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## G = C2×C4×C3⋊F5order 480 = 25·3·5

### Direct product of C2×C4 and C3⋊F5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C15 — C2×C4×C3⋊F5
 Chief series C1 — C5 — C15 — C3×D5 — C6×D5 — C2×C3⋊F5 — C22×C3⋊F5 — C2×C4×C3⋊F5
 Lower central C15 — C2×C4×C3⋊F5
 Upper central C1 — C2×C4

Generators and relations for C2×C4×C3⋊F5
G = < a,b,c,d,e | a2=b4=c3=d5=e4=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=c-1, ede-1=d3 >

Subgroups: 908 in 216 conjugacy classes, 97 normal (27 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C6, C2×C4, C2×C4, C23, D5, C10, C10, Dic3, C12, C12, C2×C6, C2×C6, C15, C42, C22×C4, Dic5, C20, F5, D10, D10, C2×C10, C2×Dic3, C2×C12, C2×C12, C22×C6, C3×D5, C30, C30, C2×C42, C4×D5, C2×Dic5, C2×C20, C2×F5, C22×D5, C4×Dic3, C22×Dic3, C22×C12, C3×Dic5, C60, C3⋊F5, C6×D5, C6×D5, C2×C30, C4×F5, C2×C4×D5, C22×F5, C2×C4×Dic3, D5×C12, C6×Dic5, C2×C60, C2×C3⋊F5, D5×C2×C6, C2×C4×F5, C4×C3⋊F5, D5×C2×C12, C22×C3⋊F5, C2×C4×C3⋊F5
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, Dic3, D6, C42, C22×C4, F5, C4×S3, C2×Dic3, C22×S3, C2×C42, C2×F5, C4×Dic3, S3×C2×C4, C22×Dic3, C3⋊F5, C4×F5, C22×F5, C2×C4×Dic3, C2×C3⋊F5, C2×C4×F5, C4×C3⋊F5, C22×C3⋊F5, C2×C4×C3⋊F5

Smallest permutation representation of C2×C4×C3⋊F5
On 120 points
Generators in S120
(1 64)(2 65)(3 61)(4 62)(5 63)(6 66)(7 67)(8 68)(9 69)(10 70)(11 71)(12 72)(13 73)(14 74)(15 75)(16 76)(17 77)(18 78)(19 79)(20 80)(21 81)(22 82)(23 83)(24 84)(25 85)(26 86)(27 87)(28 88)(29 89)(30 90)(31 91)(32 92)(33 93)(34 94)(35 95)(36 96)(37 97)(38 98)(39 99)(40 100)(41 101)(42 102)(43 103)(44 104)(45 105)(46 106)(47 107)(48 108)(49 109)(50 110)(51 111)(52 112)(53 113)(54 114)(55 115)(56 116)(57 117)(58 118)(59 119)(60 120)
(1 49 19 34)(2 50 20 35)(3 46 16 31)(4 47 17 32)(5 48 18 33)(6 51 21 36)(7 52 22 37)(8 53 23 38)(9 54 24 39)(10 55 25 40)(11 56 26 41)(12 57 27 42)(13 58 28 43)(14 59 29 44)(15 60 30 45)(61 106 76 91)(62 107 77 92)(63 108 78 93)(64 109 79 94)(65 110 80 95)(66 111 81 96)(67 112 82 97)(68 113 83 98)(69 114 84 99)(70 115 85 100)(71 116 86 101)(72 117 87 102)(73 118 88 103)(74 119 89 104)(75 120 90 105)
(1 9 14)(2 10 15)(3 6 11)(4 7 12)(5 8 13)(16 21 26)(17 22 27)(18 23 28)(19 24 29)(20 25 30)(31 36 41)(32 37 42)(33 38 43)(34 39 44)(35 40 45)(46 51 56)(47 52 57)(48 53 58)(49 54 59)(50 55 60)(61 66 71)(62 67 72)(63 68 73)(64 69 74)(65 70 75)(76 81 86)(77 82 87)(78 83 88)(79 84 89)(80 85 90)(91 96 101)(92 97 102)(93 98 103)(94 99 104)(95 100 105)(106 111 116)(107 112 117)(108 113 118)(109 114 119)(110 115 120)
(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 49 19 34)(2 46 18 32)(3 48 17 35)(4 50 16 33)(5 47 20 31)(6 58 22 45)(7 60 21 43)(8 57 25 41)(9 59 24 44)(10 56 23 42)(11 53 27 40)(12 55 26 38)(13 52 30 36)(14 54 29 39)(15 51 28 37)(61 108 77 95)(62 110 76 93)(63 107 80 91)(64 109 79 94)(65 106 78 92)(66 118 82 105)(67 120 81 103)(68 117 85 101)(69 119 84 104)(70 116 83 102)(71 113 87 100)(72 115 86 98)(73 112 90 96)(74 114 89 99)(75 111 88 97)

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

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

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

72 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 4C 4D 4E 4F 4G 4H 4I ··· 4X 5 6A 6B 6C 6D 6E 6F 6G 10A 10B 10C 12A 12B 12C 12D 12E 12F 12G 12H 15A 15B 20A 20B 20C 20D 30A ··· 30F 60A ··· 60H order 1 2 2 2 2 2 2 2 3 4 4 4 4 4 4 4 4 4 ··· 4 5 6 6 6 6 6 6 6 10 10 10 12 12 12 12 12 12 12 12 15 15 20 20 20 20 30 ··· 30 60 ··· 60 size 1 1 1 1 5 5 5 5 2 1 1 1 1 5 5 5 5 15 ··· 15 4 2 2 2 10 10 10 10 4 4 4 2 2 2 2 10 10 10 10 4 4 4 4 4 4 4 ··· 4 4 ··· 4

72 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 type + + + + + - + - - + + + + image C1 C2 C2 C2 C4 C4 C4 C4 S3 Dic3 D6 Dic3 Dic3 D6 C4×S3 F5 C2×F5 C2×F5 C3⋊F5 C4×F5 C2×C3⋊F5 C2×C3⋊F5 C4×C3⋊F5 kernel C2×C4×C3⋊F5 C4×C3⋊F5 D5×C2×C12 C22×C3⋊F5 D5×C12 C6×Dic5 C2×C60 C2×C3⋊F5 C2×C4×D5 C4×D5 C4×D5 C2×Dic5 C2×C20 C22×D5 D10 C2×C12 C12 C2×C6 C2×C4 C6 C4 C22 C2 # reps 1 4 1 2 4 2 2 16 1 2 2 1 1 1 8 1 2 1 2 4 4 2 8

Matrix representation of C2×C4×C3⋊F5 in GL6(𝔽61)

 60 0 0 0 0 0 0 60 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 11 0 0 0 0 0 0 11 0 0 0 0 0 0 11 0 0 0 0 0 0 11
,
 45 25 0 0 0 0 5 15 0 0 0 0 0 0 27 0 55 55 0 0 6 33 6 0 0 0 0 6 33 6 0 0 55 55 0 27
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 60 60 60 60
,
 18 44 0 0 0 0 55 43 0 0 0 0 0 0 11 0 0 0 0 0 0 0 0 11 0 0 0 11 0 0 0 0 50 50 50 50

G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,11],[45,5,0,0,0,0,25,15,0,0,0,0,0,0,27,6,0,55,0,0,0,33,6,55,0,0,55,6,33,0,0,0,55,0,6,27],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,60,0,0,1,0,0,60,0,0,0,1,0,60,0,0,0,0,1,60],[18,55,0,0,0,0,44,43,0,0,0,0,0,0,11,0,0,50,0,0,0,0,11,50,0,0,0,0,0,50,0,0,0,11,0,50] >;

C2×C4×C3⋊F5 in GAP, Magma, Sage, TeX

C_2\times C_4\times C_3\rtimes F_5
% in TeX

G:=Group("C2xC4xC3:F5");
// GroupNames label

G:=SmallGroup(480,1063);
// by ID

G=gap.SmallGroup(480,1063);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,100,2693,14118,2379]);
// Polycyclic

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

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