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G = C3×D4⋊D10order 480 = 25·3·5

Direct product of C3 and D4⋊D10

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

Aliases: C3×D4⋊D10, C60.220D4, C60.206C23, D4⋊D56C6, D44(C6×D5), Q8⋊D56C6, Q85(C6×D5), (C2×D20)⋊10C6, (C6×D20)⋊26C2, (C3×D4)⋊26D10, (C3×Q8)⋊23D10, C10.59(C6×D4), C20.49(C3×D4), (C2×C30).85D4, C4.Dic59C6, C1538(C8⋊C22), D20.11(C2×C6), C30.416(C2×D4), (C2×C12).246D10, (D4×C15)⋊28C22, C20.17(C22×C6), (Q8×C15)⋊25C22, C12.117(C5⋊D4), (C2×C60).302C22, (C3×D20).50C22, C12.206(C22×D5), C55(C3×C8⋊C22), C4.17(D5×C2×C6), C52C84(C2×C6), (C3×C4○D4)⋊6D5, C4○D43(C3×D5), (C5×C4○D4)⋊5C6, (C5×D4)⋊4(C2×C6), (C5×Q8)⋊6(C2×C6), (C3×D4⋊D5)⋊14C2, (C15×C4○D4)⋊6C2, (C3×Q8⋊D5)⋊14C2, (C2×C10).8(C3×D4), (C2×C4).17(C6×D5), C2.23(C6×C5⋊D4), C4.24(C3×C5⋊D4), (C2×C20).39(C2×C6), C6.144(C2×C5⋊D4), C22.5(C3×C5⋊D4), (C3×C52C8)⋊26C22, (C2×C6).41(C5⋊D4), (C3×C4.Dic5)⋊21C2, SmallGroup(480,742)

Series: Derived Chief Lower central Upper central

C1C20 — C3×D4⋊D10
C1C5C10C20C60C3×D20C6×D20 — C3×D4⋊D10
C5C10C20 — C3×D4⋊D10
C1C6C2×C12C3×C4○D4

Generators and relations for C3×D4⋊D10
 G = < a,b,c,d,e | a3=b4=c2=d10=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe=b-1, bd=db, dcd-1=b2c, ece=b-1c, ede=d-1 >

Subgroups: 512 in 136 conjugacy classes, 58 normal (42 characteristic)
C1, C2, C2 [×4], C3, C4 [×2], C4, C22, C22 [×5], C5, C6, C6 [×4], C8 [×2], C2×C4, C2×C4, D4, D4 [×4], Q8, C23, D5 [×2], C10, C10 [×2], C12 [×2], C12, C2×C6, C2×C6 [×5], C15, M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, C20 [×2], C20, D10 [×4], C2×C10, C2×C10, C24 [×2], C2×C12, C2×C12, C3×D4, C3×D4 [×4], C3×Q8, C22×C6, C3×D5 [×2], C30, C30 [×2], C8⋊C22, C52C8 [×2], D20 [×2], D20, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C22×D5, C3×M4(2), C3×D8 [×2], C3×SD16 [×2], C6×D4, C3×C4○D4, C60 [×2], C60, C6×D5 [×4], C2×C30, C2×C30, C4.Dic5, D4⋊D5 [×2], Q8⋊D5 [×2], C2×D20, C5×C4○D4, C3×C8⋊C22, C3×C52C8 [×2], C3×D20 [×2], C3×D20, C2×C60, C2×C60, D4×C15, D4×C15, Q8×C15, D5×C2×C6, D4⋊D10, C3×C4.Dic5, C3×D4⋊D5 [×2], C3×Q8⋊D5 [×2], C6×D20, C15×C4○D4, C3×D4⋊D10
Quotients: C1, C2 [×7], C3, C22 [×7], C6 [×7], D4 [×2], C23, D5, C2×C6 [×7], C2×D4, D10 [×3], C3×D4 [×2], C22×C6, C3×D5, C8⋊C22, C5⋊D4 [×2], C22×D5, C6×D4, C6×D5 [×3], C2×C5⋊D4, C3×C8⋊C22, C3×C5⋊D4 [×2], D5×C2×C6, D4⋊D10, C6×C5⋊D4, C3×D4⋊D10

Smallest permutation representation of C3×D4⋊D10
On 120 points
Generators in S120
(1 43 23)(2 44 24)(3 45 25)(4 41 21)(5 42 22)(6 46 26)(7 47 27)(8 48 28)(9 49 29)(10 50 30)(11 53 33)(12 54 34)(13 55 35)(14 51 31)(15 52 32)(16 59 39)(17 60 40)(18 56 36)(19 57 37)(20 58 38)(61 107 87)(62 108 88)(63 109 89)(64 110 90)(65 101 81)(66 102 82)(67 103 83)(68 104 84)(69 105 85)(70 106 86)(71 117 97)(72 118 98)(73 119 99)(74 120 100)(75 111 91)(76 112 92)(77 113 93)(78 114 94)(79 115 95)(80 116 96)
(1 17 7 12)(2 18 8 13)(3 19 9 14)(4 20 10 15)(5 16 6 11)(21 38 30 32)(22 39 26 33)(23 40 27 34)(24 36 28 35)(25 37 29 31)(41 58 50 52)(42 59 46 53)(43 60 47 54)(44 56 48 55)(45 57 49 51)(61 71 66 76)(62 72 67 77)(63 73 68 78)(64 74 69 79)(65 75 70 80)(81 91 86 96)(82 92 87 97)(83 93 88 98)(84 94 89 99)(85 95 90 100)(101 111 106 116)(102 112 107 117)(103 113 108 118)(104 114 109 119)(105 115 110 120)
(1 76)(2 72)(3 78)(4 74)(5 80)(6 75)(7 71)(8 77)(9 73)(10 79)(11 65)(12 61)(13 67)(14 63)(15 69)(16 70)(17 66)(18 62)(19 68)(20 64)(21 100)(22 96)(23 92)(24 98)(25 94)(26 91)(27 97)(28 93)(29 99)(30 95)(31 89)(32 85)(33 81)(34 87)(35 83)(36 88)(37 84)(38 90)(39 86)(40 82)(41 120)(42 116)(43 112)(44 118)(45 114)(46 111)(47 117)(48 113)(49 119)(50 115)(51 109)(52 105)(53 101)(54 107)(55 103)(56 108)(57 104)(58 110)(59 106)(60 102)
(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 5)(2 4)(6 7)(8 10)(11 17)(12 16)(13 20)(14 19)(15 18)(21 24)(22 23)(26 27)(28 30)(31 37)(32 36)(33 40)(34 39)(35 38)(41 44)(42 43)(46 47)(48 50)(51 57)(52 56)(53 60)(54 59)(55 58)(61 80)(62 79)(63 78)(64 77)(65 76)(66 75)(67 74)(68 73)(69 72)(70 71)(81 92)(82 91)(83 100)(84 99)(85 98)(86 97)(87 96)(88 95)(89 94)(90 93)(101 112)(102 111)(103 120)(104 119)(105 118)(106 117)(107 116)(108 115)(109 114)(110 113)

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

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

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

93 conjugacy classes

class 1 2A2B2C2D2E3A3B4A4B4C5A5B6A6B6C6D6E6F6G6H6I6J8A8B10A10B10C···10H12A12B12C12D12E12F15A15B15C15D20A20B20C20D20E···20J24A24B24C24D30A30B30C30D30E···30P60A···60H60I···60T
order1222223344455666666666688101010···10121212121212151515152020202020···20242424243030303030···3060···6060···60
size112420201122422112244202020202020224···4222244222222224···42020202022224···42···24···4

93 irreducible representations

dim11111111111122222222222222224444
type++++++++++++++
imageC1C2C2C2C2C2C3C6C6C6C6C6D4D4D5D10D10D10C3×D4C3×D4C3×D5C5⋊D4C5⋊D4C6×D5C6×D5C6×D5C3×C5⋊D4C3×C5⋊D4C8⋊C22C3×C8⋊C22D4⋊D10C3×D4⋊D10
kernelC3×D4⋊D10C3×C4.Dic5C3×D4⋊D5C3×Q8⋊D5C6×D20C15×C4○D4D4⋊D10C4.Dic5D4⋊D5Q8⋊D5C2×D20C5×C4○D4C60C2×C30C3×C4○D4C2×C12C3×D4C3×Q8C20C2×C10C4○D4C12C2×C6C2×C4D4Q8C4C22C15C5C3C1
# reps11221122442211222222444444881248

Matrix representation of C3×D4⋊D10 in GL6(𝔽241)

1500000
0150000
001000
000100
000010
000001
,
24000000
02400000
0019715600
00884400
001281284185
001130156200
,
010000
100000
0021312879126
00019847194
008288156113
0082235113156
,
24000000
02400000
00018900
00515100
0023948152
003191189189
,
24000000
010000
001000
005024000
001594241119
0068156156200

G:=sub<GL(6,GF(241))| [15,0,0,0,0,0,0,15,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,197,88,128,113,0,0,156,44,128,0,0,0,0,0,41,156,0,0,0,0,85,200],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,213,0,82,82,0,0,128,198,88,235,0,0,79,47,156,113,0,0,126,194,113,156],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,0,51,239,3,0,0,189,51,48,191,0,0,0,0,1,189,0,0,0,0,52,189],[240,0,0,0,0,0,0,1,0,0,0,0,0,0,1,50,159,68,0,0,0,240,42,156,0,0,0,0,41,156,0,0,0,0,119,200] >;

C3×D4⋊D10 in GAP, Magma, Sage, TeX

C_3\times D_4\rtimes D_{10}
% in TeX

G:=Group("C3xD4:D10");
// GroupNames label

G:=SmallGroup(480,742);
// by ID

G=gap.SmallGroup(480,742);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,590,555,2524,648,102,18822]);
// Polycyclic

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

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