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G = C3×C4○D20order 240 = 24·3·5

Direct product of C3 and C4○D20

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

Aliases: C3×C4○D20, D205C6, Dic105C6, C12.60D10, C30.41C23, C60.63C22, (C2×C60)⋊9C2, (C2×C20)⋊4C6, (C4×D5)⋊4C6, (C2×C12)⋊7D5, C5⋊D43C6, (D5×C12)⋊9C2, C4.16(C6×D5), (C3×D20)⋊11C2, C1512(C4○D4), C20.12(C2×C6), D10.1(C2×C6), (C2×C6).20D10, C22.2(C6×D5), C10.4(C22×C6), Dic5.2(C2×C6), C6.41(C22×D5), (C3×Dic10)⋊11C2, (C2×C30).40C22, (C6×D5).16C22, (C3×Dic5).18C22, C51(C3×C4○D4), C2.5(D5×C2×C6), (C2×C4)⋊3(C3×D5), (C3×C5⋊D4)⋊7C2, (C2×C10).11(C2×C6), SmallGroup(240,158)

Series: Derived Chief Lower central Upper central

C1C10 — C3×C4○D20
C1C5C10C30C6×D5D5×C12 — C3×C4○D20
C5C10 — C3×C4○D20
C1C12C2×C12

Generators and relations for C3×C4○D20
 G = < a,b,c,d | a3=b4=d2=1, c10=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c9 >

Subgroups: 212 in 80 conjugacy classes, 46 normal (30 characteristic)
C1, C2, C2 [×3], C3, C4 [×2], C4 [×2], C22, C22 [×2], C5, C6, C6 [×3], C2×C4, C2×C4 [×2], D4 [×3], Q8, D5 [×2], C10, C10, C12 [×2], C12 [×2], C2×C6, C2×C6 [×2], C15, C4○D4, Dic5 [×2], C20 [×2], D10 [×2], C2×C10, C2×C12, C2×C12 [×2], C3×D4 [×3], C3×Q8, C3×D5 [×2], C30, C30, Dic10, C4×D5 [×2], D20, C5⋊D4 [×2], C2×C20, C3×C4○D4, C3×Dic5 [×2], C60 [×2], C6×D5 [×2], C2×C30, C4○D20, C3×Dic10, D5×C12 [×2], C3×D20, C3×C5⋊D4 [×2], C2×C60, C3×C4○D20
Quotients: C1, C2 [×7], C3, C22 [×7], C6 [×7], C23, D5, C2×C6 [×7], C4○D4, D10 [×3], C22×C6, C3×D5, C22×D5, C3×C4○D4, C6×D5 [×3], C4○D20, D5×C2×C6, C3×C4○D20

Smallest permutation representation of C3×C4○D20
On 120 points
Generators in S120
(1 95 38)(2 96 39)(3 97 40)(4 98 21)(5 99 22)(6 100 23)(7 81 24)(8 82 25)(9 83 26)(10 84 27)(11 85 28)(12 86 29)(13 87 30)(14 88 31)(15 89 32)(16 90 33)(17 91 34)(18 92 35)(19 93 36)(20 94 37)(41 115 79)(42 116 80)(43 117 61)(44 118 62)(45 119 63)(46 120 64)(47 101 65)(48 102 66)(49 103 67)(50 104 68)(51 105 69)(52 106 70)(53 107 71)(54 108 72)(55 109 73)(56 110 74)(57 111 75)(58 112 76)(59 113 77)(60 114 78)
(1 105 11 115)(2 106 12 116)(3 107 13 117)(4 108 14 118)(5 109 15 119)(6 110 16 120)(7 111 17 101)(8 112 18 102)(9 113 19 103)(10 114 20 104)(21 54 31 44)(22 55 32 45)(23 56 33 46)(24 57 34 47)(25 58 35 48)(26 59 36 49)(27 60 37 50)(28 41 38 51)(29 42 39 52)(30 43 40 53)(61 97 71 87)(62 98 72 88)(63 99 73 89)(64 100 74 90)(65 81 75 91)(66 82 76 92)(67 83 77 93)(68 84 78 94)(69 85 79 95)(70 86 80 96)
(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 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 11)(21 34)(22 33)(23 32)(24 31)(25 30)(26 29)(27 28)(35 40)(36 39)(37 38)(41 60)(42 59)(43 58)(44 57)(45 56)(46 55)(47 54)(48 53)(49 52)(50 51)(61 76)(62 75)(63 74)(64 73)(65 72)(66 71)(67 70)(68 69)(77 80)(78 79)(81 88)(82 87)(83 86)(84 85)(89 100)(90 99)(91 98)(92 97)(93 96)(94 95)(101 108)(102 107)(103 106)(104 105)(109 120)(110 119)(111 118)(112 117)(113 116)(114 115)

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

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

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

C3×C4○D20 is a maximal subgroup of
C60.96D4  C60.97D4  D20.3Dic3  D20.2Dic3  D20.34D6  D2021D6  D2019D6  D20.37D6  D20.31D6  C60.63D4  D20.38D6  D20.39D6  D2024D6  D2026D6  D2029D6  C3×D5×C4○D4
C3×C4○D20 is a maximal quotient of
C12×Dic10  C12×D20  C12×C5⋊D4

78 conjugacy classes

class 1 2A2B2C2D3A3B4A4B4C4D4E5A5B6A6B6C6D6E6F6G6H10A···10F12A12B12C12D12E12F12G12H12I12J15A15B15C15D20A···20H30A···30L60A···60P
order122223344444556666666610···10121212121212121212121515151520···2030···3060···60
size1121010111121010221122101010102···21111221010101022222···22···22···2

78 irreducible representations

dim1111111111112222222222
type+++++++++
imageC1C2C2C2C2C2C3C6C6C6C6C6D5C4○D4D10D10C3×D5C3×C4○D4C6×D5C6×D5C4○D20C3×C4○D20
kernelC3×C4○D20C3×Dic10D5×C12C3×D20C3×C5⋊D4C2×C60C4○D20Dic10C4×D5D20C5⋊D4C2×C20C2×C12C15C12C2×C6C2×C4C5C4C22C3C1
# reps11212122424222424484816

Matrix representation of C3×C4○D20 in GL2(𝔽61) generated by

470
047
,
110
011
,
732
292
,
732
2954
G:=sub<GL(2,GF(61))| [47,0,0,47],[11,0,0,11],[7,29,32,2],[7,29,32,54] >;

C3×C4○D20 in GAP, Magma, Sage, TeX

C_3\times C_4\circ D_{20}
% in TeX

G:=Group("C3xC4oD20");
// GroupNames label

G:=SmallGroup(240,158);
// by ID

G=gap.SmallGroup(240,158);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-2,-5,151,506,6917]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^4=d^2=1,c^10=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^2*c^9>;
// generators/relations

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