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## G = C3×C22.F5order 240 = 24·3·5

### Direct product of C3 and C22.F5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C3×C22.F5
 Chief series C1 — C5 — C10 — Dic5 — C3×Dic5 — C3×C5⋊C8 — C3×C22.F5
 Lower central C5 — C10 — C3×C22.F5
 Upper central C1 — C6 — C2×C6

Generators and relations for C3×C22.F5
G = < a,b,c,d,e | a3=b2=c2=d5=1, e4=c, ab=ba, ac=ca, ad=da, ae=ea, ebe-1=bc=cb, bd=db, cd=dc, ce=ec, ede-1=d3 >

Smallest permutation representation of C3×C22.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 51)(10 87 52)(11 88 53)(12 81 54)(13 82 55)(14 83 56)(15 84 49)(16 85 50)(17 94 119)(18 95 120)(19 96 113)(20 89 114)(21 90 115)(22 91 116)(23 92 117)(24 93 118)(25 45 66)(26 46 67)(27 47 68)(28 48 69)(29 41 70)(30 42 71)(31 43 72)(32 44 65)(33 108 74)(34 109 75)(35 110 76)(36 111 77)(37 112 78)(38 105 79)(39 106 80)(40 107 73)
(2 6)(4 8)(10 14)(12 16)(18 22)(20 24)(25 29)(27 31)(33 37)(35 39)(41 45)(43 47)(50 54)(52 56)(57 61)(59 63)(66 70)(68 72)(74 78)(76 80)(81 85)(83 87)(89 93)(91 95)(97 101)(99 103)(106 110)(108 112)(114 118)(116 120)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)(33 37)(34 38)(35 39)(36 40)(41 45)(42 46)(43 47)(44 48)(49 53)(50 54)(51 55)(52 56)(57 61)(58 62)(59 63)(60 64)(65 69)(66 70)(67 71)(68 72)(73 77)(74 78)(75 79)(76 80)(81 85)(82 86)(83 87)(84 88)(89 93)(90 94)(91 95)(92 96)(97 101)(98 102)(99 103)(100 104)(105 109)(106 110)(107 111)(108 112)(113 117)(114 118)(115 119)(116 120)
(1 109 46 23 53)(2 24 110 54 47)(3 55 17 48 111)(4 41 56 112 18)(5 105 42 19 49)(6 20 106 50 43)(7 51 21 44 107)(8 45 52 108 22)(9 90 65 73 98)(10 74 91 99 66)(11 100 75 67 92)(12 68 101 93 76)(13 94 69 77 102)(14 78 95 103 70)(15 104 79 71 96)(16 72 97 89 80)(25 87 33 116 59)(26 117 88 60 34)(27 61 118 35 81)(28 36 62 82 119)(29 83 37 120 63)(30 113 84 64 38)(31 57 114 39 85)(32 40 58 86 115)
(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)(9,86,51)(10,87,52)(11,88,53)(12,81,54)(13,82,55)(14,83,56)(15,84,49)(16,85,50)(17,94,119)(18,95,120)(19,96,113)(20,89,114)(21,90,115)(22,91,116)(23,92,117)(24,93,118)(25,45,66)(26,46,67)(27,47,68)(28,48,69)(29,41,70)(30,42,71)(31,43,72)(32,44,65)(33,108,74)(34,109,75)(35,110,76)(36,111,77)(37,112,78)(38,105,79)(39,106,80)(40,107,73), (2,6)(4,8)(10,14)(12,16)(18,22)(20,24)(25,29)(27,31)(33,37)(35,39)(41,45)(43,47)(50,54)(52,56)(57,61)(59,63)(66,70)(68,72)(74,78)(76,80)(81,85)(83,87)(89,93)(91,95)(97,101)(99,103)(106,110)(108,112)(114,118)(116,120), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(41,45)(42,46)(43,47)(44,48)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64)(65,69)(66,70)(67,71)(68,72)(73,77)(74,78)(75,79)(76,80)(81,85)(82,86)(83,87)(84,88)(89,93)(90,94)(91,95)(92,96)(97,101)(98,102)(99,103)(100,104)(105,109)(106,110)(107,111)(108,112)(113,117)(114,118)(115,119)(116,120), (1,109,46,23,53)(2,24,110,54,47)(3,55,17,48,111)(4,41,56,112,18)(5,105,42,19,49)(6,20,106,50,43)(7,51,21,44,107)(8,45,52,108,22)(9,90,65,73,98)(10,74,91,99,66)(11,100,75,67,92)(12,68,101,93,76)(13,94,69,77,102)(14,78,95,103,70)(15,104,79,71,96)(16,72,97,89,80)(25,87,33,116,59)(26,117,88,60,34)(27,61,118,35,81)(28,36,62,82,119)(29,83,37,120,63)(30,113,84,64,38)(31,57,114,39,85)(32,40,58,86,115), (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)(9,86,51)(10,87,52)(11,88,53)(12,81,54)(13,82,55)(14,83,56)(15,84,49)(16,85,50)(17,94,119)(18,95,120)(19,96,113)(20,89,114)(21,90,115)(22,91,116)(23,92,117)(24,93,118)(25,45,66)(26,46,67)(27,47,68)(28,48,69)(29,41,70)(30,42,71)(31,43,72)(32,44,65)(33,108,74)(34,109,75)(35,110,76)(36,111,77)(37,112,78)(38,105,79)(39,106,80)(40,107,73), (2,6)(4,8)(10,14)(12,16)(18,22)(20,24)(25,29)(27,31)(33,37)(35,39)(41,45)(43,47)(50,54)(52,56)(57,61)(59,63)(66,70)(68,72)(74,78)(76,80)(81,85)(83,87)(89,93)(91,95)(97,101)(99,103)(106,110)(108,112)(114,118)(116,120), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(41,45)(42,46)(43,47)(44,48)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64)(65,69)(66,70)(67,71)(68,72)(73,77)(74,78)(75,79)(76,80)(81,85)(82,86)(83,87)(84,88)(89,93)(90,94)(91,95)(92,96)(97,101)(98,102)(99,103)(100,104)(105,109)(106,110)(107,111)(108,112)(113,117)(114,118)(115,119)(116,120), (1,109,46,23,53)(2,24,110,54,47)(3,55,17,48,111)(4,41,56,112,18)(5,105,42,19,49)(6,20,106,50,43)(7,51,21,44,107)(8,45,52,108,22)(9,90,65,73,98)(10,74,91,99,66)(11,100,75,67,92)(12,68,101,93,76)(13,94,69,77,102)(14,78,95,103,70)(15,104,79,71,96)(16,72,97,89,80)(25,87,33,116,59)(26,117,88,60,34)(27,61,118,35,81)(28,36,62,82,119)(29,83,37,120,63)(30,113,84,64,38)(31,57,114,39,85)(32,40,58,86,115), (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),(9,86,51),(10,87,52),(11,88,53),(12,81,54),(13,82,55),(14,83,56),(15,84,49),(16,85,50),(17,94,119),(18,95,120),(19,96,113),(20,89,114),(21,90,115),(22,91,116),(23,92,117),(24,93,118),(25,45,66),(26,46,67),(27,47,68),(28,48,69),(29,41,70),(30,42,71),(31,43,72),(32,44,65),(33,108,74),(34,109,75),(35,110,76),(36,111,77),(37,112,78),(38,105,79),(39,106,80),(40,107,73)], [(2,6),(4,8),(10,14),(12,16),(18,22),(20,24),(25,29),(27,31),(33,37),(35,39),(41,45),(43,47),(50,54),(52,56),(57,61),(59,63),(66,70),(68,72),(74,78),(76,80),(81,85),(83,87),(89,93),(91,95),(97,101),(99,103),(106,110),(108,112),(114,118),(116,120)], [(1,5),(2,6),(3,7),(4,8),(9,13),(10,14),(11,15),(12,16),(17,21),(18,22),(19,23),(20,24),(25,29),(26,30),(27,31),(28,32),(33,37),(34,38),(35,39),(36,40),(41,45),(42,46),(43,47),(44,48),(49,53),(50,54),(51,55),(52,56),(57,61),(58,62),(59,63),(60,64),(65,69),(66,70),(67,71),(68,72),(73,77),(74,78),(75,79),(76,80),(81,85),(82,86),(83,87),(84,88),(89,93),(90,94),(91,95),(92,96),(97,101),(98,102),(99,103),(100,104),(105,109),(106,110),(107,111),(108,112),(113,117),(114,118),(115,119),(116,120)], [(1,109,46,23,53),(2,24,110,54,47),(3,55,17,48,111),(4,41,56,112,18),(5,105,42,19,49),(6,20,106,50,43),(7,51,21,44,107),(8,45,52,108,22),(9,90,65,73,98),(10,74,91,99,66),(11,100,75,67,92),(12,68,101,93,76),(13,94,69,77,102),(14,78,95,103,70),(15,104,79,71,96),(16,72,97,89,80),(25,87,33,116,59),(26,117,88,60,34),(27,61,118,35,81),(28,36,62,82,119),(29,83,37,120,63),(30,113,84,64,38),(31,57,114,39,85),(32,40,58,86,115)], [(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×C22.F5 is a maximal subgroup of   Dic5.D12  Dic5.4D12  D15⋊C8⋊C2  D152M4(2)

42 conjugacy classes

 class 1 2A 2B 3A 3B 4A 4B 4C 5 6A 6B 6C 6D 8A 8B 8C 8D 10A 10B 10C 12A 12B 12C 12D 12E 12F 15A 15B 24A ··· 24H 30A ··· 30F order 1 2 2 3 3 4 4 4 5 6 6 6 6 8 8 8 8 10 10 10 12 12 12 12 12 12 15 15 24 ··· 24 30 ··· 30 size 1 1 2 1 1 5 5 10 4 1 1 2 2 10 10 10 10 4 4 4 5 5 5 5 10 10 4 4 10 ··· 10 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 C22.F5 C6×F5 C3×C22.F5 kernel C3×C22.F5 C3×C5⋊C8 C6×Dic5 C22.F5 C3×Dic5 C2×C30 C5⋊C8 C2×Dic5 Dic5 C2×C10 C15 C5 C2×C6 C6 C22 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×C22.F5 in GL6(𝔽241)

 225 0 0 0 0 0 0 225 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 240 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 240 0 0 0 0 0 0 240 0 0 0 0 0 0 1 0 0 0 0 0 0 1 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 0 240 1 0 0 0 0 240 0 1 0 0 0 240 0 0 1 0 0 240 0 0 0
,
 0 1 0 0 0 0 64 0 0 0 0 0 0 0 181 52 64 72 0 0 4 124 177 12 0 0 117 64 229 76 0 0 169 128 60 189

G:=sub<GL(6,GF(241))| [225,0,0,0,0,0,0,225,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,240,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,240,240,240,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[0,64,0,0,0,0,1,0,0,0,0,0,0,0,181,4,117,169,0,0,52,124,64,128,0,0,64,177,229,60,0,0,72,12,76,189] >;

C3×C22.F5 in GAP, Magma, Sage, TeX

C_3\times C_2^2.F_5
% in TeX

G:=Group("C3xC2^2.F5");
// GroupNames label

G:=SmallGroup(240,116);
// by ID

G=gap.SmallGroup(240,116);
# by ID

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

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

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