Copied to
clipboard

## G = C3×C4.F5order 240 = 24·3·5

### Direct product of C3 and C4.F5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C3×C4.F5
 Chief series C1 — C5 — C10 — Dic5 — C3×Dic5 — C3×C5⋊C8 — C3×C4.F5
 Lower central C5 — C10 — C3×C4.F5
 Upper central C1 — C6 — C12

Generators and relations for C3×C4.F5
G = < a,b,c,d | a3=b4=c5=1, d4=b2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c3 >

Smallest permutation representation of C3×C4.F5
On 120 points
Generators in S120
(1 100 60)(2 101 61)(3 102 62)(4 103 63)(5 104 64)(6 97 57)(7 98 58)(8 99 59)(9 86 46)(10 87 47)(11 88 48)(12 81 41)(13 82 42)(14 83 43)(15 84 44)(16 85 45)(17 95 120)(18 96 113)(19 89 114)(20 90 115)(21 91 116)(22 92 117)(23 93 118)(24 94 119)(25 110 66)(26 111 67)(27 112 68)(28 105 69)(29 106 70)(30 107 71)(31 108 72)(32 109 65)(33 56 74)(34 49 75)(35 50 76)(36 51 77)(37 52 78)(38 53 79)(39 54 80)(40 55 73)
(1 7 5 3)(2 4 6 8)(9 90 13 94)(10 95 14 91)(11 92 15 96)(12 89 16 93)(17 43 21 47)(18 48 22 44)(19 45 23 41)(20 42 24 46)(25 33 29 37)(26 38 30 34)(27 35 31 39)(28 40 32 36)(49 111 53 107)(50 108 54 112)(51 105 55 109)(52 110 56 106)(57 59 61 63)(58 64 62 60)(65 77 69 73)(66 74 70 78)(67 79 71 75)(68 76 72 80)(81 114 85 118)(82 119 86 115)(83 116 87 120)(84 113 88 117)(97 99 101 103)(98 104 102 100)
(1 49 46 22 105)(2 23 50 106 47)(3 107 24 48 51)(4 41 108 52 17)(5 53 42 18 109)(6 19 54 110 43)(7 111 20 44 55)(8 45 112 56 21)(9 92 69 100 75)(10 101 93 76 70)(11 77 102 71 94)(12 72 78 95 103)(13 96 65 104 79)(14 97 89 80 66)(15 73 98 67 90)(16 68 74 91 99)(25 83 57 114 39)(26 115 84 40 58)(27 33 116 59 85)(28 60 34 86 117)(29 87 61 118 35)(30 119 88 36 62)(31 37 120 63 81)(32 64 38 82 113)
(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)

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

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

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

C3×C4.F5 is a maximal subgroup of   D124F5  D602C4  D10.Dic6  D10.2Dic6  D12.F5  D15⋊M4(2)  Dic6.F5

42 conjugacy classes

 class 1 2A 2B 3A 3B 4A 4B 4C 5 6A 6B 6C 6D 8A 8B 8C 8D 10 12A 12B 12C 12D 12E 12F 15A 15B 20A 20B 24A ··· 24H 30A 30B 60A 60B 60C 60D order 1 2 2 3 3 4 4 4 5 6 6 6 6 8 8 8 8 10 12 12 12 12 12 12 15 15 20 20 24 ··· 24 30 30 60 60 60 60 size 1 1 10 1 1 2 5 5 4 1 1 10 10 10 10 10 10 4 2 2 5 5 5 5 4 4 4 4 10 ··· 10 4 4 4 4 4 4

42 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 4 4 4 4 4 4 type + + + + + image C1 C2 C2 C3 C4 C4 C6 C6 C12 C12 M4(2) C3×M4(2) F5 C2×F5 C3×F5 C4.F5 C6×F5 C3×C4.F5 kernel C3×C4.F5 C3×C5⋊C8 D5×C12 C4.F5 C60 C6×D5 C5⋊C8 C4×D5 C20 D10 C15 C5 C12 C6 C4 C3 C2 C1 # reps 1 2 1 2 2 2 4 2 4 4 2 4 1 1 2 2 2 4

Matrix representation of C3×C4.F5 in GL4(𝔽7) generated by

 2 0 0 0 0 2 0 0 0 0 2 0 0 0 0 2
,
 5 4 4 4 5 1 0 3 2 6 4 2 4 2 5 4
,
 1 0 2 0 0 4 6 0 6 2 0 1 2 5 5 1
,
 5 3 3 2 6 6 5 1 4 2 2 4 0 5 5 1
G:=sub<GL(4,GF(7))| [2,0,0,0,0,2,0,0,0,0,2,0,0,0,0,2],[5,5,2,4,4,1,6,2,4,0,4,5,4,3,2,4],[1,0,6,2,0,4,2,5,2,6,0,5,0,0,1,1],[5,6,4,0,3,6,2,5,3,5,2,5,2,1,4,1] >;

C3×C4.F5 in GAP, Magma, Sage, TeX

C_3\times C_4.F_5
% in TeX

G:=Group("C3xC4.F5");
// GroupNames label

G:=SmallGroup(240,112);
// by ID

G=gap.SmallGroup(240,112);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-5,72,313,151,69,3461,599]);
// Polycyclic

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

Export

׿
×
𝔽