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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, C60.125D4, C60.201C23, D4⋊D55C6, (C6×D4)⋊9D5, C4○D203C6, (D4×C10)⋊2C6, D206(C2×C6), (D4×C30)⋊9C2, D4.D55C6, D4.6(C6×D5), C10.49(C6×D4), C20.15(C3×D4), C4.Dic56C6, C1535(C8⋊C22), Dic105(C2×C6), (C3×D4).35D10, (C2×C30).164D4, C30.403(C2×D4), (C2×C12).239D10, (C3×D20)⋊36C22, C12.95(C5⋊D4), C20.12(C22×C6), (C2×C60).290C22, (D4×C15).40C22, C12.201(C22×D5), (C3×Dic10)⋊32C22, C54(C3×C8⋊C22), C4.12(D5×C2×C6), C52C83(C2×C6), (C2×D4)⋊2(C3×D5), C2.9(C6×C5⋊D4), (C3×D4⋊D5)⋊13C2, (C5×D4).6(C2×C6), (C2×C4).13(C6×D5), C4.16(C3×C5⋊D4), (C3×C4○D20)⋊13C2, (C2×C20).27(C2×C6), (C3×D4.D5)⋊13C2, (C2×C10).39(C3×D4), C6.130(C2×C5⋊D4), (C3×C52C8)⋊25C22, (C2×C6).63(C5⋊D4), (C3×C4.Dic5)⋊18C2, C22.10(C3×C5⋊D4), SmallGroup(480,725)

Series: Derived Chief Lower central Upper central

C1C20 — C3×D4.D10
C1C5C10C20C60C3×D20C3×C4○D20 — C3×D4.D10
C5C10C20 — C3×D4.D10
C1C6C2×C12C6×D4

Generators and relations for C3×D4.D10
 G = < a,b,c,d,e | a3=b4=c2=1, d10=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b-1, bd=db, dcd-1=b2c, ece-1=b-1c, ede-1=d9 >

Subgroups: 416 in 136 conjugacy classes, 58 normal (38 characteristic)
C1, C2, C2 [×4], C3, C4 [×2], C4, C22, C22 [×5], C5, C6, C6 [×4], C8 [×2], C2×C4, C2×C4, D4 [×2], D4 [×3], Q8, C23, D5, C10, C10 [×3], C12 [×2], C12, C2×C6, C2×C6 [×5], C15, M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, Dic5, C20 [×2], D10, C2×C10, C2×C10 [×4], C24 [×2], C2×C12, C2×C12, C3×D4 [×2], C3×D4 [×3], C3×Q8, C22×C6, C3×D5, C30, C30 [×3], C8⋊C22, C52C8 [×2], Dic10, C4×D5, D20, C5⋊D4, C2×C20, C5×D4 [×2], C5×D4, C22×C10, C3×M4(2), C3×D8 [×2], C3×SD16 [×2], C6×D4, C3×C4○D4, C3×Dic5, C60 [×2], C6×D5, C2×C30, C2×C30 [×4], C4.Dic5, D4⋊D5 [×2], D4.D5 [×2], C4○D20, D4×C10, C3×C8⋊C22, C3×C52C8 [×2], C3×Dic10, D5×C12, C3×D20, C3×C5⋊D4, C2×C60, D4×C15 [×2], D4×C15, C22×C30, D4.D10, C3×C4.Dic5, C3×D4⋊D5 [×2], C3×D4.D5 [×2], C3×C4○D20, D4×C30, 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 42 22)(2 43 23)(3 44 24)(4 45 25)(5 46 26)(6 47 27)(7 48 28)(8 49 29)(9 50 30)(10 51 31)(11 52 32)(12 53 33)(13 54 34)(14 55 35)(15 56 36)(16 57 37)(17 58 38)(18 59 39)(19 60 40)(20 41 21)(61 115 95)(62 116 96)(63 117 97)(64 118 98)(65 119 99)(66 120 100)(67 101 81)(68 102 82)(69 103 83)(70 104 84)(71 105 85)(72 106 86)(73 107 87)(74 108 88)(75 109 89)(76 110 90)(77 111 91)(78 112 92)(79 113 93)(80 114 94)
(1 16 11 6)(2 17 12 7)(3 18 13 8)(4 19 14 9)(5 20 15 10)(21 36 31 26)(22 37 32 27)(23 38 33 28)(24 39 34 29)(25 40 35 30)(41 56 51 46)(42 57 52 47)(43 58 53 48)(44 59 54 49)(45 60 55 50)(61 66 71 76)(62 67 72 77)(63 68 73 78)(64 69 74 79)(65 70 75 80)(81 86 91 96)(82 87 92 97)(83 88 93 98)(84 89 94 99)(85 90 95 100)(101 106 111 116)(102 107 112 117)(103 108 113 118)(104 109 114 119)(105 110 115 120)
(1 11)(3 13)(5 15)(7 17)(9 19)(22 32)(24 34)(26 36)(28 38)(30 40)(42 52)(44 54)(46 56)(48 58)(50 60)(61 66)(62 77)(63 68)(64 79)(65 70)(67 72)(69 74)(71 76)(73 78)(75 80)(81 86)(82 97)(83 88)(84 99)(85 90)(87 92)(89 94)(91 96)(93 98)(95 100)(101 106)(102 117)(103 108)(104 119)(105 110)(107 112)(109 114)(111 116)(113 118)(115 120)
(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 75 11 65)(2 64 12 74)(3 73 13 63)(4 62 14 72)(5 71 15 61)(6 80 16 70)(7 69 17 79)(8 78 18 68)(9 67 19 77)(10 76 20 66)(21 100 31 90)(22 89 32 99)(23 98 33 88)(24 87 34 97)(25 96 35 86)(26 85 36 95)(27 94 37 84)(28 83 38 93)(29 92 39 82)(30 81 40 91)(41 120 51 110)(42 109 52 119)(43 118 53 108)(44 107 54 117)(45 116 55 106)(46 105 56 115)(47 114 57 104)(48 103 58 113)(49 112 59 102)(50 101 60 111)

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

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

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

93 conjugacy classes

class 1 2A2B2C2D2E3A3B4A4B4C5A5B6A6B6C6D6E6F6G6H6I6J8A8B10A···10F10G···10N12A12B12C12D12E12F15A15B15C15D20A20B20C20D24A24B24C24D30A···30L30M···30AB60A···60H
order122222334445566666666668810···1010···1012121212121215151515202020202424242430···3030···3060···60
size11244201122202211224444202020202···24···42222202022224444202020202···24···44···4

93 irreducible representations

dim111111111111222222222222224444
type++++++++++++
imageC1C2C2C2C2C2C3C6C6C6C6C6D4D4D5D10D10C3×D4C3×D4C3×D5C5⋊D4C5⋊D4C6×D5C6×D5C3×C5⋊D4C3×C5⋊D4C8⋊C22C3×C8⋊C22D4.D10C3×D4.D10
kernelC3×D4.D10C3×C4.Dic5C3×D4⋊D5C3×D4.D5C3×C4○D20D4×C30D4.D10C4.Dic5D4⋊D5D4.D5C4○D20D4×C10C60C2×C30C6×D4C2×C12C3×D4C20C2×C10C2×D4C12C2×C6C2×C4D4C4C22C15C5C3C1
# reps112211224422112242244448881248

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

22500000
02250000
001000
000100
000010
000001
,
24000000
02400000
00115500
0021324000
000024086
0000281
,
100000
1442400000
002408600
000100
000010
0000213240
,
100000
010000
001541100
00268700
00003637
0000197205
,
240770000
010000
00003637
0000197205
001541100
00268700

G:=sub<GL(6,GF(241))| [225,0,0,0,0,0,0,225,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,1,213,0,0,0,0,155,240,0,0,0,0,0,0,240,28,0,0,0,0,86,1],[1,144,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,86,1,0,0,0,0,0,0,1,213,0,0,0,0,0,240],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,154,26,0,0,0,0,11,87,0,0,0,0,0,0,36,197,0,0,0,0,37,205],[240,0,0,0,0,0,77,1,0,0,0,0,0,0,0,0,154,26,0,0,0,0,11,87,0,0,36,197,0,0,0,0,37,205,0,0] >;

C3×D4.D10 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(480,725);
// by ID

G=gap.SmallGroup(480,725);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,344,590,555,2524,648,102,18822]);
// Polycyclic

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

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