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G = C2×C3⋊D20order 240 = 24·3·5

Direct product of C2 and C3⋊D20

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

Aliases: C2×C3⋊D20, C62D20, C302D4, D106D6, Dic34D10, D308C22, C30.22C23, C155(C2×D4), C33(C2×D20), C101(C3⋊D4), (C2×Dic3)⋊4D5, (C2×C10).17D6, (C2×C6).17D10, (C22×D5)⋊3S3, (C6×D5)⋊6C22, (C10×Dic3)⋊6C2, (C22×D15)⋊4C2, C22.15(S3×D5), C6.22(C22×D5), C10.22(C22×S3), (C2×C30).16C22, (C5×Dic3)⋊7C22, (D5×C2×C6)⋊2C2, C51(C2×C3⋊D4), C2.22(C2×S3×D5), SmallGroup(240,146)

Series: Derived Chief Lower central Upper central

C1C30 — C2×C3⋊D20
C1C5C15C30C6×D5C3⋊D20 — C2×C3⋊D20
C15C30 — C2×C3⋊D20
C1C22

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)])

C2×C3⋊D20 is a maximal subgroup of
D10.4D12  Dic3⋊C4⋊D5  Dic3.D20  D30.34D4  D30.D4  Dic34D20  Dic1513D4  Dic3⋊D20  D10.16D12  C1520(C4×D4)  D10⋊D12  C127D20  D6⋊D20  (C2×Dic6)⋊D5  C12⋊D20  D302D4  D3012D4  C122D20  D64D20  D305D4  D306D4  Dic153D4  (C2×C6)⋊8D20  Dic155D4  (C2×C6)⋊D20  D3018D4  C2×S3×D20  D2014D6  C2×D5×C3⋊D4
C2×C3⋊D20 is a maximal quotient of
D2019D6  D20.31D6  D6030C22  C60.63D4  C12.D20  C60.68D4  C60.44D4  C60.47D4  C60.48D4  D3010Q8  C127D20  C12⋊D20  C122D20  C6.(C2×D20)  C6.D4⋊D5  (C2×C6)⋊8D20  (C2×C6)⋊D20  D3018D4

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⋊D4D20S3×D5C3⋊D20C2×S3×D5
kernelC2×C3⋊D20C3⋊D20C10×Dic3D5×C2×C6C22×D15C22×D5C30C2×Dic3D10C2×C10Dic3C2×C6C10C6C22C2C2
# reps14111122214248242

Matrix representation of C2×C3⋊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] >;

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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