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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 — C3×C4○D20 — C3×D4.D10
 Lower central C5 — C10 — C20 — C3×D4.D10
 Upper central C1 — C6 — C2×C12 — C6×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 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 ··· 10F 10G ··· 10N 12A 12B 12C 12D 12E 12F 15A 15B 15C 15D 20A 20B 20C 20D 24A 24B 24C 24D 30A ··· 30L 30M ··· 30AB 60A ··· 60H 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 24 24 24 24 30 ··· 30 30 ··· 30 60 ··· 60 size 1 1 2 4 4 20 1 1 2 2 20 2 2 1 1 2 2 4 4 4 4 20 20 20 20 2 ··· 2 4 ··· 4 2 2 2 2 20 20 2 2 2 2 4 4 4 4 20 20 20 20 2 ··· 2 4 ··· 4 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 4 4 4 4 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 D4 D4 D5 D10 D10 C3×D4 C3×D4 C3×D5 C5⋊D4 C5⋊D4 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×D4.D5 C3×C4○D20 D4×C30 D4.D10 C4.Dic5 D4⋊D5 D4.D5 C4○D20 D4×C10 C60 C2×C30 C6×D4 C2×C12 C3×D4 C20 C2×C10 C2×D4 C12 C2×C6 C2×C4 D4 C4 C22 C15 C5 C3 C1 # reps 1 1 2 2 1 1 2 2 4 4 2 2 1 1 2 2 4 2 2 4 4 4 4 8 8 8 1 2 4 8

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

 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 1 155 0 0 0 0 213 240 0 0 0 0 0 0 240 86 0 0 0 0 28 1
,
 1 0 0 0 0 0 144 240 0 0 0 0 0 0 240 86 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 213 240
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 154 11 0 0 0 0 26 87 0 0 0 0 0 0 36 37 0 0 0 0 197 205
,
 240 77 0 0 0 0 0 1 0 0 0 0 0 0 0 0 36 37 0 0 0 0 197 205 0 0 154 11 0 0 0 0 26 87 0 0

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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