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## G = C5×D6⋊C4order 240 = 24·3·5

### Direct product of C5 and D6⋊C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C5×D6⋊C4
 Chief series C1 — C3 — C6 — C2×C6 — C2×C30 — S3×C2×C10 — C5×D6⋊C4
 Lower central C3 — C6 — C5×D6⋊C4
 Upper central C1 — C2×C10 — C2×C20

Generators and relations for C5×D6⋊C4
G = < a,b,c,d | a5=b6=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, dcd-1=b3c >

Subgroups: 152 in 68 conjugacy classes, 34 normal (30 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C22, C22 [×4], C5, S3 [×2], C6 [×3], C2×C4, C2×C4, C23, C10 [×3], C10 [×2], Dic3, C12, D6 [×2], D6 [×2], C2×C6, C15, C22⋊C4, C20 [×2], C2×C10, C2×C10 [×4], C2×Dic3, C2×C12, C22×S3, C5×S3 [×2], C30 [×3], C2×C20, C2×C20, C22×C10, D6⋊C4, C5×Dic3, C60, S3×C10 [×2], S3×C10 [×2], C2×C30, C5×C22⋊C4, C10×Dic3, C2×C60, S3×C2×C10, C5×D6⋊C4
Quotients: C1, C2 [×3], C4 [×2], C22, C5, S3, C2×C4, D4 [×2], C10 [×3], D6, C22⋊C4, C20 [×2], C2×C10, C4×S3, D12, C3⋊D4, C5×S3, C2×C20, C5×D4 [×2], D6⋊C4, S3×C10, C5×C22⋊C4, S3×C20, C5×D12, C5×C3⋊D4, C5×D6⋊C4

Smallest permutation representation of C5×D6⋊C4
On 120 points
Generators in S120
(1 53 41 29 17)(2 54 42 30 18)(3 49 37 25 13)(4 50 38 26 14)(5 51 39 27 15)(6 52 40 28 16)(7 55 43 31 19)(8 56 44 32 20)(9 57 45 33 21)(10 58 46 34 22)(11 59 47 35 23)(12 60 48 36 24)(61 109 97 85 73)(62 110 98 86 74)(63 111 99 87 75)(64 112 100 88 76)(65 113 101 89 77)(66 114 102 90 78)(67 115 103 91 79)(68 116 104 92 80)(69 117 105 93 81)(70 118 106 94 82)(71 119 107 95 83)(72 120 108 96 84)
(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 6)(2 5)(3 4)(7 8)(9 12)(10 11)(13 14)(15 18)(16 17)(19 20)(21 24)(22 23)(25 26)(27 30)(28 29)(31 32)(33 36)(34 35)(37 38)(39 42)(40 41)(43 44)(45 48)(46 47)(49 50)(51 54)(52 53)(55 56)(57 60)(58 59)(61 65)(62 64)(67 71)(68 70)(73 77)(74 76)(79 83)(80 82)(85 89)(86 88)(91 95)(92 94)(97 101)(98 100)(103 107)(104 106)(109 113)(110 112)(115 119)(116 118)
(1 71 11 65)(2 72 12 66)(3 67 7 61)(4 68 8 62)(5 69 9 63)(6 70 10 64)(13 79 19 73)(14 80 20 74)(15 81 21 75)(16 82 22 76)(17 83 23 77)(18 84 24 78)(25 91 31 85)(26 92 32 86)(27 93 33 87)(28 94 34 88)(29 95 35 89)(30 96 36 90)(37 103 43 97)(38 104 44 98)(39 105 45 99)(40 106 46 100)(41 107 47 101)(42 108 48 102)(49 115 55 109)(50 116 56 110)(51 117 57 111)(52 118 58 112)(53 119 59 113)(54 120 60 114)

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

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

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

90 conjugacy classes

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

90 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 C4 C5 C10 C10 C10 C20 S3 D4 D6 C4×S3 D12 C3⋊D4 C5×S3 C5×D4 S3×C10 S3×C20 C5×D12 C5×C3⋊D4 kernel C5×D6⋊C4 C10×Dic3 C2×C60 S3×C2×C10 S3×C10 D6⋊C4 C2×Dic3 C2×C12 C22×S3 D6 C2×C20 C30 C2×C10 C10 C10 C10 C2×C4 C6 C22 C2 C2 C2 # reps 1 1 1 1 4 4 4 4 4 16 1 2 1 2 2 2 4 8 4 8 8 8

Matrix representation of C5×D6⋊C4 in GL4(𝔽61) generated by

 9 0 0 0 0 9 0 0 0 0 9 0 0 0 0 9
,
 60 0 0 0 0 60 0 0 0 0 0 60 0 0 1 1
,
 60 0 0 0 23 1 0 0 0 0 0 60 0 0 60 0
,
 53 55 0 0 21 8 0 0 0 0 52 43 0 0 18 9
G:=sub<GL(4,GF(61))| [9,0,0,0,0,9,0,0,0,0,9,0,0,0,0,9],[60,0,0,0,0,60,0,0,0,0,0,1,0,0,60,1],[60,23,0,0,0,1,0,0,0,0,0,60,0,0,60,0],[53,21,0,0,55,8,0,0,0,0,52,18,0,0,43,9] >;

C5×D6⋊C4 in GAP, Magma, Sage, TeX

C_5\times D_6\rtimes C_4
% in TeX

G:=Group("C5xD6:C4");
// GroupNames label

G:=SmallGroup(240,59);
// by ID

G=gap.SmallGroup(240,59);
# by ID

G:=PCGroup([6,-2,-2,-5,-2,-2,-3,505,127,5765]);
// Polycyclic

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

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