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## G = C3×D4⋊D10order 480 = 25·3·5

### Direct product of C3 and D4⋊D10

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C3×D4⋊D10
 Chief series C1 — C5 — C10 — C20 — C60 — C3×D20 — C6×D20 — C3×D4⋊D10
 Lower central C5 — C10 — C20 — C3×D4⋊D10
 Upper central C1 — C6 — C2×C12 — C3×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 2A 2B 2C 2D 2E 3A 3B 4A 4B 4C 5A 5B 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 8A 8B 10A 10B 10C ··· 10H 12A 12B 12C 12D 12E 12F 15A 15B 15C 15D 20A 20B 20C 20D 20E ··· 20J 24A 24B 24C 24D 30A 30B 30C 30D 30E ··· 30P 60A ··· 60H 60I ··· 60T order 1 2 2 2 2 2 3 3 4 4 4 5 5 6 6 6 6 6 6 6 6 6 6 8 8 10 10 10 ··· 10 12 12 12 12 12 12 15 15 15 15 20 20 20 20 20 ··· 20 24 24 24 24 30 30 30 30 30 ··· 30 60 ··· 60 60 ··· 60 size 1 1 2 4 20 20 1 1 2 2 4 2 2 1 1 2 2 4 4 20 20 20 20 20 20 2 2 4 ··· 4 2 2 2 2 4 4 2 2 2 2 2 2 2 2 4 ··· 4 20 20 20 20 2 2 2 2 4 ··· 4 2 ··· 2 4 ··· 4

93 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 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 D4 D4 D5 D10 D10 D10 C3×D4 C3×D4 C3×D5 C5⋊D4 C5⋊D4 C6×D5 C6×D5 C6×D5 C3×C5⋊D4 C3×C5⋊D4 C8⋊C22 C3×C8⋊C22 D4⋊D10 C3×D4⋊D10 kernel C3×D4⋊D10 C3×C4.Dic5 C3×D4⋊D5 C3×Q8⋊D5 C6×D20 C15×C4○D4 D4⋊D10 C4.Dic5 D4⋊D5 Q8⋊D5 C2×D20 C5×C4○D4 C60 C2×C30 C3×C4○D4 C2×C12 C3×D4 C3×Q8 C20 C2×C10 C4○D4 C12 C2×C6 C2×C4 D4 Q8 C4 C22 C15 C5 C3 C1 # reps 1 1 2 2 1 1 2 2 4 4 2 2 1 1 2 2 2 2 2 2 4 4 4 4 4 4 8 8 1 2 4 8

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

 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 0 197 156 0 0 0 0 88 44 0 0 0 0 128 128 41 85 0 0 113 0 156 200
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 213 128 79 126 0 0 0 198 47 194 0 0 82 88 156 113 0 0 82 235 113 156
,
 240 0 0 0 0 0 0 240 0 0 0 0 0 0 0 189 0 0 0 0 51 51 0 0 0 0 239 48 1 52 0 0 3 191 189 189
,
 240 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 50 240 0 0 0 0 159 42 41 119 0 0 68 156 156 200

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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