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## G = C2×A4⋊Dic5order 480 = 25·3·5

### Direct product of C2 and A4⋊Dic5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C5×A4 — C2×A4⋊Dic5
 Chief series C1 — C22 — C2×C10 — C5×A4 — C10×A4 — A4⋊Dic5 — C2×A4⋊Dic5
 Lower central C5×A4 — C2×A4⋊Dic5
 Upper central C1 — C22

Generators and relations for C2×A4⋊Dic5
G = < a,b,c,d,e,f | a2=b2=c2=d3=e10=1, f2=e5, ab=ba, ac=ca, ad=da, ae=ea, af=fa, dbd-1=fbf-1=bc=cb, be=eb, dcd-1=b, ce=ec, cf=fc, de=ed, fdf-1=d-1, fef-1=e-1 >

Subgroups: 760 in 126 conjugacy classes, 33 normal (19 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C5, C6, C2×C4, C23, C23, C23, C10, C10, C10, Dic3, A4, C2×C6, C15, C22⋊C4, C22×C4, C24, Dic5, C2×C10, C2×C10, C2×Dic3, C2×A4, C2×A4, C30, C2×C22⋊C4, C2×Dic5, C22×C10, C22×C10, C22×C10, A4⋊C4, C22×A4, Dic15, C5×A4, C2×C30, C23.D5, C22×Dic5, C23×C10, C2×A4⋊C4, C2×Dic15, C10×A4, C10×A4, C2×C23.D5, A4⋊Dic5, A4×C2×C10, C2×A4⋊Dic5
Quotients: C1, C2, C4, C22, S3, C2×C4, D5, Dic3, D6, Dic5, D10, C2×Dic3, S4, D15, C2×Dic5, A4⋊C4, C2×S4, Dic15, D30, C2×A4⋊C4, C2×Dic15, C5⋊S4, A4⋊Dic5, C2×C5⋊S4, C2×A4⋊Dic5

Smallest permutation representation of C2×A4⋊Dic5
On 120 points
Generators in S120
(1 31)(2 32)(3 33)(4 34)(5 35)(6 36)(7 37)(8 38)(9 39)(10 40)(11 41)(12 42)(13 43)(14 44)(15 45)(16 46)(17 47)(18 48)(19 49)(20 50)(21 51)(22 52)(23 53)(24 54)(25 55)(26 56)(27 57)(28 58)(29 59)(30 60)(61 91)(62 92)(63 93)(64 94)(65 95)(66 96)(67 97)(68 98)(69 99)(70 100)(71 101)(72 102)(73 103)(74 104)(75 105)(76 106)(77 107)(78 108)(79 109)(80 110)(81 111)(82 112)(83 113)(84 114)(85 115)(86 116)(87 117)(88 118)(89 119)(90 120)
(11 16)(12 17)(13 18)(14 19)(15 20)(21 26)(22 27)(23 28)(24 29)(25 30)(41 46)(42 47)(43 48)(44 49)(45 50)(51 56)(52 57)(53 58)(54 59)(55 60)(61 66)(62 67)(63 68)(64 69)(65 70)(71 76)(72 77)(73 78)(74 79)(75 80)(91 96)(92 97)(93 98)(94 99)(95 100)(101 106)(102 107)(103 108)(104 109)(105 110)
(1 6)(2 7)(3 8)(4 9)(5 10)(21 26)(22 27)(23 28)(24 29)(25 30)(31 36)(32 37)(33 38)(34 39)(35 40)(51 56)(52 57)(53 58)(54 59)(55 60)(61 66)(62 67)(63 68)(64 69)(65 70)(81 86)(82 87)(83 88)(84 89)(85 90)(91 96)(92 97)(93 98)(94 99)(95 100)(111 116)(112 117)(113 118)(114 119)(115 120)
(1 11 21)(2 12 22)(3 13 23)(4 14 24)(5 15 25)(6 16 26)(7 17 27)(8 18 28)(9 19 29)(10 20 30)(31 41 51)(32 42 52)(33 43 53)(34 44 54)(35 45 55)(36 46 56)(37 47 57)(38 48 58)(39 49 59)(40 50 60)(61 81 71)(62 82 72)(63 83 73)(64 84 74)(65 85 75)(66 86 76)(67 87 77)(68 88 78)(69 89 79)(70 90 80)(91 111 101)(92 112 102)(93 113 103)(94 114 104)(95 115 105)(96 116 106)(97 117 107)(98 118 108)(99 119 109)(100 120 110)
(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 64 6 69)(2 63 7 68)(3 62 8 67)(4 61 9 66)(5 70 10 65)(11 74 16 79)(12 73 17 78)(13 72 18 77)(14 71 19 76)(15 80 20 75)(21 84 26 89)(22 83 27 88)(23 82 28 87)(24 81 29 86)(25 90 30 85)(31 94 36 99)(32 93 37 98)(33 92 38 97)(34 91 39 96)(35 100 40 95)(41 104 46 109)(42 103 47 108)(43 102 48 107)(44 101 49 106)(45 110 50 105)(51 114 56 119)(52 113 57 118)(53 112 58 117)(54 111 59 116)(55 120 60 115)

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

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

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

52 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A ··· 4H 5A 5B 6A 6B 6C 10A ··· 10F 10G ··· 10N 15A 15B 15C 15D 30A ··· 30L order 1 2 2 2 2 2 2 2 3 4 ··· 4 5 5 6 6 6 10 ··· 10 10 ··· 10 15 15 15 15 30 ··· 30 size 1 1 1 1 3 3 3 3 8 30 ··· 30 2 2 8 8 8 2 ··· 2 6 ··· 6 8 8 8 8 8 ··· 8

52 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 2 2 3 3 3 6 6 6 type + + + + + - + - + + - + + + + - + image C1 C2 C2 C4 S3 D5 Dic3 D6 Dic5 D10 D15 Dic15 D30 S4 A4⋊C4 C2×S4 C5⋊S4 A4⋊Dic5 C2×C5⋊S4 kernel C2×A4⋊Dic5 A4⋊Dic5 A4×C2×C10 C10×A4 C23×C10 C22×A4 C22×C10 C22×C10 C2×A4 C2×A4 C24 C23 C23 C2×C10 C10 C10 C22 C2 C2 # reps 1 2 1 4 1 2 2 1 4 2 4 8 4 2 4 2 2 4 2

Matrix representation of C2×A4⋊Dic5 in GL7(𝔽61)

 60 0 0 0 0 0 0 0 60 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 60 0 0 0 0 0 0 0 60 0 0 0 0 0 60 0 1
,
 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 60 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 60
,
 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 56 0 0 0 0 0 25 59 0 0 0 0 0 0 0 1 2 0 0 0 0 0 0 60 60 0 0 0 0 0 1 0
,
 43 1 0 0 0 0 0 42 1 0 0 0 0 0 0 0 60 0 0 0 0 0 0 0 60 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
,
 31 16 0 0 0 0 0 39 30 0 0 0 0 0 0 0 44 7 0 0 0 0 0 37 17 0 0 0 0 0 0 0 60 0 2 0 0 0 0 0 60 60 0 0 0 0 0 0 1

G:=sub<GL(7,GF(61))| [60,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,60,0,60,0,0,0,0,0,60,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,60,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,60],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,25,0,0,0,0,0,56,59,0,0,0,0,0,0,0,1,0,0,0,0,0,0,2,60,1,0,0,0,0,0,60,0],[43,42,0,0,0,0,0,1,1,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[31,39,0,0,0,0,0,16,30,0,0,0,0,0,0,0,44,37,0,0,0,0,0,7,17,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,60,0,0,0,0,0,2,60,1] >;

C2×A4⋊Dic5 in GAP, Magma, Sage, TeX

C_2\times A_4\rtimes {\rm Dic}_5
% in TeX

G:=Group("C2xA4:Dic5");
// GroupNames label

G:=SmallGroup(480,1033);
// by ID

G=gap.SmallGroup(480,1033);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-5,-2,2,28,451,3364,10085,1286,5886,2232]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^3=e^10=1,f^2=e^5,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,d*b*d^-1=f*b*f^-1=b*c=c*b,b*e=e*b,d*c*d^-1=b,c*e=e*c,c*f=f*c,d*e=e*d,f*d*f^-1=d^-1,f*e*f^-1=e^-1>;
// generators/relations

׿
×
𝔽