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## G = C3×D10⋊C4order 240 = 24·3·5

### Direct product of C3 and D10⋊C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C3×D10⋊C4
 Chief series C1 — C5 — C10 — C2×C10 — C2×C30 — D5×C2×C6 — C3×D10⋊C4
 Lower central C5 — C10 — C3×D10⋊C4
 Upper central C1 — C2×C6 — C2×C12

Generators and relations for C3×D10⋊C4
G = < a,b,c,d | a3=b10=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, dcd-1=b5c >

Subgroups: 212 in 68 conjugacy classes, 34 normal (30 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, C6, C6, C2×C4, C2×C4, C23, D5, C10, C12, C2×C6, C2×C6, C15, C22⋊C4, Dic5, C20, D10, D10, C2×C10, C2×C12, C2×C12, C22×C6, C3×D5, C30, C2×Dic5, C2×C20, C22×D5, C3×C22⋊C4, C3×Dic5, C60, C6×D5, C6×D5, C2×C30, D10⋊C4, C6×Dic5, C2×C60, D5×C2×C6, C3×D10⋊C4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, D5, C12, C2×C6, C22⋊C4, D10, C2×C12, C3×D4, C3×D5, C4×D5, D20, C5⋊D4, C3×C22⋊C4, C6×D5, D10⋊C4, D5×C12, C3×D20, C3×C5⋊D4, C3×D10⋊C4

Smallest permutation representation of C3×D10⋊C4
On 120 points
Generators in S120
(1 47 27)(2 48 28)(3 49 29)(4 50 30)(5 41 21)(6 42 22)(7 43 23)(8 44 24)(9 45 25)(10 46 26)(11 51 31)(12 52 32)(13 53 33)(14 54 34)(15 55 35)(16 56 36)(17 57 37)(18 58 38)(19 59 39)(20 60 40)(61 101 81)(62 102 82)(63 103 83)(64 104 84)(65 105 85)(66 106 86)(67 107 87)(68 108 88)(69 109 89)(70 110 90)(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 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 10)(2 9)(3 8)(4 7)(5 6)(11 12)(13 20)(14 19)(15 18)(16 17)(21 22)(23 30)(24 29)(25 28)(26 27)(31 32)(33 40)(34 39)(35 38)(36 37)(41 42)(43 50)(44 49)(45 48)(46 47)(51 52)(53 60)(54 59)(55 58)(56 57)(61 67)(62 66)(63 65)(68 70)(71 77)(72 76)(73 75)(78 80)(81 87)(82 86)(83 85)(88 90)(91 97)(92 96)(93 95)(98 100)(101 107)(102 106)(103 105)(108 110)(111 117)(112 116)(113 115)(118 120)
(1 77 17 67)(2 78 18 68)(3 79 19 69)(4 80 20 70)(5 71 11 61)(6 72 12 62)(7 73 13 63)(8 74 14 64)(9 75 15 65)(10 76 16 66)(21 91 31 81)(22 92 32 82)(23 93 33 83)(24 94 34 84)(25 95 35 85)(26 96 36 86)(27 97 37 87)(28 98 38 88)(29 99 39 89)(30 100 40 90)(41 111 51 101)(42 112 52 102)(43 113 53 103)(44 114 54 104)(45 115 55 105)(46 116 56 106)(47 117 57 107)(48 118 58 108)(49 119 59 109)(50 120 60 110)

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

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

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

78 conjugacy classes

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

78 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C3 C4 C6 C6 C6 C12 D4 D5 D10 C3×D4 C3×D5 C4×D5 D20 C5⋊D4 C6×D5 D5×C12 C3×D20 C3×C5⋊D4 kernel C3×D10⋊C4 C6×Dic5 C2×C60 D5×C2×C6 D10⋊C4 C6×D5 C2×Dic5 C2×C20 C22×D5 D10 C30 C2×C12 C2×C6 C10 C2×C4 C6 C6 C6 C22 C2 C2 C2 # reps 1 1 1 1 2 4 2 2 2 8 2 2 2 4 4 4 4 4 4 8 8 8

Matrix representation of C3×D10⋊C4 in GL3(𝔽61) generated by

 1 0 0 0 47 0 0 0 47
,
 1 0 0 0 18 44 0 18 0
,
 60 0 0 0 60 1 0 0 1
,
 11 0 0 0 31 16 0 1 30
G:=sub<GL(3,GF(61))| [1,0,0,0,47,0,0,0,47],[1,0,0,0,18,18,0,44,0],[60,0,0,0,60,0,0,1,1],[11,0,0,0,31,1,0,16,30] >;

C3×D10⋊C4 in GAP, Magma, Sage, TeX

C_3\times D_{10}\rtimes C_4
% in TeX

G:=Group("C3xD10:C4");
// GroupNames label

G:=SmallGroup(240,43);
// by ID

G=gap.SmallGroup(240,43);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-5,313,79,6917]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^10=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,b*d=d*b,d*c*d^-1=b^5*c>;
// generators/relations

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