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## G = C2×C5⋊D12order 240 = 24·3·5

### Direct product of C2 and C5⋊D12

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — C2×C5⋊D12
 Chief series C1 — C5 — C15 — C30 — C3×Dic5 — C5⋊D12 — C2×C5⋊D12
 Lower central C15 — C30 — C2×C5⋊D12
 Upper central C1 — C22

Generators and relations for C2×C5⋊D12
G = < a,b,c,d | a2=b5=c12=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 528 in 108 conjugacy classes, 40 normal (18 characteristic)
C1, C2, C2 [×2], C2 [×4], C3, C4 [×2], C22, C22 [×8], C5, S3 [×4], C6, C6 [×2], C2×C4, D4 [×4], C23 [×2], D5 [×2], C10, C10 [×2], C10 [×2], C12 [×2], D6 [×2], D6 [×6], C2×C6, C15, C2×D4, Dic5 [×2], D10 [×4], C2×C10, C2×C10 [×4], D12 [×4], C2×C12, C22×S3, C22×S3, C5×S3 [×2], D15 [×2], C30, C30 [×2], C2×Dic5, C5⋊D4 [×4], C22×D5, C22×C10, C2×D12, C3×Dic5 [×2], S3×C10 [×2], S3×C10 [×2], D30 [×2], D30 [×2], C2×C30, C2×C5⋊D4, C5⋊D12 [×4], C6×Dic5, S3×C2×C10, C22×D15, C2×C5⋊D12
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], D12 [×2], C22×S3, C5⋊D4 [×2], C22×D5, C2×D12, S3×D5, C2×C5⋊D4, C5⋊D12 [×2], C2×S3×D5, C2×C5⋊D12

Smallest permutation representation of C2×C5⋊D12
On 120 points
Generators in S120
(1 55)(2 56)(3 57)(4 58)(5 59)(6 60)(7 49)(8 50)(9 51)(10 52)(11 53)(12 54)(13 117)(14 118)(15 119)(16 120)(17 109)(18 110)(19 111)(20 112)(21 113)(22 114)(23 115)(24 116)(25 96)(26 85)(27 86)(28 87)(29 88)(30 89)(31 90)(32 91)(33 92)(34 93)(35 94)(36 95)(37 99)(38 100)(39 101)(40 102)(41 103)(42 104)(43 105)(44 106)(45 107)(46 108)(47 97)(48 98)(61 73)(62 74)(63 75)(64 76)(65 77)(66 78)(67 79)(68 80)(69 81)(70 82)(71 83)(72 84)
(1 48 80 14 28)(2 29 15 81 37)(3 38 82 16 30)(4 31 17 83 39)(5 40 84 18 32)(6 33 19 73 41)(7 42 74 20 34)(8 35 21 75 43)(9 44 76 22 36)(10 25 23 77 45)(11 46 78 24 26)(12 27 13 79 47)(49 104 62 112 93)(50 94 113 63 105)(51 106 64 114 95)(52 96 115 65 107)(53 108 66 116 85)(54 86 117 67 97)(55 98 68 118 87)(56 88 119 69 99)(57 100 70 120 89)(58 90 109 71 101)(59 102 72 110 91)(60 92 111 61 103)
(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 9)(2 8)(3 7)(4 6)(10 12)(13 77)(14 76)(15 75)(16 74)(17 73)(18 84)(19 83)(20 82)(21 81)(22 80)(23 79)(24 78)(25 47)(26 46)(27 45)(28 44)(29 43)(30 42)(31 41)(32 40)(33 39)(34 38)(35 37)(36 48)(49 57)(50 56)(51 55)(52 54)(58 60)(61 109)(62 120)(63 119)(64 118)(65 117)(66 116)(67 115)(68 114)(69 113)(70 112)(71 111)(72 110)(85 108)(86 107)(87 106)(88 105)(89 104)(90 103)(91 102)(92 101)(93 100)(94 99)(95 98)(96 97)

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

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

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

42 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 5A 5B 6A 6B 6C 10A ··· 10F 10G ··· 10N 12A 12B 12C 12D 15A 15B 30A ··· 30F order 1 2 2 2 2 2 2 2 3 4 4 5 5 6 6 6 10 ··· 10 10 ··· 10 12 12 12 12 15 15 30 ··· 30 size 1 1 1 1 6 6 30 30 2 10 10 2 2 2 2 2 2 ··· 2 6 ··· 6 10 10 10 10 4 4 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 D12 C5⋊D4 S3×D5 C5⋊D12 C2×S3×D5 kernel C2×C5⋊D12 C5⋊D12 C6×Dic5 S3×C2×C10 C22×D15 C2×Dic5 C30 C22×S3 Dic5 C2×C10 D6 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×C5⋊D12 in GL4(𝔽61) generated by

 60 0 0 0 0 60 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 43 60 0 0 1 0
,
 0 60 0 0 1 1 0 0 0 0 31 17 0 0 8 30
,
 0 1 0 0 1 0 0 0 0 0 1 0 0 0 43 60
G:=sub<GL(4,GF(61))| [60,0,0,0,0,60,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,43,1,0,0,60,0],[0,1,0,0,60,1,0,0,0,0,31,8,0,0,17,30],[0,1,0,0,1,0,0,0,0,0,1,43,0,0,0,60] >;

C2×C5⋊D12 in GAP, Magma, Sage, TeX

C_2\times C_5\rtimes D_{12}
% in TeX

G:=Group("C2xC5:D12");
// GroupNames label

G:=SmallGroup(240,147);
// by ID

G=gap.SmallGroup(240,147);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-5,48,121,490,6917]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^5=c^12=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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