D40⋊C2
 Upper central C1 — C6 — C12 — C3×SD16

Generators and relations for C3×D40⋊C2
G = < a,b,c,d | a3=b40=c2=d2=1, ab=ba, ac=ca, ad=da, cbc=b-1, dbd=b11, cd=dc >

Subgroups: 576 in 136 conjugacy classes, 54 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C8, C8, C2×C4, D4, D4, Q8, C23, D5, C10, C10, C12, C12, C2×C6, C15, M4(2), D8, SD16, SD16, C2×D4, C4○D4, Dic5, C20, C20, D10, D10, C2×C10, C24, C24, C2×C12, C3×D4, C3×D4, C3×Q8, C22×C6, C3×D5, C30, C30, C8⋊C22, C52C8, C40, C4×D5, C4×D5, D20, D20, C5⋊D4, C5×D4, C5×Q8, C22×D5, C3×M4(2), C3×D8, C3×SD16, C3×SD16, C6×D4, C3×C4○D4, C3×Dic5, C60, C60, C6×D5, C6×D5, C2×C30, C8⋊D5, D40, D4⋊D5, Q8⋊D5, C5×SD16, D4×D5, Q82D5, C3×C8⋊C22, C3×C52C8, C120, D5×C12, D5×C12, C3×D20, C3×D20, C3×C5⋊D4, D4×C15, Q8×C15, D5×C2×C6, D40⋊C2, C3×C8⋊D5, C3×D40, C3×D4⋊D5, C3×Q8⋊D5, C15×SD16, C3×D4×D5, C3×Q82D5, C3×D40⋊C2
Quotients: C1, C2, C3, C22, C6, D4, C23, D5, C2×C6, C2×D4, D10, C3×D4, C22×C6, C3×D5, C8⋊C22, C22×D5, C6×D4, C6×D5, D4×D5, C3×C8⋊C22, D5×C2×C6, D40⋊C2, C3×D4×D5, C3×D40⋊C2

Smallest permutation representation of C3×D40⋊C2
On 120 points
Generators in S120
(1 90 66)(2 91 67)(3 92 68)(4 93 69)(5 94 70)(6 95 71)(7 96 72)(8 97 73)(9 98 74)(10 99 75)(11 100 76)(12 101 77)(13 102 78)(14 103 79)(15 104 80)(16 105 41)(17 106 42)(18 107 43)(19 108 44)(20 109 45)(21 110 46)(22 111 47)(23 112 48)(24 113 49)(25 114 50)(26 115 51)(27 116 52)(28 117 53)(29 118 54)(30 119 55)(31 120 56)(32 81 57)(33 82 58)(34 83 59)(35 84 60)(36 85 61)(37 86 62)(38 87 63)(39 88 64)(40 89 65)
(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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 40)(17 39)(18 38)(19 37)(20 36)(21 35)(22 34)(23 33)(24 32)(25 31)(26 30)(27 29)(41 65)(42 64)(43 63)(44 62)(45 61)(46 60)(47 59)(48 58)(49 57)(50 56)(51 55)(52 54)(66 80)(67 79)(68 78)(69 77)(70 76)(71 75)(72 74)(81 113)(82 112)(83 111)(84 110)(85 109)(86 108)(87 107)(88 106)(89 105)(90 104)(91 103)(92 102)(93 101)(94 100)(95 99)(96 98)(114 120)(115 119)(116 118)
(1 31)(3 13)(4 24)(5 35)(7 17)(8 28)(9 39)(11 21)(12 32)(15 25)(16 36)(19 29)(20 40)(23 33)(27 37)(41 61)(42 72)(44 54)(45 65)(46 76)(48 58)(49 69)(50 80)(52 62)(53 73)(56 66)(57 77)(60 70)(64 74)(68 78)(81 101)(82 112)(84 94)(85 105)(86 116)(88 98)(89 109)(90 120)(92 102)(93 113)(96 106)(97 117)(100 110)(104 114)(108 118)

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

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

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

75 conjugacy classes

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

75 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 C6 C6 D4 D4 D5 D10 D10 D10 C3×D4 C3×D4 C3×D5 C6×D5 C6×D5 C6×D5 C8⋊C22 D4×D5 C3×C8⋊C22 D40⋊C2 C3×D4×D5 C3×D40⋊C2 kernel C3×D40⋊C2 C3×C8⋊D5 C3×D40 C3×D4⋊D5 C3×Q8⋊D5 C15×SD16 C3×D4×D5 C3×Q8⋊2D5 D40⋊C2 C8⋊D5 D40 D4⋊D5 Q8⋊D5 C5×SD16 D4×D5 Q8⋊2D5 C3×Dic5 C6×D5 C3×SD16 C24 C3×D4 C3×Q8 Dic5 D10 SD16 C8 D4 Q8 C15 C6 C5 C3 C2 C1 # reps 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 1 1 2 2 2 2 2 2 4 4 4 4 1 2 2 4 4 8

Matrix representation of C3×D40⋊C2 in GL4(𝔽241) generated by

 225 0 0 0 0 225 0 0 0 0 225 0 0 0 0 225
,
 210 104 31 137 137 104 104 137 210 104 210 104 137 104 137 104
,
 0 0 240 189 0 0 0 1 240 189 0 0 0 1 0 0
,
 0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0
G:=sub<GL(4,GF(241))| [225,0,0,0,0,225,0,0,0,0,225,0,0,0,0,225],[210,137,210,137,104,104,104,104,31,104,210,137,137,137,104,104],[0,0,240,0,0,0,189,1,240,0,0,0,189,1,0,0],[0,0,1,0,0,0,0,1,1,0,0,0,0,1,0,0] >;

C3×D40⋊C2 in GAP, Magma, Sage, TeX

C_3\times D_{40}\rtimes C_2
% in TeX

G:=Group("C3xD40:C2");
// GroupNames label

G:=SmallGroup(480,707);
// by ID

G=gap.SmallGroup(480,707);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,1094,303,268,1271,648,102,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^40=c^2=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d=b^11,c*d=d*c>;
// generators/relations

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