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## G = C3×C4○D20order 240 = 24·3·5

### Direct product of C3 and C4○D20

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C3×C4○D20
 Chief series C1 — C5 — C10 — C30 — C6×D5 — D5×C12 — C3×C4○D20
 Lower central C5 — C10 — C3×C4○D20
 Upper central C1 — C12 — C2×C12

Generators and relations for C3×C4○D20
G = < a,b,c,d | a3=b4=d2=1, c10=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c9 >

Subgroups: 212 in 80 conjugacy classes, 46 normal (30 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C2×C4, C2×C4, D4, Q8, D5, C10, C10, C12, C12, C2×C6, C2×C6, C15, C4○D4, Dic5, C20, D10, C2×C10, C2×C12, C2×C12, C3×D4, C3×Q8, C3×D5, C30, C30, Dic10, C4×D5, D20, C5⋊D4, C2×C20, C3×C4○D4, C3×Dic5, C60, C6×D5, C2×C30, C4○D20, C3×Dic10, D5×C12, C3×D20, C3×C5⋊D4, C2×C60, C3×C4○D20
Quotients: C1, C2, C3, C22, C6, C23, D5, C2×C6, C4○D4, D10, C22×C6, C3×D5, C22×D5, C3×C4○D4, C6×D5, C4○D20, D5×C2×C6, C3×C4○D20

Smallest permutation representation of C3×C4○D20
On 120 points
Generators in S120
(1 82 39)(2 83 40)(3 84 21)(4 85 22)(5 86 23)(6 87 24)(7 88 25)(8 89 26)(9 90 27)(10 91 28)(11 92 29)(12 93 30)(13 94 31)(14 95 32)(15 96 33)(16 97 34)(17 98 35)(18 99 36)(19 100 37)(20 81 38)(41 105 65)(42 106 66)(43 107 67)(44 108 68)(45 109 69)(46 110 70)(47 111 71)(48 112 72)(49 113 73)(50 114 74)(51 115 75)(52 116 76)(53 117 77)(54 118 78)(55 119 79)(56 120 80)(57 101 61)(58 102 62)(59 103 63)(60 104 64)
(1 73 11 63)(2 74 12 64)(3 75 13 65)(4 76 14 66)(5 77 15 67)(6 78 16 68)(7 79 17 69)(8 80 18 70)(9 61 19 71)(10 62 20 72)(21 115 31 105)(22 116 32 106)(23 117 33 107)(24 118 34 108)(25 119 35 109)(26 120 36 110)(27 101 37 111)(28 102 38 112)(29 103 39 113)(30 104 40 114)(41 84 51 94)(42 85 52 95)(43 86 53 96)(44 87 54 97)(45 88 55 98)(46 89 56 99)(47 90 57 100)(48 91 58 81)(49 92 59 82)(50 93 60 83)
(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 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 11)(21 36)(22 35)(23 34)(24 33)(25 32)(26 31)(27 30)(28 29)(37 40)(38 39)(41 56)(42 55)(43 54)(44 53)(45 52)(46 51)(47 50)(48 49)(57 60)(58 59)(61 64)(62 63)(65 80)(66 79)(67 78)(68 77)(69 76)(70 75)(71 74)(72 73)(81 82)(83 100)(84 99)(85 98)(86 97)(87 96)(88 95)(89 94)(90 93)(91 92)(101 104)(102 103)(105 120)(106 119)(107 118)(108 117)(109 116)(110 115)(111 114)(112 113)

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

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

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

C3×C4○D20 is a maximal subgroup of
C60.96D4  C60.97D4  D20.3Dic3  D20.2Dic3  D20.34D6  D2021D6  D2019D6  D20.37D6  D20.31D6  C60.63D4  D20.38D6  D20.39D6  D2024D6  D2026D6  D2029D6  C3×D5×C4○D4
C3×C4○D20 is a maximal quotient of
C12×Dic10  C12×D20  C12×C5⋊D4

78 conjugacy classes

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

78 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 D5 C4○D4 D10 D10 C3×D5 C3×C4○D4 C6×D5 C6×D5 C4○D20 C3×C4○D20 kernel C3×C4○D20 C3×Dic10 D5×C12 C3×D20 C3×C5⋊D4 C2×C60 C4○D20 Dic10 C4×D5 D20 C5⋊D4 C2×C20 C2×C12 C15 C12 C2×C6 C2×C4 C5 C4 C22 C3 C1 # reps 1 1 2 1 2 1 2 2 4 2 4 2 2 2 4 2 4 4 8 4 8 16

Matrix representation of C3×C4○D20 in GL2(𝔽61) generated by

 47 0 0 47
,
 11 0 0 11
,
 7 32 29 2
,
 7 32 29 54
G:=sub<GL(2,GF(61))| [47,0,0,47],[11,0,0,11],[7,29,32,2],[7,29,32,54] >;

C3×C4○D20 in GAP, Magma, Sage, TeX

C_3\times C_4\circ D_{20}
% in TeX

G:=Group("C3xC4oD20");
// GroupNames label

G:=SmallGroup(240,158);
// by ID

G=gap.SmallGroup(240,158);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-2,-5,151,506,6917]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^4=d^2=1,c^10=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^2*c^9>;
// generators/relations

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