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## G = C2×C3⋊D20order 240 = 24·3·5

### Direct product of C2 and C3⋊D20

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — C2×C3⋊D20
 Chief series C1 — C5 — C15 — C30 — C6×D5 — C3⋊D20 — C2×C3⋊D20
 Lower central C15 — C30 — C2×C3⋊D20
 Upper central C1 — C22

Generators and relations for C2×C3⋊D20
G = < a,b,c,d | a2=b3=c20=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 560 in 108 conjugacy classes, 40 normal (18 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C5, S3, C6, C6, C6, C2×C4, D4, C23, D5, C10, C10, Dic3, D6, C2×C6, C2×C6, C15, C2×D4, C20, D10, D10, C2×C10, C2×Dic3, C3⋊D4, C22×S3, C22×C6, C3×D5, D15, C30, C30, D20, C2×C20, C22×D5, C22×D5, C2×C3⋊D4, C5×Dic3, C6×D5, C6×D5, D30, D30, C2×C30, C2×D20, C3⋊D20, C10×Dic3, D5×C2×C6, C22×D15, C2×C3⋊D20
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, D10, C3⋊D4, C22×S3, D20, C22×D5, C2×C3⋊D4, S3×D5, C2×D20, C3⋊D20, C2×S3×D5, C2×C3⋊D20

Smallest permutation representation of C2×C3⋊D20
On 120 points
Generators in S120
(1 45)(2 46)(3 47)(4 48)(5 49)(6 50)(7 51)(8 52)(9 53)(10 54)(11 55)(12 56)(13 57)(14 58)(15 59)(16 60)(17 41)(18 42)(19 43)(20 44)(21 118)(22 119)(23 120)(24 101)(25 102)(26 103)(27 104)(28 105)(29 106)(30 107)(31 108)(32 109)(33 110)(34 111)(35 112)(36 113)(37 114)(38 115)(39 116)(40 117)(61 90)(62 91)(63 92)(64 93)(65 94)(66 95)(67 96)(68 97)(69 98)(70 99)(71 100)(72 81)(73 82)(74 83)(75 84)(76 85)(77 86)(78 87)(79 88)(80 89)
(1 37 99)(2 100 38)(3 39 81)(4 82 40)(5 21 83)(6 84 22)(7 23 85)(8 86 24)(9 25 87)(10 88 26)(11 27 89)(12 90 28)(13 29 91)(14 92 30)(15 31 93)(16 94 32)(17 33 95)(18 96 34)(19 35 97)(20 98 36)(41 110 66)(42 67 111)(43 112 68)(44 69 113)(45 114 70)(46 71 115)(47 116 72)(48 73 117)(49 118 74)(50 75 119)(51 120 76)(52 77 101)(53 102 78)(54 79 103)(55 104 80)(56 61 105)(57 106 62)(58 63 107)(59 108 64)(60 65 109)
(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 20)(17 19)(21 89)(22 88)(23 87)(24 86)(25 85)(26 84)(27 83)(28 82)(29 81)(30 100)(31 99)(32 98)(33 97)(34 96)(35 95)(36 94)(37 93)(38 92)(39 91)(40 90)(41 43)(44 60)(45 59)(46 58)(47 57)(48 56)(49 55)(50 54)(51 53)(61 117)(62 116)(63 115)(64 114)(65 113)(66 112)(67 111)(68 110)(69 109)(70 108)(71 107)(72 106)(73 105)(74 104)(75 103)(76 102)(77 101)(78 120)(79 119)(80 118)

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

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

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

42 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 5A 5B 6A 6B 6C 6D 6E 6F 6G 10A ··· 10F 15A 15B 20A ··· 20H 30A ··· 30F order 1 2 2 2 2 2 2 2 3 4 4 5 5 6 6 6 6 6 6 6 10 ··· 10 15 15 20 ··· 20 30 ··· 30 size 1 1 1 1 10 10 30 30 2 6 6 2 2 2 2 2 10 10 10 10 2 ··· 2 4 4 6 ··· 6 4 ··· 4

42 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 S3 D4 D5 D6 D6 D10 D10 C3⋊D4 D20 S3×D5 C3⋊D20 C2×S3×D5 kernel C2×C3⋊D20 C3⋊D20 C10×Dic3 D5×C2×C6 C22×D15 C22×D5 C30 C2×Dic3 D10 C2×C10 Dic3 C2×C6 C10 C6 C22 C2 C2 # reps 1 4 1 1 1 1 2 2 2 1 4 2 4 8 2 4 2

Matrix representation of C2×C3⋊D20 in GL4(𝔽61) generated by

 1 0 0 0 0 1 0 0 0 0 60 0 0 0 0 60
,
 60 60 0 0 1 0 0 0 0 0 1 0 0 0 0 1
,
 52 43 0 0 52 9 0 0 0 0 17 1 0 0 16 1
,
 60 0 0 0 1 1 0 0 0 0 0 43 0 0 44 0
G:=sub<GL(4,GF(61))| [1,0,0,0,0,1,0,0,0,0,60,0,0,0,0,60],[60,1,0,0,60,0,0,0,0,0,1,0,0,0,0,1],[52,52,0,0,43,9,0,0,0,0,17,16,0,0,1,1],[60,1,0,0,0,1,0,0,0,0,0,44,0,0,43,0] >;

C2×C3⋊D20 in GAP, Magma, Sage, TeX

C_2\times C_3\rtimes D_{20}
% in TeX

G:=Group("C2xC3:D20");
// GroupNames label

G:=SmallGroup(240,146);
// by ID

G=gap.SmallGroup(240,146);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-5,121,55,490,6917]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^3=c^20=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

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