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## G = C2×D6⋊F5order 480 = 25·3·5

### Direct product of C2 and D6⋊F5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — C2×D6⋊F5
 Chief series C1 — C5 — C15 — C3×D5 — C6×D5 — C6×F5 — D6⋊F5 — C2×D6⋊F5
 Lower central C15 — C30 — C2×D6⋊F5
 Upper central C1 — C22

Generators and relations for C2×D6⋊F5
G = < a,b,c,d,e | a2=b6=c2=d5=e4=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece-1=b3c, ede-1=d3 >

Subgroups: 1716 in 264 conjugacy classes, 70 normal (30 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C5, S3, C6, C6, C6, C2×C4, C23, D5, D5, D5, C10, C10, C10, Dic3, C12, D6, D6, C2×C6, C2×C6, C15, C22⋊C4, C22×C4, C24, F5, D10, D10, D10, C2×C10, C2×C10, C2×Dic3, C2×C12, C22×S3, C22×S3, C22×C6, C5×S3, C3×D5, C3×D5, D15, C30, C30, C2×C22⋊C4, C2×F5, C2×F5, C22×D5, C22×D5, C22×C10, D6⋊C4, C22×Dic3, C22×C12, S3×C23, C3×F5, C3⋊F5, S3×D5, C6×D5, C6×D5, S3×C10, S3×C10, D30, D30, C2×C30, C22⋊F5, C22×F5, C22×F5, C23×D5, C2×D6⋊C4, C6×F5, C6×F5, C2×C3⋊F5, C2×C3⋊F5, C2×S3×D5, C2×S3×D5, D5×C2×C6, S3×C2×C10, C22×D15, C2×C22⋊F5, D6⋊F5, C2×C6×F5, C22×C3⋊F5, C22×S3×D5, C2×D6⋊F5
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22⋊C4, C22×C4, C2×D4, F5, C4×S3, D12, C3⋊D4, C22×S3, C2×C22⋊C4, C2×F5, D6⋊C4, S3×C2×C4, C2×D12, C2×C3⋊D4, C22⋊F5, C22×F5, C2×D6⋊C4, S3×F5, C2×C22⋊F5, D6⋊F5, C2×S3×F5, C2×D6⋊F5

Smallest permutation representation of C2×D6⋊F5
On 120 points
Generators in S120
(1 55)(2 56)(3 57)(4 58)(5 59)(6 60)(7 109)(8 110)(9 111)(10 112)(11 113)(12 114)(13 105)(14 106)(15 107)(16 108)(17 103)(18 104)(19 101)(20 102)(21 97)(22 98)(23 99)(24 100)(25 91)(26 92)(27 93)(28 94)(29 95)(30 96)(31 69)(32 70)(33 71)(34 72)(35 67)(36 68)(37 65)(38 66)(39 61)(40 62)(41 63)(42 64)(43 76)(44 77)(45 78)(46 73)(47 74)(48 75)(49 82)(50 83)(51 84)(52 79)(53 80)(54 81)(85 118)(86 119)(87 120)(88 115)(89 116)(90 117)
(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)
(1 115)(2 120)(3 119)(4 118)(5 117)(6 116)(7 53)(8 52)(9 51)(10 50)(11 49)(12 54)(13 47)(14 46)(15 45)(16 44)(17 43)(18 48)(19 36)(20 35)(21 34)(22 33)(23 32)(24 31)(25 40)(26 39)(27 38)(28 37)(29 42)(30 41)(55 88)(56 87)(57 86)(58 85)(59 90)(60 89)(61 92)(62 91)(63 96)(64 95)(65 94)(66 93)(67 102)(68 101)(69 100)(70 99)(71 98)(72 97)(73 106)(74 105)(75 104)(76 103)(77 108)(78 107)(79 110)(80 109)(81 114)(82 113)(83 112)(84 111)
(1 39 51 32 47)(2 40 52 33 48)(3 41 53 34 43)(4 42 54 35 44)(5 37 49 36 45)(6 38 50 31 46)(7 21 17 119 30)(8 22 18 120 25)(9 23 13 115 26)(10 24 14 116 27)(11 19 15 117 28)(12 20 16 118 29)(55 61 84 70 74)(56 62 79 71 75)(57 63 80 72 76)(58 64 81 67 77)(59 65 82 68 78)(60 66 83 69 73)(85 95 114 102 108)(86 96 109 97 103)(87 91 110 98 104)(88 92 111 99 105)(89 93 112 100 106)(90 94 113 101 107)
(1 116 4 119)(2 117 5 120)(3 118 6 115)(7 47 24 42)(8 48 19 37)(9 43 20 38)(10 44 21 39)(11 45 22 40)(12 46 23 41)(13 34 29 50)(14 35 30 51)(15 36 25 52)(16 31 26 53)(17 32 27 54)(18 33 28 49)(55 89 58 86)(56 90 59 87)(57 85 60 88)(61 112 77 97)(62 113 78 98)(63 114 73 99)(64 109 74 100)(65 110 75 101)(66 111 76 102)(67 96 84 106)(68 91 79 107)(69 92 80 108)(70 93 81 103)(71 94 82 104)(72 95 83 105)

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

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

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

48 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 3 4A 4B 4C 4D 4E 4F 4G 4H 5 6A 6B 6C 6D 6E 6F 6G 10A 10B 10C 10D 10E 10F 10G 12A ··· 12H 15 30A 30B 30C order 1 2 2 2 2 2 2 2 2 2 2 2 3 4 4 4 4 4 4 4 4 5 6 6 6 6 6 6 6 10 10 10 10 10 10 10 12 ··· 12 15 30 30 30 size 1 1 1 1 5 5 5 5 6 6 30 30 2 10 10 10 10 30 30 30 30 4 2 2 2 10 10 10 10 4 4 4 12 12 12 12 10 ··· 10 8 8 8 8

48 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 8 8 8 type + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C4 C4 C4 S3 D4 D6 D6 C4×S3 D12 C3⋊D4 C4×S3 F5 C2×F5 C2×F5 C22⋊F5 S3×F5 D6⋊F5 C2×S3×F5 kernel C2×D6⋊F5 D6⋊F5 C2×C6×F5 C22×C3⋊F5 C22×S3×D5 C2×S3×D5 S3×C2×C10 C22×D15 C22×F5 C6×D5 C2×F5 C22×D5 D10 D10 D10 C2×C10 C22×S3 D6 C2×C6 C6 C22 C2 C2 # reps 1 4 1 1 1 4 2 2 1 4 2 1 2 4 4 2 1 2 1 4 1 2 1

Matrix representation of C2×D6⋊F5 in GL8(𝔽61)

 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 60 0 0 0 0 0 0 0 0 60 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 1 0 0 0 0 0 0 60 60 0 0 0 0 0 0 0 0 60 0 0 0 0 0 0 0 0 60 0 0 0 0 0 0 0 0 60 0 0 0 0 0 0 0 0 60 0 0 0 0 0 0 0 0 60 0 0 0 0 0 0 0 0 60
,
 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 2 0 0 0 0 0 0 0 60 0 0 0 0 0 0 0 0 7 0 14 14 0 0 0 0 47 54 47 0 0 0 0 0 0 47 54 47 0 0 0 0 14 14 0 7
,
 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 60 60 60 60
,
 50 0 0 0 0 0 0 0 0 50 0 0 0 0 0 0 0 0 50 39 0 0 0 0 0 0 11 11 0 0 0 0 0 0 0 0 7 0 14 14 0 0 0 0 14 14 0 7 0 0 0 0 47 54 47 0 0 0 0 0 54 7 7 54

G:=sub<GL(8,GF(61))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,60,0,0,0,0,0,0,1,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,2,60,0,0,0,0,0,0,0,0,7,47,0,14,0,0,0,0,0,54,47,14,0,0,0,0,14,47,54,0,0,0,0,0,14,0,47,7],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,60,0,0,0,0,1,0,0,60,0,0,0,0,0,1,0,60,0,0,0,0,0,0,1,60],[50,0,0,0,0,0,0,0,0,50,0,0,0,0,0,0,0,0,50,11,0,0,0,0,0,0,39,11,0,0,0,0,0,0,0,0,7,14,47,54,0,0,0,0,0,14,54,7,0,0,0,0,14,0,47,7,0,0,0,0,14,7,0,54] >;

C2×D6⋊F5 in GAP, Magma, Sage, TeX

C_2\times D_6\rtimes F_5
% in TeX

G:=Group("C2xD6:F5");
// GroupNames label

G:=SmallGroup(480,1000);
// by ID

G=gap.SmallGroup(480,1000);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,253,1356,9414,2379]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^6=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^3*c,e*d*e^-1=d^3>;
// generators/relations

׿
×
𝔽