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G = C2×C5⋊D12order 240 = 24·3·5

Direct product of C2 and C5⋊D12

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2×C5⋊D12, C303D4, D66D10, C102D12, Dic54D6, D309C22, C30.23C23, C156(C2×D4), C53(C2×D12), C61(C5⋊D4), (C2×Dic5)⋊4S3, (C6×Dic5)⋊6C2, (C2×C10).18D6, (C2×C6).18D10, (C22×S3)⋊2D5, (S3×C10)⋊6C22, (C22×D15)⋊5C2, C22.16(S3×D5), C6.23(C22×D5), C10.23(C22×S3), (C2×C30).17C22, (C3×Dic5)⋊7C22, C31(C2×C5⋊D4), (S3×C2×C10)⋊2C2, C2.23(C2×S3×D5), SmallGroup(240,147)

Series: Derived Chief Lower central Upper central

C1C30 — C2×C5⋊D12
C1C5C15C30C3×Dic5C5⋊D12 — C2×C5⋊D12
C15C30 — C2×C5⋊D12
C1C22

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)])

C2×C5⋊D12 is a maximal subgroup of
Dic5.D12  (C2×C20).D6  Dic5.8D12  D6⋊Dic5⋊C2  D30.35D4  Dic54D12  Dic1514D4  Dic5⋊D12  D30⋊D4  D6.D20  D30.7D4  C1522(C4×D4)  D10⋊D12  C20⋊D12  D6⋊D20  C606D4  D3012D4  C202D12  D304D4  D305D4  D307D4  Dic154D4  (C2×C10)⋊4D12  Dic155D4  (C2×C10)⋊11D12  D3019D4  C2×D5×D12  D1214D10  C2×S3×C5⋊D4
C2×C5⋊D12 is a maximal quotient of
C20.60D12  D6036C22  C60.38D4  C20.D12  D12.33D10  C60.45D4  C60.69D4  C60.70D4  Dic5⋊Dic6  D309Q8  C20⋊D12  C606D4  C202D12  C10.(C2×D12)  (C2×C10).D12  (C2×C10)⋊4D12  (C2×C10)⋊11D12  D3019D4

42 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A4B5A5B6A6B6C10A···10F10G···10N12A12B12C12D15A15B30A···30F
order122222223445566610···1010···1012121212151530···30
size111166303021010222222···26···610101010444···4

42 irreducible representations

dim11111222222222444
type++++++++++++++++
imageC1C2C2C2C2S3D4D5D6D6D10D10D12C5⋊D4S3×D5C5⋊D12C2×S3×D5
kernelC2×C5⋊D12C5⋊D12C6×Dic5S3×C2×C10C22×D15C2×Dic5C30C22×S3Dic5C2×C10D6C2×C6C10C6C22C2C2
# reps14111122214248242

Matrix representation of C2×C5⋊D12 in GL4(𝔽61) generated by

60000
06000
0010
0001
,
1000
0100
004360
0010
,
06000
1100
003117
00830
,
0100
1000
0010
004360
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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