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G = C4⋊F53S3order 480 = 25·3·5

The semidirect product of C4⋊F5 and S3 acting through Inn(C4⋊F5)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C4⋊F53S3, D6⋊F5.C2, (C4×S3)⋊1F5, (S3×C20)⋊1C4, (C4×D15)⋊1C4, C60⋊C43C2, (C2×F5).1D6, D6.5(C2×F5), C4.19(S3×F5), C20.19(C4×S3), C60.19(C2×C4), D30.C24C4, (Dic3×F5)⋊1C2, (S3×Dic5)⋊4C4, (C4×D5).64D6, D30.5(C2×C4), C12.13(C2×F5), C6.2(C22×F5), C30.2(C22×C4), (C6×F5).1C22, C151(C42⋊C2), Dic15.7(C2×C4), Dic5.25(C4×S3), Dic3.11(C2×F5), (C6×D5).22C23, D5.2(D42S3), D5.1(Q83S3), (D5×C12).50C22, D10.25(C22×S3), (D5×Dic3).12C22, C32(D10.C23), C5⋊(C4⋊C47S3), C2.7(C2×S3×F5), (C3×C4⋊F5)⋊3C2, C10.2(S3×C2×C4), (C4×S3×D5).2C2, (S3×C10).5(C2×C4), (C2×C3⋊F5).2C22, (C2×S3×D5).14C22, (C3×D5).2(C4○D4), (C5×Dic3).7(C2×C4), (C3×Dic5).20(C2×C4), SmallGroup(480,983)

Series: Derived Chief Lower central Upper central

C1C30 — C4⋊F53S3
C1C5C15C3×D5C6×D5C6×F5Dic3×F5 — C4⋊F53S3
C15C30 — C4⋊F53S3
C1C2C4

Generators and relations for C4⋊F53S3
 G = < a,b,c,d,e | a4=b5=c4=d3=e2=1, ab=ba, cac-1=a-1, ad=da, ae=ea, cbc-1=b3, bd=db, be=eb, cd=dc, ece=a2c, ede=d-1 >

Subgroups: 820 in 152 conjugacy classes, 50 normal (40 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C6, C2×C4, C23, D5, D5, C10, C10, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C15, C42, C22⋊C4, C4⋊C4, C22×C4, Dic5, Dic5, C20, C20, F5, D10, D10, C2×C10, C4×S3, C4×S3, C2×Dic3, C2×C12, C22×S3, C5×S3, C3×D5, D15, C30, C42⋊C2, C4×D5, C4×D5, C2×Dic5, C2×C20, C2×F5, C2×F5, C22×D5, C4×Dic3, C4⋊Dic3, D6⋊C4, C3×C4⋊C4, S3×C2×C4, C5×Dic3, C3×Dic5, Dic15, C60, C3×F5, C3⋊F5, S3×D5, C6×D5, S3×C10, D30, C4×F5, C4⋊F5, C4⋊F5, C22⋊F5, C2×C4×D5, C4⋊C47S3, D5×Dic3, S3×Dic5, D30.C2, D5×C12, S3×C20, C4×D15, C6×F5, C2×C3⋊F5, C2×S3×D5, D10.C23, Dic3×F5, D6⋊F5, C3×C4⋊F5, C60⋊C4, C4×S3×D5, C4⋊F53S3
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, C22×C4, C4○D4, F5, C4×S3, C22×S3, C42⋊C2, C2×F5, S3×C2×C4, D42S3, Q83S3, C22×F5, C4⋊C47S3, S3×F5, D10.C23, C2×S3×F5, C4⋊F53S3

Smallest permutation representation of C4⋊F53S3
On 120 points
Generators in S120
(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 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)
(2 3 5 4)(6 8 7 10)(11 13 12 15)(16 18 17 20)(21 23 22 25)(26 28 27 30)(31 48 32 50)(33 47 35 46)(34 49)(36 53 37 55)(38 52 40 51)(39 54)(41 58 42 60)(43 57 45 56)(44 59)(61 63 62 65)(66 68 67 70)(71 73 72 75)(76 78 77 80)(81 83 82 85)(86 88 87 90)(91 108 92 110)(93 107 95 106)(94 109)(96 113 97 115)(98 112 100 111)(99 114)(101 118 102 120)(103 117 105 116)(104 119)
(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 71 66)(62 72 67)(63 73 68)(64 74 69)(65 75 70)(76 86 81)(77 87 82)(78 88 83)(79 89 84)(80 90 85)(91 101 96)(92 102 97)(93 103 98)(94 104 99)(95 105 100)(106 116 111)(107 117 112)(108 118 113)(109 119 114)(110 120 115)
(1 94)(2 95)(3 91)(4 92)(5 93)(6 96)(7 97)(8 98)(9 99)(10 100)(11 101)(12 102)(13 103)(14 104)(15 105)(16 106)(17 107)(18 108)(19 109)(20 110)(21 111)(22 112)(23 113)(24 114)(25 115)(26 116)(27 117)(28 118)(29 119)(30 120)(31 76)(32 77)(33 78)(34 79)(35 80)(36 81)(37 82)(38 83)(39 84)(40 85)(41 86)(42 87)(43 88)(44 89)(45 90)(46 61)(47 62)(48 63)(49 64)(50 65)(51 66)(52 67)(53 68)(54 69)(55 70)(56 71)(57 72)(58 73)(59 74)(60 75)

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

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

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

42 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D···4H4I4J4K4L4M4N 5 6A6B6C10A10B10C12A12B···12F 15 20A20B20C20D 30 60A60B
order12222234444···444444456661010101212···121520202020306060
size1155630223310···1015153030303042101041212420···208441212888

42 irreducible representations

dim11111111112222224444444888
type+++++++++++++-+++
imageC1C2C2C2C2C2C4C4C4C4S3D6D6C4○D4C4×S3C4×S3F5C2×F5C2×F5C2×F5D42S3Q83S3D10.C23S3×F5C2×S3×F5C4⋊F53S3
kernelC4⋊F53S3Dic3×F5D6⋊F5C3×C4⋊F5C60⋊C4C4×S3×D5S3×Dic5D30.C2S3×C20C4×D15C4⋊F5C4×D5C2×F5C3×D5Dic5C20C4×S3Dic3C12D6D5D5C3C4C2C1
# reps12211122221124221111114112

Matrix representation of C4⋊F53S3 in GL8(𝔽61)

159000000
160000000
00100000
00010000
00001000
00000100
00000010
00000001
,
10000000
01000000
00100000
00010000
000060100
000060010
000060001
000060000
,
500000000
5011000000
005000000
000500000
00000010
00001000
00000001
00000100
,
10000000
01000000
000600000
001600000
00001000
00000100
00000010
00000001
,
5022000000
5011000000
0012190000
0031490000
000060000
000006000
000000600
000000060

G:=sub<GL(8,GF(61))| [1,1,0,0,0,0,0,0,59,60,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,60,60,60,60,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0],[50,50,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,50,0,0,0,0,0,0,0,0,50,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,60,60,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[50,50,0,0,0,0,0,0,22,11,0,0,0,0,0,0,0,0,12,31,0,0,0,0,0,0,19,49,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60] >;

C4⋊F53S3 in GAP, Magma, Sage, TeX

C_4\rtimes F_5\rtimes_3S_3
% in TeX

G:=Group("C4:F5:3S3");
// GroupNames label

G:=SmallGroup(480,983);
// by ID

G=gap.SmallGroup(480,983);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,120,422,100,1356,9414,2379]);
// Polycyclic

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

׿
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