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G = C2xC3:D20order 240 = 24·3·5

Direct product of C2 and C3:D20

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

Aliases: C2xC3:D20, C6:2D20, C30:2D4, D10:6D6, Dic3:4D10, D30:8C22, C30.22C23, C15:5(C2xD4), C3:3(C2xD20), C10:1(C3:D4), (C2xDic3):4D5, (C2xC10).17D6, (C2xC6).17D10, (C22xD5):3S3, (C6xD5):6C22, (C10xDic3):6C2, (C22xD15):4C2, C22.15(S3xD5), C6.22(C22xD5), C10.22(C22xS3), (C2xC30).16C22, (C5xDic3):7C22, (D5xC2xC6):2C2, C5:1(C2xC3:D4), C2.22(C2xS3xD5), SmallGroup(240,146)

Series: Derived Chief Lower central Upper central

C1C30 — C2xC3:D20
C1C5C15C30C6xD5C3:D20 — C2xC3:D20
C15C30 — C2xC3:D20
C1C22

Generators and relations for C2xC3: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, C2xC4, D4, C23, D5, C10, C10, Dic3, D6, C2xC6, C2xC6, C15, C2xD4, C20, D10, D10, C2xC10, C2xDic3, C3:D4, C22xS3, C22xC6, C3xD5, D15, C30, C30, D20, C2xC20, C22xD5, C22xD5, C2xC3:D4, C5xDic3, C6xD5, C6xD5, D30, D30, C2xC30, C2xD20, C3:D20, C10xDic3, D5xC2xC6, C22xD15, C2xC3:D20
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2xD4, D10, C3:D4, C22xS3, D20, C22xD5, C2xC3:D4, S3xD5, C2xD20, C3:D20, C2xS3xD5, C2xC3:D20

Smallest permutation representation of C2xC3: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)]])

C2xC3:D20 is a maximal subgroup of
D10.4D12  Dic3:C4:D5  Dic3.D20  D30.34D4  D30.D4  Dic3:4D20  Dic15:13D4  Dic3:D20  D10.16D12  C15:20(C4xD4)  D10:D12  C12:7D20  D6:D20  (C2xDic6):D5  C12:D20  D30:2D4  D30:12D4  C12:2D20  D6:4D20  D30:5D4  D30:6D4  Dic15:3D4  (C2xC6):8D20  Dic15:5D4  (C2xC6):D20  D30:18D4  C2xS3xD20  D20:14D6  C2xD5xC3:D4
C2xC3:D20 is a maximal quotient of
D20:19D6  D20.31D6  D60:30C22  C60.63D4  C12.D20  C60.68D4  C60.44D4  C60.47D4  C60.48D4  D30:10Q8  C12:7D20  C12:D20  C12:2D20  C6.(C2xD20)  C6.D4:D5  (C2xC6):8D20  (C2xC6):D20  D30:18D4

42 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A4B5A5B6A6B6C6D6E6F6G10A···10F15A15B20A···20H30A···30F
order1222222234455666666610···10151520···2030···30
size11111010303026622222101010102···2446···64···4

42 irreducible representations

dim11111222222222444
type++++++++++++++++
imageC1C2C2C2C2S3D4D5D6D6D10D10C3:D4D20S3xD5C3:D20C2xS3xD5
kernelC2xC3:D20C3:D20C10xDic3D5xC2xC6C22xD15C22xD5C30C2xDic3D10C2xC10Dic3C2xC6C10C6C22C2C2
# reps14111122214248242

Matrix representation of C2xC3:D20 in GL4(F61) generated by

1000
0100
00600
00060
,
606000
1000
0010
0001
,
524300
52900
00171
00161
,
60000
1100
00043
00440
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] >;

C2xC3: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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x
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Z
F
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