Copied to
clipboard

## G = C6×C5⋊D4order 240 = 24·3·5

### Direct product of C6 and C5⋊D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C6×C5⋊D4
 Chief series C1 — C5 — C10 — C30 — C6×D5 — D5×C2×C6 — C6×C5⋊D4
 Lower central C5 — C10 — C6×C5⋊D4
 Upper central C1 — C2×C6 — C22×C6

Generators and relations for C6×C5⋊D4
G = < a,b,c,d | a6=b5=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 292 in 108 conjugacy classes, 54 normal (22 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C5, C6, C6, C6, C2×C4, D4, C23, C23, D5, C10, C10, C10, C12, C2×C6, C2×C6, C2×C6, C15, C2×D4, Dic5, D10, D10, C2×C10, C2×C10, C2×C10, C2×C12, C3×D4, C22×C6, C22×C6, C3×D5, C30, C30, C30, C2×Dic5, C5⋊D4, C22×D5, C22×C10, C6×D4, C3×Dic5, C6×D5, C6×D5, C2×C30, C2×C30, C2×C30, C2×C5⋊D4, C6×Dic5, C3×C5⋊D4, D5×C2×C6, C22×C30, C6×C5⋊D4
Quotients: C1, C2, C3, C22, C6, D4, C23, D5, C2×C6, C2×D4, D10, C3×D4, C22×C6, C3×D5, C5⋊D4, C22×D5, C6×D4, C6×D5, C2×C5⋊D4, C3×C5⋊D4, D5×C2×C6, C6×C5⋊D4

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

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

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

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

78 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 4A 4B 5A 5B 6A ··· 6F 6G 6H 6I 6J 6K 6L 6M 6N 10A ··· 10N 12A 12B 12C 12D 15A 15B 15C 15D 30A ··· 30AB order 1 2 2 2 2 2 2 2 3 3 4 4 5 5 6 ··· 6 6 6 6 6 6 6 6 6 10 ··· 10 12 12 12 12 15 15 15 15 30 ··· 30 size 1 1 1 1 2 2 10 10 1 1 10 10 2 2 1 ··· 1 2 2 2 2 10 10 10 10 2 ··· 2 10 10 10 10 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 type + + + + + + + + image C1 C2 C2 C2 C2 C3 C6 C6 C6 C6 D4 D5 D10 C3×D4 C3×D5 C5⋊D4 C6×D5 C3×C5⋊D4 kernel C6×C5⋊D4 C6×Dic5 C3×C5⋊D4 D5×C2×C6 C22×C30 C2×C5⋊D4 C2×Dic5 C5⋊D4 C22×D5 C22×C10 C30 C22×C6 C2×C6 C10 C23 C6 C22 C2 # reps 1 1 4 1 1 2 2 8 2 2 2 2 6 4 4 8 12 16

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

 48 0 0 0 60 0 0 0 60
,
 1 0 0 0 0 60 0 1 43
,
 60 0 0 0 31 8 0 17 30
,
 60 0 0 0 1 43 0 0 60
G:=sub<GL(3,GF(61))| [48,0,0,0,60,0,0,0,60],[1,0,0,0,0,1,0,60,43],[60,0,0,0,31,17,0,8,30],[60,0,0,0,1,0,0,43,60] >;

C6×C5⋊D4 in GAP, Magma, Sage, TeX

C_6\times C_5\rtimes D_4
% in TeX

G:=Group("C6xC5:D4");
// GroupNames label

G:=SmallGroup(240,164);
// by ID

G=gap.SmallGroup(240,164);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-2,-5,506,6917]);
// Polycyclic

G:=Group<a,b,c,d|a^6=b^5=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

׿
×
𝔽