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## G = C3×C8⋊D10order 480 = 25·3·5

### Direct product of C3 and C8⋊D10

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C3×C8⋊D10
 Chief series C1 — C5 — C10 — C20 — C60 — C3×D20 — C6×D20 — C3×C8⋊D10
 Lower central C5 — C10 — C20 — C3×C8⋊D10
 Upper central C1 — C6 — C2×C12 — C3×M4(2)

Generators and relations for C3×C8⋊D10
G = < a,b,c,d | a3=b8=c10=d2=1, ab=ba, ac=ca, ad=da, cbc-1=b5, dbd=b-1, dcd=c-1 >

Subgroups: 608 in 136 conjugacy classes, 58 normal (38 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C10, C12, C12, C2×C6, C2×C6, C15, M4(2), D8, SD16, C2×D4, C4○D4, Dic5, C20, D10, C2×C10, C24, C2×C12, C2×C12, C3×D4, C3×Q8, C22×C6, C3×D5, C30, C30, C8⋊C22, C40, Dic10, C4×D5, D20, D20, D20, C5⋊D4, C2×C20, C22×D5, C3×M4(2), C3×D8, C3×SD16, C6×D4, C3×C4○D4, C3×Dic5, C60, C6×D5, C2×C30, C40⋊C2, D40, C5×M4(2), C2×D20, C4○D20, C3×C8⋊C22, C120, C3×Dic10, D5×C12, C3×D20, C3×D20, C3×D20, C3×C5⋊D4, C2×C60, D5×C2×C6, C8⋊D10, C3×C40⋊C2, C3×D40, C15×M4(2), C6×D20, C3×C4○D20, C3×C8⋊D10
Quotients: C1, C2, C3, C22, C6, D4, C23, D5, C2×C6, C2×D4, D10, C3×D4, C22×C6, C3×D5, C8⋊C22, D20, C22×D5, C6×D4, C6×D5, C2×D20, C3×C8⋊C22, C3×D20, D5×C2×C6, C8⋊D10, C6×D20, C3×C8⋊D10

Smallest permutation representation of C3×C8⋊D10
On 120 points
Generators in S120
(1 41 21)(2 42 22)(3 43 23)(4 44 24)(5 45 25)(6 49 29)(7 50 30)(8 46 26)(9 47 27)(10 48 28)(11 55 35)(12 51 31)(13 52 32)(14 53 33)(15 54 34)(16 56 36)(17 57 37)(18 58 38)(19 59 39)(20 60 40)(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)(71 111 91)(72 112 92)(73 113 93)(74 114 94)(75 115 95)(76 116 96)(77 117 97)(78 118 98)(79 119 99)(80 120 100)
(1 72 16 66 8 77 12 61)(2 78 17 62 9 73 13 67)(3 74 18 68 10 79 14 63)(4 80 19 64 6 75 15 69)(5 76 20 70 7 71 11 65)(21 92 36 82 26 97 31 87)(22 98 37 88 27 93 32 83)(23 94 38 84 28 99 33 89)(24 100 39 90 29 95 34 85)(25 96 40 86 30 91 35 81)(41 112 56 102 46 117 51 107)(42 118 57 108 47 113 52 103)(43 114 58 104 48 119 53 109)(44 120 59 110 49 115 54 105)(45 116 60 106 50 111 55 101)
(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 9)(7 8)(11 16)(12 20)(13 19)(14 18)(15 17)(21 25)(22 24)(26 30)(27 29)(31 40)(32 39)(33 38)(34 37)(35 36)(41 45)(42 44)(46 50)(47 49)(51 60)(52 59)(53 58)(54 57)(55 56)(61 76)(62 75)(63 74)(64 73)(65 72)(66 71)(67 80)(68 79)(69 78)(70 77)(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)| (1,41,21)(2,42,22)(3,43,23)(4,44,24)(5,45,25)(6,49,29)(7,50,30)(8,46,26)(9,47,27)(10,48,28)(11,55,35)(12,51,31)(13,52,32)(14,53,33)(15,54,34)(16,56,36)(17,57,37)(18,58,38)(19,59,39)(20,60,40)(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)(71,111,91)(72,112,92)(73,113,93)(74,114,94)(75,115,95)(76,116,96)(77,117,97)(78,118,98)(79,119,99)(80,120,100), (1,72,16,66,8,77,12,61)(2,78,17,62,9,73,13,67)(3,74,18,68,10,79,14,63)(4,80,19,64,6,75,15,69)(5,76,20,70,7,71,11,65)(21,92,36,82,26,97,31,87)(22,98,37,88,27,93,32,83)(23,94,38,84,28,99,33,89)(24,100,39,90,29,95,34,85)(25,96,40,86,30,91,35,81)(41,112,56,102,46,117,51,107)(42,118,57,108,47,113,52,103)(43,114,58,104,48,119,53,109)(44,120,59,110,49,115,54,105)(45,116,60,106,50,111,55,101), (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)(6,9)(7,8)(11,16)(12,20)(13,19)(14,18)(15,17)(21,25)(22,24)(26,30)(27,29)(31,40)(32,39)(33,38)(34,37)(35,36)(41,45)(42,44)(46,50)(47,49)(51,60)(52,59)(53,58)(54,57)(55,56)(61,76)(62,75)(63,74)(64,73)(65,72)(66,71)(67,80)(68,79)(69,78)(70,77)(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( (1,41,21)(2,42,22)(3,43,23)(4,44,24)(5,45,25)(6,49,29)(7,50,30)(8,46,26)(9,47,27)(10,48,28)(11,55,35)(12,51,31)(13,52,32)(14,53,33)(15,54,34)(16,56,36)(17,57,37)(18,58,38)(19,59,39)(20,60,40)(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)(71,111,91)(72,112,92)(73,113,93)(74,114,94)(75,115,95)(76,116,96)(77,117,97)(78,118,98)(79,119,99)(80,120,100), (1,72,16,66,8,77,12,61)(2,78,17,62,9,73,13,67)(3,74,18,68,10,79,14,63)(4,80,19,64,6,75,15,69)(5,76,20,70,7,71,11,65)(21,92,36,82,26,97,31,87)(22,98,37,88,27,93,32,83)(23,94,38,84,28,99,33,89)(24,100,39,90,29,95,34,85)(25,96,40,86,30,91,35,81)(41,112,56,102,46,117,51,107)(42,118,57,108,47,113,52,103)(43,114,58,104,48,119,53,109)(44,120,59,110,49,115,54,105)(45,116,60,106,50,111,55,101), (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)(6,9)(7,8)(11,16)(12,20)(13,19)(14,18)(15,17)(21,25)(22,24)(26,30)(27,29)(31,40)(32,39)(33,38)(34,37)(35,36)(41,45)(42,44)(46,50)(47,49)(51,60)(52,59)(53,58)(54,57)(55,56)(61,76)(62,75)(63,74)(64,73)(65,72)(66,71)(67,80)(68,79)(69,78)(70,77)(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([[(1,41,21),(2,42,22),(3,43,23),(4,44,24),(5,45,25),(6,49,29),(7,50,30),(8,46,26),(9,47,27),(10,48,28),(11,55,35),(12,51,31),(13,52,32),(14,53,33),(15,54,34),(16,56,36),(17,57,37),(18,58,38),(19,59,39),(20,60,40),(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),(71,111,91),(72,112,92),(73,113,93),(74,114,94),(75,115,95),(76,116,96),(77,117,97),(78,118,98),(79,119,99),(80,120,100)], [(1,72,16,66,8,77,12,61),(2,78,17,62,9,73,13,67),(3,74,18,68,10,79,14,63),(4,80,19,64,6,75,15,69),(5,76,20,70,7,71,11,65),(21,92,36,82,26,97,31,87),(22,98,37,88,27,93,32,83),(23,94,38,84,28,99,33,89),(24,100,39,90,29,95,34,85),(25,96,40,86,30,91,35,81),(41,112,56,102,46,117,51,107),(42,118,57,108,47,113,52,103),(43,114,58,104,48,119,53,109),(44,120,59,110,49,115,54,105),(45,116,60,106,50,111,55,101)], [(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),(6,9),(7,8),(11,16),(12,20),(13,19),(14,18),(15,17),(21,25),(22,24),(26,30),(27,29),(31,40),(32,39),(33,38),(34,37),(35,36),(41,45),(42,44),(46,50),(47,49),(51,60),(52,59),(53,58),(54,57),(55,56),(61,76),(62,75),(63,74),(64,73),(65,72),(66,71),(67,80),(68,79),(69,78),(70,77),(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 ··· 6J 8A 8B 10A 10B 10C 10D 12A 12B 12C 12D 12E 12F 15A 15B 15C 15D 20A 20B 20C 20D 20E 20F 24A 24B 24C 24D 30A 30B 30C 30D 30E 30F 30G 30H 40A ··· 40H 60A ··· 60H 60I 60J 60K 60L 120A ··· 120P order 1 2 2 2 2 2 3 3 4 4 4 5 5 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 30 30 40 ··· 40 60 ··· 60 60 60 60 60 120 ··· 120 size 1 1 2 20 20 20 1 1 2 2 20 2 2 1 1 2 2 20 ··· 20 4 4 2 2 4 4 2 2 2 2 20 20 2 2 2 2 2 2 2 2 4 4 4 4 4 4 2 2 2 2 4 4 4 4 4 ··· 4 2 ··· 2 4 4 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 D20 D20 C6×D5 C6×D5 C3×D20 C3×D20 C8⋊C22 C3×C8⋊C22 C8⋊D10 C3×C8⋊D10 kernel C3×C8⋊D10 C3×C40⋊C2 C3×D40 C15×M4(2) C6×D20 C3×C4○D20 C8⋊D10 C40⋊C2 D40 C5×M4(2) C2×D20 C4○D20 C60 C2×C30 C3×M4(2) C24 C2×C12 C20 C2×C10 M4(2) C12 C2×C6 C8 C2×C4 C4 C22 C15 C5 C3 C1 # reps 1 2 2 1 1 1 2 4 4 2 2 2 1 1 2 4 2 2 2 4 4 4 8 4 8 8 1 2 4 8

Matrix representation of C3×C8⋊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
,
 44 156 0 0 0 0 88 197 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 197 85 0 0 0 0 153 44 0 0
,
 190 52 0 0 0 0 190 0 0 0 0 0 0 0 51 189 0 0 0 0 51 0 0 0 0 0 0 0 190 52 0 0 0 0 190 0
,
 1 240 0 0 0 0 0 240 0 0 0 0 0 0 1 240 0 0 0 0 0 240 0 0 0 0 0 0 197 200 0 0 0 0 153 44

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],[44,88,0,0,0,0,156,197,0,0,0,0,0,0,0,0,197,153,0,0,0,0,85,44,0,0,1,0,0,0,0,0,0,1,0,0],[190,190,0,0,0,0,52,0,0,0,0,0,0,0,51,51,0,0,0,0,189,0,0,0,0,0,0,0,190,190,0,0,0,0,52,0],[1,0,0,0,0,0,240,240,0,0,0,0,0,0,1,0,0,0,0,0,240,240,0,0,0,0,0,0,197,153,0,0,0,0,200,44] >;

C3×C8⋊D10 in GAP, Magma, Sage, TeX

C_3\times C_8\rtimes D_{10}
% in TeX

G:=Group("C3xC8:D10");
// GroupNames label

G:=SmallGroup(480,701);
// by ID

G=gap.SmallGroup(480,701);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,590,555,142,2524,102,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^8=c^10=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^5,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

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