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## G = C3×D4⋊F5order 480 = 25·3·5

### Direct product of C3 and D4⋊F5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C3×D4⋊F5
 Chief series C1 — C5 — C10 — C20 — C4×D5 — D5×C12 — C3×C4.F5 — C3×D4⋊F5
 Lower central C5 — C10 — C20 — C3×D4⋊F5
 Upper central C1 — C6 — C12 — C3×D4

Generators and relations for C3×D4⋊F5
G = < a,b,c,d,e | a3=b4=c2=d5=e4=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece-1=b-1c, ede-1=d3 >

Subgroups: 328 in 88 conjugacy classes, 32 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C8, C2×C4, D4, D4, Q8, D5, C10, C10, C12, C12, C2×C6, C15, C42, M4(2), C4○D4, Dic5, Dic5, C20, F5, D10, C2×C10, C24, C2×C12, C3×D4, C3×D4, C3×Q8, C3×D5, C30, C30, C4≀C2, C5⋊C8, Dic10, C4×D5, C2×Dic5, C5⋊D4, C5×D4, C2×F5, C4×C12, C3×M4(2), C3×C4○D4, C3×Dic5, C3×Dic5, C60, C3×F5, C6×D5, C2×C30, C4.F5, C4×F5, D42D5, C3×C4≀C2, C3×C5⋊C8, C3×Dic10, D5×C12, C6×Dic5, C3×C5⋊D4, D4×C15, C6×F5, D4⋊F5, C3×C4.F5, C12×F5, C3×D42D5, C3×D4⋊F5
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, C12, C2×C6, C22⋊C4, F5, C2×C12, C3×D4, C4≀C2, C2×F5, C3×C22⋊C4, C3×F5, C22⋊F5, C3×C4≀C2, C6×F5, D4⋊F5, C3×C22⋊F5, C3×D4⋊F5

Smallest permutation representation of C3×D4⋊F5
On 120 points
Generators in S120
(1 44 24)(2 45 25)(3 41 21)(4 42 22)(5 43 23)(6 46 26)(7 47 27)(8 48 28)(9 49 29)(10 50 30)(11 51 31)(12 52 32)(13 53 33)(14 54 34)(15 55 35)(16 56 36)(17 57 37)(18 58 38)(19 59 39)(20 60 40)(61 101 81)(62 102 82)(63 103 83)(64 104 84)(65 105 85)(66 106 86)(67 107 87)(68 108 88)(69 109 89)(70 110 90)(71 111 91)(72 112 92)(73 113 93)(74 114 94)(75 115 95)(76 116 96)(77 117 97)(78 118 98)(79 119 99)(80 120 100)
(1 19 9 14)(2 20 10 15)(3 16 6 11)(4 17 7 12)(5 18 8 13)(21 36 26 31)(22 37 27 32)(23 38 28 33)(24 39 29 34)(25 40 30 35)(41 56 46 51)(42 57 47 52)(43 58 48 53)(44 59 49 54)(45 60 50 55)(61 71 66 76)(62 72 67 77)(63 73 68 78)(64 74 69 79)(65 75 70 80)(81 91 86 96)(82 92 87 97)(83 93 88 98)(84 94 89 99)(85 95 90 100)(101 111 106 116)(102 112 107 117)(103 113 108 118)(104 114 109 119)(105 115 110 120)
(1 79)(2 80)(3 76)(4 77)(5 78)(6 71)(7 72)(8 73)(9 74)(10 75)(11 61)(12 62)(13 63)(14 64)(15 65)(16 66)(17 67)(18 68)(19 69)(20 70)(21 96)(22 97)(23 98)(24 99)(25 100)(26 91)(27 92)(28 93)(29 94)(30 95)(31 81)(32 82)(33 83)(34 84)(35 85)(36 86)(37 87)(38 88)(39 89)(40 90)(41 116)(42 117)(43 118)(44 119)(45 120)(46 111)(47 112)(48 113)(49 114)(50 115)(51 101)(52 102)(53 103)(54 104)(55 105)(56 106)(57 107)(58 108)(59 109)(60 110)
(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 14 9 19)(2 11 8 17)(3 13 7 20)(4 15 6 18)(5 12 10 16)(21 33 27 40)(22 35 26 38)(23 32 30 36)(24 34 29 39)(25 31 28 37)(41 53 47 60)(42 55 46 58)(43 52 50 56)(44 54 49 59)(45 51 48 57)(61 63 62 65)(66 68 67 70)(71 73 72 75)(76 78 77 80)(81 83 82 85)(86 88 87 90)(91 93 92 95)(96 98 97 100)(101 103 102 105)(106 108 107 110)(111 113 112 115)(116 118 117 120)

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

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

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

57 conjugacy classes

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

57 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 4 4 4 8 8 type + + + + + + + + + - image C1 C2 C2 C2 C3 C4 C4 C6 C6 C6 C12 C12 D4 D4 C3×D4 C3×D4 C4≀C2 C3×C4≀C2 F5 C2×F5 C3×F5 C22⋊F5 C6×F5 C3×C22⋊F5 D4⋊F5 C3×D4⋊F5 kernel C3×D4⋊F5 C3×C4.F5 C12×F5 C3×D4⋊2D5 D4⋊F5 C3×Dic10 D4×C15 C4.F5 C4×F5 D4⋊2D5 Dic10 C5×D4 C3×Dic5 C6×D5 Dic5 D10 C15 C5 C3×D4 C12 D4 C6 C4 C2 C3 C1 # reps 1 1 1 1 2 2 2 2 2 2 4 4 1 1 2 2 4 8 1 1 2 2 2 4 1 2

Matrix representation of C3×D4⋊F5 in GL8(𝔽241)

 15 0 0 0 0 0 0 0 0 15 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 240 0 0 0 0 0 0 0 0 240 0 0 0 0 0 0 0 0 177 0 0 0 0 0 0 0 144 64 0 0 0 0 0 0 0 0 240 0 0 0 0 0 0 0 0 240 0 0 0 0 0 0 0 0 240 0 0 0 0 0 0 0 0 240
,
 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 119 79 0 0 0 0 0 0 16 122 0 0 0 0 0 0 0 0 124 0 7 7 0 0 0 0 234 117 234 0 0 0 0 0 0 234 117 234 0 0 0 0 7 7 0 124
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 240 240 240 240
,
 177 0 0 0 0 0 0 0 0 64 0 0 0 0 0 0 0 0 64 0 0 0 0 0 0 0 140 1 0 0 0 0 0 0 0 0 240 0 0 0 0 0 0 0 0 0 0 240 0 0 0 0 0 240 0 0 0 0 0 0 1 1 1 1

G:=sub<GL(8,GF(241))| [15,0,0,0,0,0,0,0,0,15,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,177,144,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,119,16,0,0,0,0,0,0,79,122,0,0,0,0,0,0,0,0,124,234,0,7,0,0,0,0,0,117,234,7,0,0,0,0,7,234,117,0,0,0,0,0,7,0,234,124],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,240,0,0,0,0,1,0,0,240,0,0,0,0,0,1,0,240,0,0,0,0,0,0,1,240],[177,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,64,140,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,240,0,0,1,0,0,0,0,0,0,240,1,0,0,0,0,0,0,0,1,0,0,0,0,0,240,0,1] >;

C3×D4⋊F5 in GAP, Magma, Sage, TeX

C_3\times D_4\rtimes F_5
% in TeX

G:=Group("C3xD4:F5");
// GroupNames label

G:=SmallGroup(480,288);
// by ID

G=gap.SmallGroup(480,288);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-5,84,365,136,2524,1271,102,9414,1595]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^4=c^2=d^5=e^4=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=b^-1*c,e*d*e^-1=d^3>;
// generators/relations

׿
×
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