Copied to
clipboard

G = C3×D40⋊C2order 480 = 25·3·5

Direct product of C3 and D40⋊C2

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

Aliases: C3×D40⋊C2, D406C6, C2418D10, C12018C22, C60.194C23, C83(C6×D5), C403(C2×C6), D4⋊D53C6, (D4×D5)⋊3C6, Q8⋊D52C6, Q83(C6×D5), D202(C2×C6), C8⋊D51C6, D4.3(C6×D5), (C3×D40)⋊14C2, (C3×Q8)⋊15D10, Q82D54C6, SD161(C3×D5), (C3×SD16)⋊5D5, (C5×SD16)⋊1C6, (C6×D5).72D4, C10.31(C6×D4), C6.185(D4×D5), C1532(C8⋊C22), (C15×SD16)⋊5C2, (C3×D4).27D10, D10.14(C3×D4), C30.344(C2×D4), C20.5(C22×C6), (C3×D20)⋊19C22, Dic5.17(C3×D4), (C3×Dic5).78D4, (Q8×C15)⋊15C22, (D5×C12).77C22, (D4×C15).27C22, C12.194(C22×D5), C4.5(D5×C2×C6), (C3×D4×D5)⋊10C2, C53(C3×C8⋊C22), C2.19(C3×D4×D5), C52C82(C2×C6), (C5×Q8)⋊3(C2×C6), (C3×D4⋊D5)⋊11C2, (C3×C8⋊D5)⋊5C2, (C3×Q8⋊D5)⋊10C2, (C5×D4).3(C2×C6), (C4×D5).2(C2×C6), (C3×Q82D5)⋊8C2, (C3×C52C8)⋊24C22, SmallGroup(480,707)

Series: Derived Chief Lower central Upper central

C1C20 — C3×D40⋊C2
C1C5C10C20C60D5×C12C3×D4×D5 — C3×D40⋊C2
C5C10C20 — C3×D40⋊C2
C1C6C12C3×SD16

Generators and relations for C3×D40⋊C2
 G = < a,b,c,d | a3=b40=c2=d2=1, ab=ba, ac=ca, ad=da, cbc=b-1, dbd=b11, cd=dc >

Subgroups: 576 in 136 conjugacy classes, 54 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C8, C8, C2×C4, D4, D4, Q8, C23, D5, C10, C10, C12, C12, C2×C6, C15, M4(2), D8, SD16, SD16, C2×D4, C4○D4, Dic5, C20, C20, D10, D10, C2×C10, C24, C24, C2×C12, C3×D4, C3×D4, C3×Q8, C22×C6, C3×D5, C30, C30, C8⋊C22, C52C8, C40, C4×D5, C4×D5, D20, D20, C5⋊D4, C5×D4, C5×Q8, C22×D5, C3×M4(2), C3×D8, C3×SD16, C3×SD16, C6×D4, C3×C4○D4, C3×Dic5, C60, C60, C6×D5, C6×D5, C2×C30, C8⋊D5, D40, D4⋊D5, Q8⋊D5, C5×SD16, D4×D5, Q82D5, C3×C8⋊C22, C3×C52C8, C120, D5×C12, D5×C12, C3×D20, C3×D20, C3×C5⋊D4, D4×C15, Q8×C15, D5×C2×C6, D40⋊C2, C3×C8⋊D5, C3×D40, C3×D4⋊D5, C3×Q8⋊D5, C15×SD16, C3×D4×D5, C3×Q82D5, C3×D40⋊C2
Quotients: C1, C2, C3, C22, C6, D4, C23, D5, C2×C6, C2×D4, D10, C3×D4, C22×C6, C3×D5, C8⋊C22, C22×D5, C6×D4, C6×D5, D4×D5, C3×C8⋊C22, D5×C2×C6, D40⋊C2, C3×D4×D5, C3×D40⋊C2

Smallest permutation representation of C3×D40⋊C2
On 120 points
Generators in S120
(1 90 66)(2 91 67)(3 92 68)(4 93 69)(5 94 70)(6 95 71)(7 96 72)(8 97 73)(9 98 74)(10 99 75)(11 100 76)(12 101 77)(13 102 78)(14 103 79)(15 104 80)(16 105 41)(17 106 42)(18 107 43)(19 108 44)(20 109 45)(21 110 46)(22 111 47)(23 112 48)(24 113 49)(25 114 50)(26 115 51)(27 116 52)(28 117 53)(29 118 54)(30 119 55)(31 120 56)(32 81 57)(33 82 58)(34 83 59)(35 84 60)(36 85 61)(37 86 62)(38 87 63)(39 88 64)(40 89 65)
(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 40)(17 39)(18 38)(19 37)(20 36)(21 35)(22 34)(23 33)(24 32)(25 31)(26 30)(27 29)(41 65)(42 64)(43 63)(44 62)(45 61)(46 60)(47 59)(48 58)(49 57)(50 56)(51 55)(52 54)(66 80)(67 79)(68 78)(69 77)(70 76)(71 75)(72 74)(81 113)(82 112)(83 111)(84 110)(85 109)(86 108)(87 107)(88 106)(89 105)(90 104)(91 103)(92 102)(93 101)(94 100)(95 99)(96 98)(114 120)(115 119)(116 118)
(1 31)(3 13)(4 24)(5 35)(7 17)(8 28)(9 39)(11 21)(12 32)(15 25)(16 36)(19 29)(20 40)(23 33)(27 37)(41 61)(42 72)(44 54)(45 65)(46 76)(48 58)(49 69)(50 80)(52 62)(53 73)(56 66)(57 77)(60 70)(64 74)(68 78)(81 101)(82 112)(84 94)(85 105)(86 116)(88 98)(89 109)(90 120)(92 102)(93 113)(96 106)(97 117)(100 110)(104 114)(108 118)

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

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

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

75 conjugacy classes

class 1 2A2B2C2D2E3A3B4A4B4C5A5B6A6B6C6D6E6F6G6H6I6J8A8B10A10B10C10D12A12B12C12D12E12F15A15B15C15D20A20B20C20D24A24B24C24D30A30B30C30D30E30F30G30H40A40B40C40D60A60B60C60D60E60F60G60H120A···120H
order1222223344455666666666688101010101212121212121515151520202020242424243030303030303030404040406060606060606060120···120
size11410202011241022114410102020202042022882244101022224488442020222288884444444488884···4

75 irreducible representations

dim1111111111111111222222222222444444
type+++++++++++++++++
imageC1C2C2C2C2C2C2C2C3C6C6C6C6C6C6C6D4D4D5D10D10D10C3×D4C3×D4C3×D5C6×D5C6×D5C6×D5C8⋊C22D4×D5C3×C8⋊C22D40⋊C2C3×D4×D5C3×D40⋊C2
kernelC3×D40⋊C2C3×C8⋊D5C3×D40C3×D4⋊D5C3×Q8⋊D5C15×SD16C3×D4×D5C3×Q82D5D40⋊C2C8⋊D5D40D4⋊D5Q8⋊D5C5×SD16D4×D5Q82D5C3×Dic5C6×D5C3×SD16C24C3×D4C3×Q8Dic5D10SD16C8D4Q8C15C6C5C3C2C1
# reps1111111122222222112222224444122448

Matrix representation of C3×D40⋊C2 in GL4(𝔽241) generated by

225000
022500
002250
000225
,
21010431137
137104104137
210104210104
137104137104
,
00240189
0001
24018900
0100
,
0010
0001
1000
0100
G:=sub<GL(4,GF(241))| [225,0,0,0,0,225,0,0,0,0,225,0,0,0,0,225],[210,137,210,137,104,104,104,104,31,104,210,137,137,137,104,104],[0,0,240,0,0,0,189,1,240,0,0,0,189,1,0,0],[0,0,1,0,0,0,0,1,1,0,0,0,0,1,0,0] >;

C3×D40⋊C2 in GAP, Magma, Sage, TeX

C_3\times D_{40}\rtimes C_2
% in TeX

G:=Group("C3xD40:C2");
// GroupNames label

G:=SmallGroup(480,707);
// by ID

G=gap.SmallGroup(480,707);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,1094,303,268,1271,648,102,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^40=c^2=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d=b^11,c*d=d*c>;
// generators/relations

׿
×
𝔽