Copied to
clipboard

## 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, C3, C4, C4, C22, C22, C5, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C10, C10, Dic3, C12, D6, C2×C6, C15, C42, M4(2), C4○D4, C20, C20, 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, C5×D4, C5×Q8, C4×Dic3, C3×M4(2), C4○D12, C5×Dic3, C60, 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, C4, C22, C5, S3, C2×C4, D4, C10, D6, C22⋊C4, C20, C2×C10, C4×S3, D12, C3⋊D4, C5×S3, C4≀C2, C2×C20, C5×D4, 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 51 47 25 20)(2 52 48 26 21)(3 53 37 27 22)(4 54 38 28 23)(5 55 39 29 24)(6 56 40 30 13)(7 57 41 31 14)(8 58 42 32 15)(9 59 43 33 16)(10 60 44 34 17)(11 49 45 35 18)(12 50 46 36 19)(61 110 103 96 81)(62 111 104 85 82)(63 112 105 86 83)(64 113 106 87 84)(65 114 107 88 73)(66 115 108 89 74)(67 116 97 90 75)(68 117 98 91 76)(69 118 99 92 77)(70 119 100 93 78)(71 120 101 94 79)(72 109 102 95 80)
(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 68)(2 67)(3 66)(4 65)(5 64)(6 63)(7 62)(8 61)(9 72)(10 71)(11 70)(12 69)(13 83)(14 82)(15 81)(16 80)(17 79)(18 78)(19 77)(20 76)(21 75)(22 74)(23 73)(24 84)(25 91)(26 90)(27 89)(28 88)(29 87)(30 86)(31 85)(32 96)(33 95)(34 94)(35 93)(36 92)(37 108)(38 107)(39 106)(40 105)(41 104)(42 103)(43 102)(44 101)(45 100)(46 99)(47 98)(48 97)(49 119)(50 118)(51 117)(52 116)(53 115)(54 114)(55 113)(56 112)(57 111)(58 110)(59 109)(60 120)
(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 28 31 34)(26 33 32 27)(29 36 35 30)(37 48 43 42)(38 41 44 47)(39 46 45 40)(49 56 55 50)(51 54 57 60)(52 59 58 53)(61 65)(62 70)(64 68)(67 71)(73 81)(75 79)(76 84)(78 82)(85 93)(87 91)(88 96)(90 94)(97 101)(98 106)(100 104)(103 107)(110 114)(111 119)(113 117)(116 120)

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

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

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

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

׿
×
𝔽