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## G = C2×D10⋊D6order 480 = 25·3·5

### Direct product of C2 and D10⋊D6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — C2×D10⋊D6
 Chief series C1 — C5 — C15 — C30 — C6×D5 — C2×S3×D5 — C22×S3×D5 — C2×D10⋊D6
 Lower central C15 — C30 — C2×D10⋊D6
 Upper central C1 — C22 — C23

Generators and relations for C2×D10⋊D6
G = < a,b,c,d,e | a2=b10=c2=d6=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe=b-1, bd=db, dcd-1=b5c, ece=b3c, ede=d-1 >

Subgroups: 2940 in 472 conjugacy classes, 124 normal (36 characteristic)
C1, C2, C2 [×2], C2 [×12], C3, C4 [×4], C22, C22 [×2], C22 [×36], C5, S3 [×8], C6, C6 [×2], C6 [×4], C2×C4 [×6], D4 [×16], C23, C23 [×20], D5 [×8], C10, C10 [×2], C10 [×4], Dic3 [×2], C12 [×2], D6 [×2], D6 [×28], C2×C6, C2×C6 [×2], C2×C6 [×6], C15, C22×C4, C2×D4 [×12], C24 [×2], Dic5 [×2], C20 [×2], D10 [×2], D10 [×28], C2×C10, C2×C10 [×2], C2×C10 [×6], C4×S3 [×4], D12 [×4], C2×Dic3, C3⋊D4 [×4], C3⋊D4 [×4], C2×C12, C3×D4 [×4], C22×S3, C22×S3 [×18], C22×C6, C22×C6, C5×S3 [×2], C3×D5 [×2], D15 [×4], D15 [×2], C30, C30 [×2], C30 [×2], C22×D4, C4×D5 [×4], D20 [×4], C2×Dic5, C5⋊D4 [×4], C5⋊D4 [×4], C2×C20, C5×D4 [×4], C22×D5, C22×D5 [×18], C22×C10, C22×C10, S3×C2×C4, C2×D12, S3×D4 [×8], C2×C3⋊D4, C2×C3⋊D4, C6×D4, S3×C23 [×2], C5×Dic3 [×2], C3×Dic5 [×2], S3×D5 [×8], C6×D5 [×2], C6×D5 [×2], S3×C10 [×2], S3×C10 [×2], D30 [×8], D30 [×10], C2×C30, C2×C30 [×2], C2×C30 [×2], C2×C4×D5, C2×D20, D4×D5 [×8], C2×C5⋊D4, C2×C5⋊D4, D4×C10, C23×D5 [×2], C2×S3×D4, D30.C2 [×4], C3⋊D20 [×4], C5⋊D12 [×4], C6×Dic5, C3×C5⋊D4 [×4], C10×Dic3, C5×C3⋊D4 [×4], C2×S3×D5 [×4], C2×S3×D5 [×4], D5×C2×C6, S3×C2×C10, C22×D15 [×2], C22×D15 [×4], C22×D15 [×4], C22×C30, C2×D4×D5, C2×D30.C2, C2×C3⋊D20, C2×C5⋊D12, D10⋊D6 [×8], C6×C5⋊D4, C10×C3⋊D4, C22×S3×D5, C23×D15, C2×D10⋊D6
Quotients: C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D5, D6 [×7], C2×D4 [×6], C24, D10 [×7], C22×S3 [×7], C22×D4, C22×D5 [×7], S3×D4 [×2], S3×C23, S3×D5, D4×D5 [×2], C23×D5, C2×S3×D4, C2×S3×D5 [×3], C2×D4×D5, D10⋊D6 [×2], C22×S3×D5, C2×D10⋊D6

Smallest permutation representation of C2×D10⋊D6
On 120 points
Generators in S120
(1 87)(2 88)(3 89)(4 90)(5 81)(6 82)(7 83)(8 84)(9 85)(10 86)(11 65)(12 66)(13 67)(14 68)(15 69)(16 70)(17 61)(18 62)(19 63)(20 64)(21 77)(22 78)(23 79)(24 80)(25 71)(26 72)(27 73)(28 74)(29 75)(30 76)(31 91)(32 92)(33 93)(34 94)(35 95)(36 96)(37 97)(38 98)(39 99)(40 100)(41 101)(42 102)(43 103)(44 104)(45 105)(46 106)(47 107)(48 108)(49 109)(50 110)(51 111)(52 112)(53 113)(54 114)(55 115)(56 116)(57 117)(58 118)(59 119)(60 120)
(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 10)(2 9)(3 8)(4 7)(5 6)(11 15)(12 14)(16 20)(17 19)(21 26)(22 25)(23 24)(27 30)(28 29)(31 38)(32 37)(33 36)(34 35)(39 40)(41 43)(44 50)(45 49)(46 48)(51 57)(52 56)(53 55)(58 60)(61 63)(64 70)(65 69)(66 68)(71 78)(72 77)(73 76)(74 75)(79 80)(81 82)(83 90)(84 89)(85 88)(86 87)(91 98)(92 97)(93 96)(94 95)(99 100)(101 103)(104 110)(105 109)(106 108)(111 117)(112 116)(113 115)(118 120)
(1 70 29 57 40 45)(2 61 30 58 31 46)(3 62 21 59 32 47)(4 63 22 60 33 48)(5 64 23 51 34 49)(6 65 24 52 35 50)(7 66 25 53 36 41)(8 67 26 54 37 42)(9 68 27 55 38 43)(10 69 28 56 39 44)(11 80 112 95 110 82)(12 71 113 96 101 83)(13 72 114 97 102 84)(14 73 115 98 103 85)(15 74 116 99 104 86)(16 75 117 100 105 87)(17 76 118 91 106 88)(18 77 119 92 107 89)(19 78 120 93 108 90)(20 79 111 94 109 81)
(1 110)(2 109)(3 108)(4 107)(5 106)(6 105)(7 104)(8 103)(9 102)(10 101)(11 40)(12 39)(13 38)(14 37)(15 36)(16 35)(17 34)(18 33)(19 32)(20 31)(21 120)(22 119)(23 118)(24 117)(25 116)(26 115)(27 114)(28 113)(29 112)(30 111)(41 86)(42 85)(43 84)(44 83)(45 82)(46 81)(47 90)(48 89)(49 88)(50 87)(51 76)(52 75)(53 74)(54 73)(55 72)(56 71)(57 80)(58 79)(59 78)(60 77)(61 94)(62 93)(63 92)(64 91)(65 100)(66 99)(67 98)(68 97)(69 96)(70 95)

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

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

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

66 conjugacy classes

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

66 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 S3 D4 D5 D6 D6 D6 D6 D10 D10 D10 D10 S3×D4 S3×D5 D4×D5 C2×S3×D5 D10⋊D6 kernel C2×D10⋊D6 C2×D30.C2 C2×C3⋊D20 C2×C5⋊D12 D10⋊D6 C6×C5⋊D4 C10×C3⋊D4 C22×S3×D5 C23×D15 C2×C5⋊D4 D30 C2×C3⋊D4 C2×Dic5 C5⋊D4 C22×D5 C22×C10 C2×Dic3 C3⋊D4 C22×S3 C22×C6 C10 C23 C6 C22 C2 # reps 1 1 1 1 8 1 1 1 1 1 4 2 1 4 1 1 2 8 2 2 2 2 4 6 8

Matrix representation of C2×D10⋊D6 in GL6(𝔽61)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 60 0 0 0 0 0 0 60 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 17 0 0 0 0 43 43 0 0 0 0 0 0 60 0 0 0 0 0 0 60
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 42 60 0 0 0 0 0 0 60 0 0 0 0 0 53 1
,
 0 60 0 0 0 0 1 1 0 0 0 0 0 0 60 0 0 0 0 0 0 60 0 0 0 0 0 0 60 46 0 0 0 0 0 1
,
 0 60 0 0 0 0 60 0 0 0 0 0 0 0 43 44 0 0 0 0 19 18 0 0 0 0 0 0 1 15 0 0 0 0 0 60

G:=sub<GL(6,GF(61))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,43,0,0,0,0,17,43,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,42,0,0,0,0,0,60,0,0,0,0,0,0,60,53,0,0,0,0,0,1],[0,1,0,0,0,0,60,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,46,1],[0,60,0,0,0,0,60,0,0,0,0,0,0,0,43,19,0,0,0,0,44,18,0,0,0,0,0,0,1,0,0,0,0,0,15,60] >;

C2×D10⋊D6 in GAP, Magma, Sage, TeX

C_2\times D_{10}\rtimes D_6
% in TeX

G:=Group("C2xD10:D6");
// GroupNames label

G:=SmallGroup(480,1124);
// by ID

G=gap.SmallGroup(480,1124);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,675,346,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^10=c^2=d^6=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^5*c,e*c*e=b^3*c,e*d*e=d^-1>;
// generators/relations

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