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## G = C3×D20⋊7C4order 480 = 25·3·5

### Direct product of C3 and D20⋊7C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C3×D20⋊7C4
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×C60 — C3×C4○D20 — C3×D20⋊7C4
 Lower central C5 — C10 — C20 — C3×D20⋊7C4
 Upper central C1 — C12 — C2×C12 — C3×M4(2)

Generators and relations for C3×D207C4
G = < a,b,c,d | a3=b20=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, dbd-1=b9, dcd-1=b3c >

Subgroups: 320 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 [×3], C20 [×2], D10, C2×C10, C24, C2×C12, C2×C12 [×2], C3×D4 [×2], C3×Q8, C3×D5, C30, C30, C4≀C2, C40, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C4×C12, C3×M4(2), C3×C4○D4, C3×Dic5 [×3], C60 [×2], C6×D5, C2×C30, C4×Dic5, C5×M4(2), C4○D20, C3×C4≀C2, C120, C3×Dic10, D5×C12, C3×D20, C6×Dic5, C3×C5⋊D4, C2×C60, D207C4, C12×Dic5, C15×M4(2), C3×C4○D20, C3×D207C4
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, D207C4, C3×D10⋊C4, C3×D207C4

Smallest permutation representation of C3×D207C4
On 120 points
Generators in S120
(1 46 31)(2 47 32)(3 48 33)(4 49 34)(5 50 35)(6 51 36)(7 52 37)(8 53 38)(9 54 39)(10 55 40)(11 56 21)(12 57 22)(13 58 23)(14 59 24)(15 60 25)(16 41 26)(17 42 27)(18 43 28)(19 44 29)(20 45 30)(61 111 86)(62 112 87)(63 113 88)(64 114 89)(65 115 90)(66 116 91)(67 117 92)(68 118 93)(69 119 94)(70 120 95)(71 101 96)(72 102 97)(73 103 98)(74 104 99)(75 105 100)(76 106 81)(77 107 82)(78 108 83)(79 109 84)(80 110 85)
(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 95)(22 94)(23 93)(24 92)(25 91)(26 90)(27 89)(28 88)(29 87)(30 86)(31 85)(32 84)(33 83)(34 82)(35 81)(36 100)(37 99)(38 98)(39 97)(40 96)(41 115)(42 114)(43 113)(44 112)(45 111)(46 110)(47 109)(48 108)(49 107)(50 106)(51 105)(52 104)(53 103)(54 102)(55 101)(56 120)(57 119)(58 118)(59 117)(60 116)
(1 16 11 6)(2 5 12 15)(3 14 13 4)(7 10 17 20)(8 19 18 9)(21 36 31 26)(22 25 32 35)(23 34 33 24)(27 30 37 40)(28 39 38 29)(41 56 51 46)(42 45 52 55)(43 54 53 44)(47 50 57 60)(48 59 58 49)(62 70)(63 79)(64 68)(65 77)(67 75)(69 73)(72 80)(74 78)(82 90)(83 99)(84 88)(85 97)(87 95)(89 93)(92 100)(94 98)(102 110)(103 119)(104 108)(105 117)(107 115)(109 113)(112 120)(114 118)

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

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

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

102 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4A 4B 4C 4D 4E 4F 4G 4H 5A 5B 6A 6B 6C 6D 6E 6F 8A 8B 10A 10B 10C 10D 12A 12B 12C 12D 12E 12F 12G ··· 12N 12O 12P 15A 15B 15C 15D 20A 20B 20C 20D 20E 20F 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 3 4 4 4 4 4 4 4 4 5 5 6 6 6 6 6 6 8 8 10 10 10 10 12 12 12 12 12 12 12 ··· 12 12 12 15 15 15 15 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 20 1 1 1 1 2 10 10 10 10 20 2 2 1 1 2 2 20 20 4 4 2 2 4 4 1 1 1 1 2 2 10 ··· 10 20 20 2 2 2 2 2 2 2 2 4 4 4 4 4 4 2 2 2 2 4 4 4 4 4 ··· 4 2 ··· 2 4 4 4 4 4 ··· 4

102 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 C3 C4 C4 C6 C6 C6 C12 C12 D4 D4 D5 D10 C3×D4 C3×D4 C3×D5 C4≀C2 C4×D5 C5⋊D4 D20 C6×D5 C3×C4≀C2 D5×C12 C3×C5⋊D4 C3×D20 D20⋊7C4 C3×D20⋊7C4 kernel C3×D20⋊7C4 C12×Dic5 C15×M4(2) C3×C4○D20 D20⋊7C4 C3×Dic10 C3×D20 C4×Dic5 C5×M4(2) C4○D20 Dic10 D20 C60 C2×C30 C3×M4(2) C2×C12 C20 C2×C10 M4(2) 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 4 8

Matrix representation of C3×D207C4 in GL4(𝔽241) generated by

 225 0 0 0 0 225 0 0 0 0 225 0 0 0 0 225
,
 240 240 0 0 191 190 0 0 0 0 64 47 0 0 0 177
,
 0 52 0 0 51 0 0 0 0 0 240 0 0 0 54 1
,
 51 52 0 0 191 190 0 0 0 0 177 155 0 0 0 1
G:=sub<GL(4,GF(241))| [225,0,0,0,0,225,0,0,0,0,225,0,0,0,0,225],[240,191,0,0,240,190,0,0,0,0,64,0,0,0,47,177],[0,51,0,0,52,0,0,0,0,0,240,54,0,0,0,1],[51,191,0,0,52,190,0,0,0,0,177,0,0,0,155,1] >;

C3×D207C4 in GAP, Magma, Sage, TeX

C_3\times D_{20}\rtimes_7C_4
% in TeX

G:=Group("C3xD20:7C4");
// GroupNames label

G:=SmallGroup(480,103);
// by ID

G=gap.SmallGroup(480,103);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-5,365,92,136,2524,1271,102,18822]);
// 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,d*b*d^-1=b^9,d*c*d^-1=b^3*c>;
// generators/relations

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