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## G = C3×D20⋊4C4order 480 = 25·3·5

### Direct product of C3 and D20⋊4C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C3×D20⋊4C4
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×C60 — C3×C4○D20 — C3×D20⋊4C4
 Lower central C5 — C10 — C20 — C3×D20⋊4C4
 Upper central C1 — C12 — C2×C12 — C4×C12

Generators and relations for C3×D204C4
G = < a,b,c,d | a3=b20=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, dcd-1=b15c >

Subgroups: 288 in 88 conjugacy classes, 38 normal (all characteristic)
C1, C2, C2 [×2], C3, C4 [×2], C4 [×3], C22, C22, C5, C6, C6 [×2], C8, C2×C4, C2×C4 [×2], D4 [×2], Q8, D5, C10, C10, C12 [×2], C12 [×3], C2×C6, C2×C6, C15, C42, M4(2), C4○D4, Dic5, C20 [×2], C20 [×2], D10, C2×C10, C24, C2×C12, C2×C12 [×2], C3×D4 [×2], C3×Q8, C3×D5, C30, C30, C4≀C2, C52C8, Dic10, C4×D5, D20, C5⋊D4, C2×C20, C2×C20, C4×C12, C3×M4(2), C3×C4○D4, C3×Dic5, C60 [×2], C60 [×2], C6×D5, C2×C30, C4.Dic5, C4×C20, C4○D20, C3×C4≀C2, C3×C52C8, C3×Dic10, D5×C12, C3×D20, C3×C5⋊D4, C2×C60, C2×C60, D204C4, C3×C4.Dic5, C4×C60, C3×C4○D20, C3×D204C4
Quotients: C1, C2 [×3], C3, C4 [×2], C22, C6 [×3], C2×C4, D4 [×2], D5, C12 [×2], C2×C6, C22⋊C4, D10, C2×C12, C3×D4 [×2], C3×D5, C4≀C2, C4×D5, D20, C5⋊D4, C3×C22⋊C4, C6×D5, D10⋊C4, C3×C4≀C2, D5×C12, C3×D20, C3×C5⋊D4, D204C4, C3×D10⋊C4, C3×D204C4

Smallest permutation representation of C3×D204C4
On 120 points
Generators in S120
(1 41 31)(2 42 32)(3 43 33)(4 44 34)(5 45 35)(6 46 36)(7 47 37)(8 48 38)(9 49 39)(10 50 40)(11 51 21)(12 52 22)(13 53 23)(14 54 24)(15 55 25)(16 56 26)(17 57 27)(18 58 28)(19 59 29)(20 60 30)(61 114 84)(62 115 85)(63 116 86)(64 117 87)(65 118 88)(66 119 89)(67 120 90)(68 101 91)(69 102 92)(70 103 93)(71 104 94)(72 105 95)(73 106 96)(74 107 97)(75 108 98)(76 109 99)(77 110 100)(78 111 81)(79 112 82)(80 113 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 80)(2 79)(3 78)(4 77)(5 76)(6 75)(7 74)(8 73)(9 72)(10 71)(11 70)(12 69)(13 68)(14 67)(15 66)(16 65)(17 64)(18 63)(19 62)(20 61)(21 93)(22 92)(23 91)(24 90)(25 89)(26 88)(27 87)(28 86)(29 85)(30 84)(31 83)(32 82)(33 81)(34 100)(35 99)(36 98)(37 97)(38 96)(39 95)(40 94)(41 113)(42 112)(43 111)(44 110)(45 109)(46 108)(47 107)(48 106)(49 105)(50 104)(51 103)(52 102)(53 101)(54 120)(55 119)(56 118)(57 117)(58 116)(59 115)(60 114)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(49 59)(50 60)(61 66 71 76)(62 67 72 77)(63 68 73 78)(64 69 74 79)(65 70 75 80)(81 86 91 96)(82 87 92 97)(83 88 93 98)(84 89 94 99)(85 90 95 100)(101 106 111 116)(102 107 112 117)(103 108 113 118)(104 109 114 119)(105 110 115 120)

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

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

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

138 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4A 4B 4C ··· 4G 4H 5A 5B 6A 6B 6C 6D 6E 6F 8A 8B 10A ··· 10F 12A 12B 12C 12D 12E ··· 12N 12O 12P 15A 15B 15C 15D 20A ··· 20X 24A 24B 24C 24D 30A ··· 30L 60A ··· 60AV order 1 2 2 2 3 3 4 4 4 ··· 4 4 5 5 6 6 6 6 6 6 8 8 10 ··· 10 12 12 12 12 12 ··· 12 12 12 15 15 15 15 20 ··· 20 24 24 24 24 30 ··· 30 60 ··· 60 size 1 1 2 20 1 1 1 1 2 ··· 2 20 2 2 1 1 2 2 20 20 20 20 2 ··· 2 1 1 1 1 2 ··· 2 20 20 2 2 2 2 2 ··· 2 20 20 20 20 2 ··· 2 2 ··· 2

138 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 2 2 type + + + + + + + + + image C1 C2 C2 C2 C3 C4 C4 C6 C6 C6 C12 C12 D4 D4 D5 D10 C3×D4 C3×D4 C3×D5 C4≀C2 C4×D5 D20 C5⋊D4 C6×D5 C3×C4≀C2 D5×C12 C3×D20 C3×C5⋊D4 D20⋊4C4 C3×D20⋊4C4 kernel C3×D20⋊4C4 C3×C4.Dic5 C4×C60 C3×C4○D20 D20⋊4C4 C3×Dic10 C3×D20 C4.Dic5 C4×C20 C4○D20 Dic10 D20 C60 C2×C30 C4×C12 C2×C12 C20 C2×C10 C42 C15 C12 C12 C2×C6 C2×C4 C5 C4 C4 C22 C3 C1 # reps 1 1 1 1 2 2 2 2 2 2 4 4 1 1 2 2 2 2 4 4 4 4 4 4 8 8 8 8 16 32

Matrix representation of C3×D204C4 in GL3(𝔽241) generated by

 15 0 0 0 1 0 0 0 1
,
 1 0 0 0 40 0 0 0 235
,
 240 0 0 0 0 235 0 40 0
,
 240 0 0 0 240 0 0 0 177
G:=sub<GL(3,GF(241))| [15,0,0,0,1,0,0,0,1],[1,0,0,0,40,0,0,0,235],[240,0,0,0,0,40,0,235,0],[240,0,0,0,240,0,0,0,177] >;

C3×D204C4 in GAP, Magma, Sage, TeX

C_3\times D_{20}\rtimes_4C_4
% in TeX

G:=Group("C3xD20:4C4");
// GroupNames label

G:=SmallGroup(480,83);
// by ID

G=gap.SmallGroup(480,83);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-5,365,92,1683,2524,102,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^20=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,b*d=d*b,d*c*d^-1=b^15*c>;
// generators/relations

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