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## G = C10×A4⋊C4order 480 = 25·3·5

### Direct product of C10 and A4⋊C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — A4 — C10×A4⋊C4
 Chief series C1 — C22 — A4 — C2×A4 — C10×A4 — C5×A4⋊C4 — C10×A4⋊C4
 Lower central A4 — C10×A4⋊C4
 Upper central C1 — C2×C10

Generators and relations for C10×A4⋊C4
G = < a,b,c,d,e | a10=b2=c2=d3=e4=1, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=ebe-1=bc=cb, dcd-1=b, ce=ec, ede-1=d-1 >

Subgroups: 376 in 126 conjugacy classes, 36 normal (20 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, C20, C2×C10, C2×C10, C2×Dic3, C2×A4, C2×A4, C30, C2×C22⋊C4, C2×C20, C22×C10, C22×C10, C22×C10, A4⋊C4, C22×A4, C5×Dic3, C5×A4, C2×C30, C5×C22⋊C4, C22×C20, C23×C10, C2×A4⋊C4, C10×Dic3, C10×A4, C10×A4, C10×C22⋊C4, C5×A4⋊C4, A4×C2×C10, C10×A4⋊C4
Quotients: C1, C2, C4, C22, C5, S3, C2×C4, C10, Dic3, D6, C20, C2×C10, C2×Dic3, S4, C5×S3, C2×C20, A4⋊C4, C2×S4, C5×Dic3, S3×C10, C2×A4⋊C4, C10×Dic3, C5×S4, C5×A4⋊C4, C10×S4, C10×A4⋊C4

Smallest permutation representation of C10×A4⋊C4
On 120 points
Generators in S120
(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)
(11 119)(12 120)(13 111)(14 112)(15 113)(16 114)(17 115)(18 116)(19 117)(20 118)(21 39)(22 40)(23 31)(24 32)(25 33)(26 34)(27 35)(28 36)(29 37)(30 38)(61 75)(62 76)(63 77)(64 78)(65 79)(66 80)(67 71)(68 72)(69 73)(70 74)(81 99)(82 100)(83 91)(84 92)(85 93)(86 94)(87 95)(88 96)(89 97)(90 98)
(1 109)(2 110)(3 101)(4 102)(5 103)(6 104)(7 105)(8 106)(9 107)(10 108)(11 119)(12 120)(13 111)(14 112)(15 113)(16 114)(17 115)(18 116)(19 117)(20 118)(41 55)(42 56)(43 57)(44 58)(45 59)(46 60)(47 51)(48 52)(49 53)(50 54)(61 75)(62 76)(63 77)(64 78)(65 79)(66 80)(67 71)(68 72)(69 73)(70 74)
(1 95 61)(2 96 62)(3 97 63)(4 98 64)(5 99 65)(6 100 66)(7 91 67)(8 92 68)(9 93 69)(10 94 70)(11 47 23)(12 48 24)(13 49 25)(14 50 26)(15 41 27)(16 42 28)(17 43 29)(18 44 30)(19 45 21)(20 46 22)(31 119 51)(32 120 52)(33 111 53)(34 112 54)(35 113 55)(36 114 56)(37 115 57)(38 116 58)(39 117 59)(40 118 60)(71 105 83)(72 106 84)(73 107 85)(74 108 86)(75 109 87)(76 110 88)(77 101 89)(78 102 90)(79 103 81)(80 104 82)
(1 20 109 118)(2 11 110 119)(3 12 101 120)(4 13 102 111)(5 14 103 112)(6 15 104 113)(7 16 105 114)(8 17 106 115)(9 18 107 116)(10 19 108 117)(21 86 39 94)(22 87 40 95)(23 88 31 96)(24 89 32 97)(25 90 33 98)(26 81 34 99)(27 82 35 100)(28 83 36 91)(29 84 37 92)(30 85 38 93)(41 80 55 66)(42 71 56 67)(43 72 57 68)(44 73 58 69)(45 74 59 70)(46 75 60 61)(47 76 51 62)(48 77 52 63)(49 78 53 64)(50 79 54 65)

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

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

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

100 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A ··· 4H 5A 5B 5C 5D 6A 6B 6C 10A ··· 10L 10M ··· 10AB 15A 15B 15C 15D 20A ··· 20AF 30A ··· 30L order 1 2 2 2 2 2 2 2 3 4 ··· 4 5 5 5 5 6 6 6 10 ··· 10 10 ··· 10 15 15 15 15 20 ··· 20 30 ··· 30 size 1 1 1 1 3 3 3 3 8 6 ··· 6 1 1 1 1 8 8 8 1 ··· 1 3 ··· 3 8 8 8 8 6 ··· 6 8 ··· 8

100 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 3 3 3 3 3 3 type + + + + - + + + image C1 C2 C2 C4 C5 C10 C10 C20 S3 Dic3 D6 C5×S3 C5×Dic3 S3×C10 S4 A4⋊C4 C2×S4 C5×S4 C5×A4⋊C4 C10×S4 kernel C10×A4⋊C4 C5×A4⋊C4 A4×C2×C10 C10×A4 C2×A4⋊C4 A4⋊C4 C22×A4 C2×A4 C23×C10 C22×C10 C22×C10 C24 C23 C23 C2×C10 C10 C10 C22 C2 C2 # reps 1 2 1 4 4 8 4 16 1 2 1 4 8 4 2 4 2 8 16 8

Matrix representation of C10×A4⋊C4 in GL5(𝔽61)

 60 0 0 0 0 0 60 0 0 0 0 0 34 0 0 0 0 0 34 0 0 0 0 0 34
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 60 0 0 0 0 0 60
,
 1 0 0 0 0 0 1 0 0 0 0 0 60 0 0 0 0 0 1 0 0 0 0 0 60
,
 0 60 0 0 0 1 60 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0
,
 51 35 0 0 0 25 10 0 0 0 0 0 0 0 50 0 0 0 50 0 0 0 50 0 0

G:=sub<GL(5,GF(61))| [60,0,0,0,0,0,60,0,0,0,0,0,34,0,0,0,0,0,34,0,0,0,0,0,34],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,60,0,0,0,0,0,60],[1,0,0,0,0,0,1,0,0,0,0,0,60,0,0,0,0,0,1,0,0,0,0,0,60],[0,1,0,0,0,60,60,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0],[51,25,0,0,0,35,10,0,0,0,0,0,0,0,50,0,0,0,50,0,0,0,50,0,0] >;

C10×A4⋊C4 in GAP, Magma, Sage, TeX

C_{10}\times A_4\rtimes C_4
% in TeX

G:=Group("C10xA4:C4");
// GroupNames label

G:=SmallGroup(480,1022);
// by ID

G=gap.SmallGroup(480,1022);
# by ID

G:=PCGroup([7,-2,-2,-5,-2,-3,-2,2,140,2804,10085,285,5886,475]);
// Polycyclic

G:=Group<a,b,c,d,e|a^10=b^2=c^2=d^3=e^4=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,d*b*d^-1=e*b*e^-1=b*c=c*b,d*c*d^-1=b,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations

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