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## G = C5×C6.D4order 240 = 24·3·5

### Direct product of C5 and C6.D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C5×C6.D4
 Chief series C1 — C3 — C6 — C2×C6 — C2×C30 — C10×Dic3 — C5×C6.D4
 Lower central C3 — C6 — C5×C6.D4
 Upper central C1 — C2×C10 — C22×C10

Generators and relations for C5×C6.D4
G = < a,b,c,d | a5=b6=c4=1, d2=b3, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=b3c-1 >

Subgroups: 120 in 68 conjugacy classes, 38 normal (18 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C5, C6, C6, C6, C2×C4, C23, C10, C10, C10, Dic3, C2×C6, C2×C6, C2×C6, C15, C22⋊C4, C20, C2×C10, C2×C10, C2×C10, C2×Dic3, C22×C6, C30, C30, C30, C2×C20, C22×C10, C6.D4, C5×Dic3, C2×C30, C2×C30, C2×C30, C5×C22⋊C4, C10×Dic3, C22×C30, C5×C6.D4
Quotients: C1, C2, C4, C22, C5, S3, C2×C4, D4, C10, Dic3, D6, C22⋊C4, C20, C2×C10, C2×Dic3, C3⋊D4, C5×S3, C2×C20, C5×D4, C6.D4, C5×Dic3, S3×C10, C5×C22⋊C4, C10×Dic3, C5×C3⋊D4, C5×C6.D4

Smallest permutation representation of C5×C6.D4
On 120 points
Generators in S120
(1 66 49 37 25)(2 61 50 38 26)(3 62 51 39 27)(4 63 52 40 28)(5 64 53 41 29)(6 65 54 42 30)(7 110 98 86 74)(8 111 99 87 75)(9 112 100 88 76)(10 113 101 89 77)(11 114 102 90 78)(12 109 97 85 73)(13 58 46 34 22)(14 59 47 35 23)(15 60 48 36 24)(16 55 43 31 19)(17 56 44 32 20)(18 57 45 33 21)(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 67 14 74)(2 72 15 73)(3 71 16 78)(4 70 17 77)(5 69 18 76)(6 68 13 75)(7 66 115 59)(8 65 116 58)(9 64 117 57)(10 63 118 56)(11 62 119 55)(12 61 120 60)(19 90 27 83)(20 89 28 82)(21 88 29 81)(22 87 30 80)(23 86 25 79)(24 85 26 84)(31 102 39 95)(32 101 40 94)(33 100 41 93)(34 99 42 92)(35 98 37 91)(36 97 38 96)(43 114 51 107)(44 113 52 106)(45 112 53 105)(46 111 54 104)(47 110 49 103)(48 109 50 108)
(1 77 4 74)(2 76 5 73)(3 75 6 78)(7 66 10 63)(8 65 11 62)(9 64 12 61)(13 71 16 68)(14 70 17 67)(15 69 18 72)(19 80 22 83)(20 79 23 82)(21 84 24 81)(25 89 28 86)(26 88 29 85)(27 87 30 90)(31 92 34 95)(32 91 35 94)(33 96 36 93)(37 101 40 98)(38 100 41 97)(39 99 42 102)(43 104 46 107)(44 103 47 106)(45 108 48 105)(49 113 52 110)(50 112 53 109)(51 111 54 114)(55 116 58 119)(56 115 59 118)(57 120 60 117)

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

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

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

90 conjugacy classes

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

90 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + + + - + image C1 C2 C2 C4 C5 C10 C10 C20 S3 D4 Dic3 D6 C3⋊D4 C5×S3 C5×D4 C5×Dic3 S3×C10 C5×C3⋊D4 kernel C5×C6.D4 C10×Dic3 C22×C30 C2×C30 C6.D4 C2×Dic3 C22×C6 C2×C6 C22×C10 C30 C2×C10 C2×C10 C10 C23 C6 C22 C22 C2 # reps 1 2 1 4 4 8 4 16 1 2 2 1 4 4 8 8 4 16

Matrix representation of C5×C6.D4 in GL3(𝔽61) generated by

 1 0 0 0 58 0 0 0 58
,
 60 0 0 0 47 0 0 0 13
,
 11 0 0 0 0 1 0 60 0
,
 50 0 0 0 0 1 0 1 0
G:=sub<GL(3,GF(61))| [1,0,0,0,58,0,0,0,58],[60,0,0,0,47,0,0,0,13],[11,0,0,0,0,60,0,1,0],[50,0,0,0,0,1,0,1,0] >;

C5×C6.D4 in GAP, Magma, Sage, TeX

C_5\times C_6.D_4
% in TeX

G:=Group("C5xC6.D4");
// GroupNames label

G:=SmallGroup(240,64);
// by ID

G=gap.SmallGroup(240,64);
# by ID

G:=PCGroup([6,-2,-2,-5,-2,-2,-3,120,505,5765]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^6=c^4=1,d^2=b^3,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=b^3*c^-1>;
// generators/relations

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