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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 [×9], C3, C4 [×2], C4 [×4], C22, C22 [×4], C22 [×10], C5, C6, C6 [×9], C2×C4, C2×C4 [×8], D4 [×4], D4 [×14], Q8 [×2], C23 [×2], C23 [×4], D5 [×4], C10, C10 [×5], C12 [×2], C12 [×4], C2×C6, C2×C6 [×4], C2×C6 [×10], C15, C2×D4, C2×D4 [×8], C4○D4 [×6], Dic5 [×4], C20 [×2], D10 [×4], D10 [×4], C2×C10, C2×C10 [×4], C2×C10 [×2], C2×C12, C2×C12 [×8], C3×D4 [×4], C3×D4 [×14], C3×Q8 [×2], C22×C6 [×2], C22×C6 [×4], C3×D5 [×4], C30, C30 [×5], 2+ 1+4, Dic10 [×2], C4×D5 [×4], D20 [×2], C2×Dic5 [×4], C5⋊D4 [×12], C2×C20, C5×D4 [×4], C22×D5 [×4], C22×C10 [×2], C6×D4, C6×D4 [×8], C3×C4○D4 [×6], C3×Dic5 [×4], C60 [×2], C6×D5 [×4], C6×D5 [×4], C2×C30, C2×C30 [×4], C2×C30 [×2], C4○D20 [×2], D4×D5 [×4], D42D5 [×4], C2×C5⋊D4 [×4], D4×C10, C3×2+ 1+4, C3×Dic10 [×2], D5×C12 [×4], C3×D20 [×2], C6×Dic5 [×4], C3×C5⋊D4 [×12], C2×C60, D4×C15 [×4], D5×C2×C6 [×4], C22×C30 [×2], D46D10, C3×C4○D20 [×2], C3×D4×D5 [×4], C3×D42D5 [×4], C6×C5⋊D4 [×4], D4×C30, C3×D46D10
Quotients: C1, C2 [×15], C3, C22 [×35], C6 [×15], C23 [×15], D5, C2×C6 [×35], C24, D10 [×7], C22×C6 [×15], C3×D5, 2+ 1+4, C22×D5 [×7], C23×C6, C6×D5 [×7], C23×D5, C3×2+ 1+4, D5×C2×C6 [×7], D46D10, D5×C22×C6, C3×D46D10

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

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

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

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