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## G = C3×D4⋊6D10order 480 = 25·3·5

### Direct product of C3 and D4⋊6D10

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C3×D4⋊6D10
 Chief series C1 — C5 — C10 — C30 — C6×D5 — D5×C2×C6 — C3×D4×D5 — C3×D4⋊6D10
 Lower central C5 — C10 — C3×D4⋊6D10
 Upper central C1 — C6 — C6×D4

Generators and relations for C3×D46D10
G = < a,b,c,d,e | a3=b4=c2=d10=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=dbd-1=b-1, be=eb, dcd-1=ece=b2c, ede=d-1 >

Subgroups: 1040 in 332 conjugacy classes, 170 normal (22 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, C5, C6, C6, C2×C4, C2×C4, D4, D4, Q8, C23, C23, D5, C10, C10, C12, C12, C2×C6, C2×C6, C2×C6, C15, C2×D4, C2×D4, C4○D4, Dic5, C20, D10, D10, C2×C10, C2×C10, C2×C10, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C22×C6, C22×C6, C3×D5, C30, C30, 2+ 1+4, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C22×D5, C22×C10, C6×D4, C6×D4, C3×C4○D4, C3×Dic5, C60, C6×D5, C6×D5, C2×C30, C2×C30, C2×C30, C4○D20, D4×D5, D42D5, C2×C5⋊D4, D4×C10, C3×2+ 1+4, C3×Dic10, D5×C12, C3×D20, C6×Dic5, C3×C5⋊D4, C2×C60, D4×C15, D5×C2×C6, C22×C30, D46D10, C3×C4○D20, C3×D4×D5, C3×D42D5, C6×C5⋊D4, D4×C30, C3×D46D10
Quotients: C1, C2, C3, C22, C6, C23, D5, C2×C6, C24, D10, C22×C6, C3×D5, 2+ 1+4, C22×D5, C23×C6, C6×D5, C23×D5, C3×2+ 1+4, D5×C2×C6, D46D10, D5×C22×C6, C3×D46D10

Smallest permutation representation of C3×D46D10
On 120 points
Generators in S120
(1 22 12)(2 23 13)(3 24 14)(4 25 15)(5 21 11)(6 28 18)(7 29 19)(8 30 20)(9 26 16)(10 27 17)(31 55 45)(32 51 41)(33 52 42)(34 53 43)(35 54 44)(36 60 50)(37 56 46)(38 57 47)(39 58 48)(40 59 49)(61 87 71)(62 88 72)(63 89 73)(64 90 74)(65 81 75)(66 82 76)(67 83 77)(68 84 78)(69 85 79)(70 86 80)(91 111 101)(92 112 102)(93 113 103)(94 114 104)(95 115 105)(96 116 106)(97 117 107)(98 118 108)(99 119 109)(100 120 110)
(1 92 10 97)(2 98 6 93)(3 94 7 99)(4 100 8 95)(5 96 9 91)(11 106 16 101)(12 102 17 107)(13 108 18 103)(14 104 19 109)(15 110 20 105)(21 116 26 111)(22 112 27 117)(23 118 28 113)(24 114 29 119)(25 120 30 115)(31 70 36 65)(32 66 37 61)(33 62 38 67)(34 68 39 63)(35 64 40 69)(41 76 46 71)(42 72 47 77)(43 78 48 73)(44 74 49 79)(45 80 50 75)(51 82 56 87)(52 88 57 83)(53 84 58 89)(54 90 59 85)(55 86 60 81)
(1 61)(2 67)(3 63)(4 69)(5 65)(6 62)(7 68)(8 64)(9 70)(10 66)(11 75)(12 71)(13 77)(14 73)(15 79)(16 80)(17 76)(18 72)(19 78)(20 74)(21 81)(22 87)(23 83)(24 89)(25 85)(26 86)(27 82)(28 88)(29 84)(30 90)(31 91)(32 97)(33 93)(34 99)(35 95)(36 96)(37 92)(38 98)(39 94)(40 100)(41 107)(42 103)(43 109)(44 105)(45 101)(46 102)(47 108)(48 104)(49 110)(50 106)(51 117)(52 113)(53 119)(54 115)(55 111)(56 112)(57 118)(58 114)(59 120)(60 116)
(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 8)(9 10)(11 12)(13 15)(16 17)(18 20)(21 22)(23 25)(26 27)(28 30)(31 37)(32 36)(33 40)(34 39)(35 38)(41 50)(42 49)(43 48)(44 47)(45 46)(51 60)(52 59)(53 58)(54 57)(55 56)(61 70)(62 69)(63 68)(64 67)(65 66)(71 80)(72 79)(73 78)(74 77)(75 76)(81 82)(83 90)(84 89)(85 88)(86 87)(91 97)(92 96)(93 95)(98 100)(101 107)(102 106)(103 105)(108 110)(111 117)(112 116)(113 115)(118 120)

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

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

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

111 conjugacy classes

 class 1 2A 2B ··· 2F 2G 2H 2I 2J 3A 3B 4A 4B 4C 4D 4E 4F 5A 5B 6A 6B 6C ··· 6L 6M ··· 6T 10A ··· 10F 10G ··· 10N 12A 12B 12C 12D 12E ··· 12L 15A 15B 15C 15D 20A 20B 20C 20D 30A ··· 30L 30M ··· 30AB 60A ··· 60H order 1 2 2 ··· 2 2 2 2 2 3 3 4 4 4 4 4 4 5 5 6 6 6 ··· 6 6 ··· 6 10 ··· 10 10 ··· 10 12 12 12 12 12 ··· 12 15 15 15 15 20 20 20 20 30 ··· 30 30 ··· 30 60 ··· 60 size 1 1 2 ··· 2 10 10 10 10 1 1 2 2 10 10 10 10 2 2 1 1 2 ··· 2 10 ··· 10 2 ··· 2 4 ··· 4 2 2 2 2 10 ··· 10 2 2 2 2 4 4 4 4 2 ··· 2 4 ··· 4 4 ··· 4

111 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 D5 D10 D10 D10 C3×D5 C6×D5 C6×D5 C6×D5 2+ 1+4 C3×2+ 1+4 D4⋊6D10 C3×D4⋊6D10 kernel C3×D4⋊6D10 C3×C4○D20 C3×D4×D5 C3×D4⋊2D5 C6×C5⋊D4 D4×C30 D4⋊6D10 C4○D20 D4×D5 D4⋊2D5 C2×C5⋊D4 D4×C10 C6×D4 C2×C12 C3×D4 C22×C6 C2×D4 C2×C4 D4 C23 C15 C5 C3 C1 # reps 1 2 4 4 4 1 2 4 8 8 8 2 2 2 8 4 4 4 16 8 1 2 4 8

Matrix representation of C3×D46D10 in GL6(𝔽61)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 47 0 0 0 0 0 0 47 0 0 0 0 0 0 47 0 0 0 0 0 0 47
,
 60 0 0 0 0 0 0 60 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 60 0 0 0 0 60 0 0 0
,
 60 0 0 0 0 0 0 60 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0
,
 0 43 0 0 0 0 17 18 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 60 0 0 0 0 0 0 60
,
 1 0 0 0 0 0 60 60 0 0 0 0 0 0 1 0 0 0 0 0 0 60 0 0 0 0 0 0 60 0 0 0 0 0 0 1

G:=sub<GL(6,GF(61))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,47,0,0,0,0,0,0,47,0,0,0,0,0,0,47,0,0,0,0,0,0,47],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,0,60,0,0,0,0,60,0,0,0,0,1,0,0,0,0,1,0,0,0],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[0,17,0,0,0,0,43,18,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[1,60,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,1] >;

C3×D46D10 in GAP, Magma, Sage, TeX

C_3\times D_4\rtimes_6D_{10}
% in TeX

G:=Group("C3xD4:6D10");
// GroupNames label

G:=SmallGroup(480,1141);
// by ID

G=gap.SmallGroup(480,1141);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-5,555,1571,18822]);
// 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=d*b*d^-1=b^-1,b*e=e*b,d*c*d^-1=e*c*e=b^2*c,e*d*e=d^-1>;
// generators/relations

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