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## G = C10×C3⋊D4order 240 = 24·3·5

### Direct product of C10 and C3⋊D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C10×C3⋊D4
 Chief series C1 — C3 — C6 — C30 — S3×C10 — S3×C2×C10 — C10×C3⋊D4
 Lower central C3 — C6 — C10×C3⋊D4
 Upper central C1 — C2×C10 — C22×C10

Generators and relations for C10×C3⋊D4
G = < a,b,c,d | a10=b3=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 216 in 108 conjugacy classes, 54 normal (22 characteristic)
C1, C2, C2 [×2], C2 [×4], C3, C4 [×2], C22, C22 [×2], C22 [×6], C5, S3 [×2], C6, C6 [×2], C6 [×2], C2×C4, D4 [×4], C23, C23, C10, C10 [×2], C10 [×4], Dic3 [×2], D6 [×2], D6 [×2], C2×C6, C2×C6 [×2], C2×C6 [×2], C15, C2×D4, C20 [×2], C2×C10, C2×C10 [×2], C2×C10 [×6], C2×Dic3, C3⋊D4 [×4], C22×S3, C22×C6, C5×S3 [×2], C30, C30 [×2], C30 [×2], C2×C20, C5×D4 [×4], C22×C10, C22×C10, C2×C3⋊D4, C5×Dic3 [×2], S3×C10 [×2], S3×C10 [×2], C2×C30, C2×C30 [×2], C2×C30 [×2], D4×C10, C10×Dic3, C5×C3⋊D4 [×4], S3×C2×C10, C22×C30, C10×C3⋊D4
Quotients: C1, C2 [×7], C22 [×7], C5, S3, D4 [×2], C23, C10 [×7], D6 [×3], C2×D4, C2×C10 [×7], C3⋊D4 [×2], C22×S3, C5×S3, C5×D4 [×2], C22×C10, C2×C3⋊D4, S3×C10 [×3], D4×C10, C5×C3⋊D4 [×2], S3×C2×C10, C10×C3⋊D4

Smallest permutation representation of C10×C3⋊D4
On 120 points
Generators in S120
(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 56 28)(2 57 29)(3 58 30)(4 59 21)(5 60 22)(6 51 23)(7 52 24)(8 53 25)(9 54 26)(10 55 27)(11 112 80)(12 113 71)(13 114 72)(14 115 73)(15 116 74)(16 117 75)(17 118 76)(18 119 77)(19 120 78)(20 111 79)(31 108 83)(32 109 84)(33 110 85)(34 101 86)(35 102 87)(36 103 88)(37 104 89)(38 105 90)(39 106 81)(40 107 82)(41 69 92)(42 70 93)(43 61 94)(44 62 95)(45 63 96)(46 64 97)(47 65 98)(48 66 99)(49 67 100)(50 68 91)
(1 20 43 104)(2 11 44 105)(3 12 45 106)(4 13 46 107)(5 14 47 108)(6 15 48 109)(7 16 49 110)(8 17 50 101)(9 18 41 102)(10 19 42 103)(21 114 97 82)(22 115 98 83)(23 116 99 84)(24 117 100 85)(25 118 91 86)(26 119 92 87)(27 120 93 88)(28 111 94 89)(29 112 95 90)(30 113 96 81)(31 60 73 65)(32 51 74 66)(33 52 75 67)(34 53 76 68)(35 54 77 69)(36 55 78 70)(37 56 79 61)(38 57 80 62)(39 58 71 63)(40 59 72 64)
(11 105)(12 106)(13 107)(14 108)(15 109)(16 110)(17 101)(18 102)(19 103)(20 104)(21 59)(22 60)(23 51)(24 52)(25 53)(26 54)(27 55)(28 56)(29 57)(30 58)(31 115)(32 116)(33 117)(34 118)(35 119)(36 120)(37 111)(38 112)(39 113)(40 114)(61 94)(62 95)(63 96)(64 97)(65 98)(66 99)(67 100)(68 91)(69 92)(70 93)(71 81)(72 82)(73 83)(74 84)(75 85)(76 86)(77 87)(78 88)(79 89)(80 90)

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

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

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

90 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 5A 5B 5C 5D 6A ··· 6G 10A ··· 10L 10M ··· 10T 10U ··· 10AB 15A 15B 15C 15D 20A ··· 20H 30A ··· 30AB order 1 2 2 2 2 2 2 2 3 4 4 5 5 5 5 6 ··· 6 10 ··· 10 10 ··· 10 10 ··· 10 15 15 15 15 20 ··· 20 30 ··· 30 size 1 1 1 1 2 2 6 6 2 6 6 1 1 1 1 2 ··· 2 1 ··· 1 2 ··· 2 6 ··· 6 2 2 2 2 6 ··· 6 2 ··· 2

90 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C2 C5 C10 C10 C10 C10 S3 D4 D6 C3⋊D4 C5×S3 C5×D4 S3×C10 C5×C3⋊D4 kernel C10×C3⋊D4 C10×Dic3 C5×C3⋊D4 S3×C2×C10 C22×C30 C2×C3⋊D4 C2×Dic3 C3⋊D4 C22×S3 C22×C6 C22×C10 C30 C2×C10 C10 C23 C6 C22 C2 # reps 1 1 4 1 1 4 4 16 4 4 1 2 3 4 4 8 12 16

Matrix representation of C10×C3⋊D4 in GL3(𝔽61) generated by

 60 0 0 0 27 0 0 0 27
,
 1 0 0 0 60 60 0 1 0
,
 60 0 0 0 9 18 0 9 52
,
 60 0 0 0 1 0 0 60 60
G:=sub<GL(3,GF(61))| [60,0,0,0,27,0,0,0,27],[1,0,0,0,60,1,0,60,0],[60,0,0,0,9,9,0,18,52],[60,0,0,0,1,60,0,0,60] >;

C10×C3⋊D4 in GAP, Magma, Sage, TeX

C_{10}\times C_3\rtimes D_4
% in TeX

G:=Group("C10xC3:D4");
// GroupNames label

G:=SmallGroup(240,174);
// by ID

G=gap.SmallGroup(240,174);
# by ID

G:=PCGroup([6,-2,-2,-2,-5,-2,-3,794,5765]);
// Polycyclic

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

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