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## G = C5×D12⋊C4order 480 = 25·3·5

### Direct product of C5 and D12⋊C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C5×D12⋊C4
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×C60 — C5×C4○D12 — C5×D12⋊C4
 Lower central C3 — C6 — C12 — C5×D12⋊C4
 Upper central C1 — C20 — C2×C20 — C5×M4(2)

Generators and relations for C5×D12⋊C4
G = < a,b,c,d | a5=b12=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, dbd-1=b5, dcd-1=b7c >

Subgroups: 228 in 88 conjugacy classes, 38 normal (all characteristic)
C1, C2, C2 [×2], C3, C4 [×2], C4 [×3], C22, C22, C5, S3, C6, C6, C8, C2×C4, C2×C4 [×2], D4 [×2], Q8, C10, C10 [×2], Dic3 [×3], C12 [×2], D6, C2×C6, C15, C42, M4(2), C4○D4, C20 [×2], C20 [×3], C2×C10, C2×C10, C24, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C5×S3, C30, C30, C4≀C2, C40, C2×C20, C2×C20 [×2], C5×D4 [×2], C5×Q8, C4×Dic3, C3×M4(2), C4○D12, C5×Dic3 [×3], C60 [×2], S3×C10, C2×C30, C4×C20, C5×M4(2), C5×C4○D4, D12⋊C4, C120, C5×Dic6, S3×C20, C5×D12, C10×Dic3, C5×C3⋊D4, C2×C60, C5×C4≀C2, Dic3×C20, C15×M4(2), C5×C4○D12, C5×D12⋊C4
Quotients: C1, C2 [×3], C4 [×2], C22, C5, S3, C2×C4, D4 [×2], C10 [×3], D6, C22⋊C4, C20 [×2], C2×C10, C4×S3, D12, C3⋊D4, C5×S3, C4≀C2, C2×C20, C5×D4 [×2], D6⋊C4, S3×C10, C5×C22⋊C4, D12⋊C4, S3×C20, C5×D12, C5×C3⋊D4, C5×C4≀C2, C5×D6⋊C4, C5×D12⋊C4

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

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

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

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

120 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 4F 4G 4H 5A 5B 5C 5D 6A 6B 8A 8B 10A 10B 10C 10D 10E 10F 10G 10H 10I 10J 10K 10L 12A 12B 12C 15A 15B 15C 15D 20A ··· 20H 20I 20J 20K 20L 20M ··· 20AB 20AC 20AD 20AE 20AF 24A 24B 24C 24D 30A 30B 30C 30D 30E 30F 30G 30H 40A ··· 40H 60A ··· 60H 60I 60J 60K 60L 120A ··· 120P order 1 2 2 2 3 4 4 4 4 4 4 4 4 5 5 5 5 6 6 8 8 10 10 10 10 10 10 10 10 10 10 10 10 12 12 12 15 15 15 15 20 ··· 20 20 20 20 20 20 ··· 20 20 20 20 20 24 24 24 24 30 30 30 30 30 30 30 30 40 ··· 40 60 ··· 60 60 60 60 60 120 ··· 120 size 1 1 2 12 2 1 1 2 6 6 6 6 12 1 1 1 1 2 4 4 4 1 1 1 1 2 2 2 2 12 12 12 12 2 2 4 2 2 2 2 1 ··· 1 2 2 2 2 6 ··· 6 12 12 12 12 4 4 4 4 2 2 2 2 4 4 4 4 4 ··· 4 2 ··· 2 4 4 4 4 4 ··· 4

120 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 2 2 2 2 2 2 4 4 type + + + + + + + + + image C1 C2 C2 C2 C4 C4 C5 C10 C10 C10 C20 C20 S3 D4 D4 D6 C4×S3 C3⋊D4 D12 C5×S3 C4≀C2 C5×D4 C5×D4 S3×C10 S3×C20 C5×C3⋊D4 C5×D12 C5×C4≀C2 D12⋊C4 C5×D12⋊C4 kernel C5×D12⋊C4 Dic3×C20 C15×M4(2) C5×C4○D12 C5×Dic6 C5×D12 D12⋊C4 C4×Dic3 C3×M4(2) C4○D12 Dic6 D12 C5×M4(2) C60 C2×C30 C2×C20 C20 C20 C2×C10 M4(2) C15 C12 C2×C6 C2×C4 C4 C4 C22 C3 C5 C1 # reps 1 1 1 1 2 2 4 4 4 4 8 8 1 1 1 1 2 2 2 4 4 4 4 4 8 8 8 16 2 8

Matrix representation of C5×D12⋊C4 in GL4(𝔽241) generated by

 91 0 0 0 0 91 0 0 0 0 205 0 0 0 0 205
,
 0 240 0 0 1 1 0 0 0 0 64 0 0 0 0 177
,
 43 142 0 0 99 198 0 0 0 0 0 1 0 0 1 0
,
 240 0 0 0 1 1 0 0 0 0 177 0 0 0 0 1
G:=sub<GL(4,GF(241))| [91,0,0,0,0,91,0,0,0,0,205,0,0,0,0,205],[0,1,0,0,240,1,0,0,0,0,64,0,0,0,0,177],[43,99,0,0,142,198,0,0,0,0,0,1,0,0,1,0],[240,1,0,0,0,1,0,0,0,0,177,0,0,0,0,1] >;

C5×D12⋊C4 in GAP, Magma, Sage, TeX

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

G:=Group("C5xD12:C4");
// GroupNames label

G:=SmallGroup(480,144);
// by ID

G=gap.SmallGroup(480,144);
# by ID

G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-3,589,148,136,4204,2111,102,15686]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^12=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d^-1=b^5,d*c*d^-1=b^7*c>;
// generators/relations

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