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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 [×2], C2 [×4], C3, C4 [×2], C22, C22 [×8], C5, S3 [×2], C6, C6 [×2], C6 [×2], C2×C4, D4 [×4], C23 [×2], D5 [×4], C10, C10 [×2], Dic3 [×2], D6 [×4], C2×C6, C2×C6 [×4], C15, C2×D4, C20 [×2], D10 [×2], D10 [×6], C2×C10, C2×Dic3, C3⋊D4 [×4], C22×S3, C22×C6, C3×D5 [×2], D15 [×2], C30, C30 [×2], D20 [×4], C2×C20, C22×D5, C22×D5, C2×C3⋊D4, C5×Dic3 [×2], C6×D5 [×2], C6×D5 [×2], D30 [×2], D30 [×2], C2×C30, C2×D20, C3⋊D20 [×4], C10×Dic3, D5×C2×C6, C22×D15, C2×C3⋊D20
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], C3⋊D4 [×2], C22×S3, D20 [×2], C22×D5, C2×C3⋊D4, S3×D5, C2×D20, C3⋊D20 [×2], C2×S3×D5, C2×C3⋊D20

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

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

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

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

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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