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## G = C5×C4○D12order 240 = 24·3·5

### Direct product of C5 and C4○D12

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C5×C4○D12
 Chief series C1 — C3 — C6 — C30 — S3×C10 — S3×C20 — C5×C4○D12
 Lower central C3 — C6 — C5×C4○D12
 Upper central C1 — C20 — C2×C20

Generators and relations for C5×C4○D12
G = < a,b,c,d | a5=b4=d2=1, c6=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c5 >

Subgroups: 152 in 80 conjugacy classes, 46 normal (30 characteristic)
C1, C2, C2 [×3], C3, C4 [×2], C4 [×2], C22, C22 [×2], C5, S3 [×2], C6, C6, C2×C4, C2×C4 [×2], D4 [×3], Q8, C10, C10 [×3], Dic3 [×2], C12 [×2], D6 [×2], C2×C6, C15, C4○D4, C20 [×2], C20 [×2], C2×C10, C2×C10 [×2], Dic6, C4×S3 [×2], D12, C3⋊D4 [×2], C2×C12, C5×S3 [×2], C30, C30, C2×C20, C2×C20 [×2], C5×D4 [×3], C5×Q8, C4○D12, C5×Dic3 [×2], C60 [×2], S3×C10 [×2], C2×C30, C5×C4○D4, C5×Dic6, S3×C20 [×2], C5×D12, C5×C3⋊D4 [×2], C2×C60, C5×C4○D12
Quotients: C1, C2 [×7], C22 [×7], C5, S3, C23, C10 [×7], D6 [×3], C4○D4, C2×C10 [×7], C22×S3, C5×S3, C22×C10, C4○D12, S3×C10 [×3], C5×C4○D4, S3×C2×C10, C5×C4○D12

Smallest permutation representation of C5×C4○D12
On 120 points
Generators in S120
(1 117 42 79 29)(2 118 43 80 30)(3 119 44 81 31)(4 120 45 82 32)(5 109 46 83 33)(6 110 47 84 34)(7 111 48 73 35)(8 112 37 74 36)(9 113 38 75 25)(10 114 39 76 26)(11 115 40 77 27)(12 116 41 78 28)(13 91 54 107 65)(14 92 55 108 66)(15 93 56 97 67)(16 94 57 98 68)(17 95 58 99 69)(18 96 59 100 70)(19 85 60 101 71)(20 86 49 102 72)(21 87 50 103 61)(22 88 51 104 62)(23 89 52 105 63)(24 90 53 106 64)
(1 85 7 91)(2 86 8 92)(3 87 9 93)(4 88 10 94)(5 89 11 95)(6 90 12 96)(13 29 19 35)(14 30 20 36)(15 31 21 25)(16 32 22 26)(17 33 23 27)(18 34 24 28)(37 108 43 102)(38 97 44 103)(39 98 45 104)(40 99 46 105)(41 100 47 106)(42 101 48 107)(49 112 55 118)(50 113 56 119)(51 114 57 120)(52 115 58 109)(53 116 59 110)(54 117 60 111)(61 75 67 81)(62 76 68 82)(63 77 69 83)(64 78 70 84)(65 79 71 73)(66 80 72 74)
(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 3)(4 12)(5 11)(6 10)(7 9)(13 15)(16 24)(17 23)(18 22)(19 21)(25 35)(26 34)(27 33)(28 32)(29 31)(38 48)(39 47)(40 46)(41 45)(42 44)(50 60)(51 59)(52 58)(53 57)(54 56)(61 71)(62 70)(63 69)(64 68)(65 67)(73 75)(76 84)(77 83)(78 82)(79 81)(85 87)(88 96)(89 95)(90 94)(91 93)(97 107)(98 106)(99 105)(100 104)(101 103)(109 115)(110 114)(111 113)(116 120)(117 119)

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

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

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

C5×C4○D12 is a maximal subgroup of
C60.98D4  C60.99D4  D12.2Dic5  D12.Dic5  D20.34D6  C60.36D4  C20.60D12  C60.38D4  D12.37D10  D12.33D10  D20.39D6  C30.C24  D2024D6  D2025D6  D2029D6  C5×S3×C4○D4
C5×C4○D12 is a maximal quotient of
C20×Dic6  C20×D12  C20×C3⋊D4

90 conjugacy classes

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

90 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 C5 C10 C10 C10 C10 C10 S3 D6 D6 C4○D4 C5×S3 C4○D12 S3×C10 S3×C10 C5×C4○D4 C5×C4○D12 kernel C5×C4○D12 C5×Dic6 S3×C20 C5×D12 C5×C3⋊D4 C2×C60 C4○D12 Dic6 C4×S3 D12 C3⋊D4 C2×C12 C2×C20 C20 C2×C10 C15 C2×C4 C5 C4 C22 C3 C1 # reps 1 1 2 1 2 1 4 4 8 4 8 4 1 2 1 2 4 4 8 4 8 16

Matrix representation of C5×C4○D12 in GL2(𝔽61) generated by

 20 0 0 20
,
 11 0 0 11
,
 23 23 38 46
,
 0 60 60 0
G:=sub<GL(2,GF(61))| [20,0,0,20],[11,0,0,11],[23,38,23,46],[0,60,60,0] >;

C5×C4○D12 in GAP, Magma, Sage, TeX

C_5\times C_4\circ D_{12}
% in TeX

G:=Group("C5xC4oD12");
// GroupNames label

G:=SmallGroup(240,168);
// by ID

G=gap.SmallGroup(240,168);
# by ID

G:=PCGroup([6,-2,-2,-2,-5,-2,-3,247,794,5765]);
// Polycyclic

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

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