direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5×S3×Q8, C20.41D6, Dic6⋊4C10, C60.48C22, C30.54C23, C3⋊2(Q8×C10), C15⋊10(C2×Q8), C4.6(S3×C10), (C3×Q8)⋊2C10, (Q8×C15)⋊8C2, (S3×C20).4C2, (C4×S3).1C10, C12.6(C2×C10), D6.5(C2×C10), (C5×Dic6)⋊10C2, C6.7(C22×C10), C10.44(C22×S3), Dic3.4(C2×C10), (S3×C10).16C22, (C5×Dic3).15C22, C2.8(S3×C2×C10), SmallGroup(240,171)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×S3×Q8
G = < a,b,c,d,e | a5=b3=c2=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >
Subgroups: 128 in 76 conjugacy classes, 50 normal (16 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C2×C4, Q8, Q8, C10, C10, Dic3, C12, D6, C15, C2×Q8, C20, C20, C2×C10, Dic6, C4×S3, C3×Q8, C5×S3, C30, C2×C20, C5×Q8, C5×Q8, S3×Q8, C5×Dic3, C60, S3×C10, Q8×C10, C5×Dic6, S3×C20, Q8×C15, C5×S3×Q8
Quotients: C1, C2, C22, C5, S3, Q8, C23, C10, D6, C2×Q8, C2×C10, C22×S3, C5×S3, C5×Q8, C22×C10, S3×Q8, S3×C10, Q8×C10, S3×C2×C10, C5×S3×Q8
►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)
(1 39 33)(2 40 34)(3 36 35)(4 37 31)(5 38 32)(6 20 13)(7 16 14)(8 17 15)(9 18 11)(10 19 12)(21 118 111)(22 119 112)(23 120 113)(24 116 114)(25 117 115)(26 41 47)(27 42 48)(28 43 49)(29 44 50)(30 45 46)(51 79 58)(52 80 59)(53 76 60)(54 77 56)(55 78 57)(61 74 67)(62 75 68)(63 71 69)(64 72 70)(65 73 66)(81 109 88)(82 110 89)(83 106 90)(84 107 86)(85 108 87)(91 104 97)(92 105 98)(93 101 99)(94 102 100)(95 103 96)
(1 26)(2 27)(3 28)(4 29)(5 30)(6 111)(7 112)(8 113)(9 114)(10 115)(11 24)(12 25)(13 21)(14 22)(15 23)(16 119)(17 120)(18 116)(19 117)(20 118)(31 44)(32 45)(33 41)(34 42)(35 43)(36 49)(37 50)(38 46)(39 47)(40 48)(51 65)(52 61)(53 62)(54 63)(55 64)(56 71)(57 72)(58 73)(59 74)(60 75)(66 79)(67 80)(68 76)(69 77)(70 78)(81 95)(82 91)(83 92)(84 93)(85 94)(86 101)(87 102)(88 103)(89 104)(90 105)(96 109)(97 110)(98 106)(99 107)(100 108)
(1 56 26 71)(2 57 27 72)(3 58 28 73)(4 59 29 74)(5 60 30 75)(6 109 21 95)(7 110 22 91)(8 106 23 92)(9 107 24 93)(10 108 25 94)(11 84 114 99)(12 85 115 100)(13 81 111 96)(14 82 112 97)(15 83 113 98)(16 89 119 104)(17 90 120 105)(18 86 116 101)(19 87 117 102)(20 88 118 103)(31 80 50 61)(32 76 46 62)(33 77 47 63)(34 78 48 64)(35 79 49 65)(36 51 43 66)(37 52 44 67)(38 53 45 68)(39 54 41 69)(40 55 42 70)
(1 86 26 101)(2 87 27 102)(3 88 28 103)(4 89 29 104)(5 90 30 105)(6 65 21 79)(7 61 22 80)(8 62 23 76)(9 63 24 77)(10 64 25 78)(11 69 114 54)(12 70 115 55)(13 66 111 51)(14 67 112 52)(15 68 113 53)(16 74 119 59)(17 75 120 60)(18 71 116 56)(19 72 117 57)(20 73 118 58)(31 110 50 91)(32 106 46 92)(33 107 47 93)(34 108 48 94)(35 109 49 95)(36 81 43 96)(37 82 44 97)(38 83 45 98)(39 84 41 99)(40 85 42 100)
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), (1,39,33)(2,40,34)(3,36,35)(4,37,31)(5,38,32)(6,20,13)(7,16,14)(8,17,15)(9,18,11)(10,19,12)(21,118,111)(22,119,112)(23,120,113)(24,116,114)(25,117,115)(26,41,47)(27,42,48)(28,43,49)(29,44,50)(30,45,46)(51,79,58)(52,80,59)(53,76,60)(54,77,56)(55,78,57)(61,74,67)(62,75,68)(63,71,69)(64,72,70)(65,73,66)(81,109,88)(82,110,89)(83,106,90)(84,107,86)(85,108,87)(91,104,97)(92,105,98)(93,101,99)(94,102,100)(95,103,96), (1,26)(2,27)(3,28)(4,29)(5,30)(6,111)(7,112)(8,113)(9,114)(10,115)(11,24)(12,25)(13,21)(14,22)(15,23)(16,119)(17,120)(18,116)(19,117)(20,118)(31,44)(32,45)(33,41)(34,42)(35,43)(36,49)(37,50)(38,46)(39,47)(40,48)(51,65)(52,61)(53,62)(54,63)(55,64)(56,71)(57,72)(58,73)(59,74)(60,75)(66,79)(67,80)(68,76)(69,77)(70,78)(81,95)(82,91)(83,92)(84,93)(85,94)(86,101)(87,102)(88,103)(89,104)(90,105)(96,109)(97,110)(98,106)(99,107)(100,108), (1,56,26,71)(2,57,27,72)(3,58,28,73)(4,59,29,74)(5,60,30,75)(6,109,21,95)(7,110,22,91)(8,106,23,92)(9,107,24,93)(10,108,25,94)(11,84,114,99)(12,85,115,100)(13,81,111,96)(14,82,112,97)(15,83,113,98)(16,89,119,104)(17,90,120,105)(18,86,116,101)(19,87,117,102)(20,88,118,103)(31,80,50,61)(32,76,46,62)(33,77,47,63)(34,78,48,64)(35,79,49,65)(36,51,43,66)(37,52,44,67)(38,53,45,68)(39,54,41,69)(40,55,42,70), (1,86,26,101)(2,87,27,102)(3,88,28,103)(4,89,29,104)(5,90,30,105)(6,65,21,79)(7,61,22,80)(8,62,23,76)(9,63,24,77)(10,64,25,78)(11,69,114,54)(12,70,115,55)(13,66,111,51)(14,67,112,52)(15,68,113,53)(16,74,119,59)(17,75,120,60)(18,71,116,56)(19,72,117,57)(20,73,118,58)(31,110,50,91)(32,106,46,92)(33,107,47,93)(34,108,48,94)(35,109,49,95)(36,81,43,96)(37,82,44,97)(38,83,45,98)(39,84,41,99)(40,85,42,100)>;
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), (1,39,33)(2,40,34)(3,36,35)(4,37,31)(5,38,32)(6,20,13)(7,16,14)(8,17,15)(9,18,11)(10,19,12)(21,118,111)(22,119,112)(23,120,113)(24,116,114)(25,117,115)(26,41,47)(27,42,48)(28,43,49)(29,44,50)(30,45,46)(51,79,58)(52,80,59)(53,76,60)(54,77,56)(55,78,57)(61,74,67)(62,75,68)(63,71,69)(64,72,70)(65,73,66)(81,109,88)(82,110,89)(83,106,90)(84,107,86)(85,108,87)(91,104,97)(92,105,98)(93,101,99)(94,102,100)(95,103,96), (1,26)(2,27)(3,28)(4,29)(5,30)(6,111)(7,112)(8,113)(9,114)(10,115)(11,24)(12,25)(13,21)(14,22)(15,23)(16,119)(17,120)(18,116)(19,117)(20,118)(31,44)(32,45)(33,41)(34,42)(35,43)(36,49)(37,50)(38,46)(39,47)(40,48)(51,65)(52,61)(53,62)(54,63)(55,64)(56,71)(57,72)(58,73)(59,74)(60,75)(66,79)(67,80)(68,76)(69,77)(70,78)(81,95)(82,91)(83,92)(84,93)(85,94)(86,101)(87,102)(88,103)(89,104)(90,105)(96,109)(97,110)(98,106)(99,107)(100,108), (1,56,26,71)(2,57,27,72)(3,58,28,73)(4,59,29,74)(5,60,30,75)(6,109,21,95)(7,110,22,91)(8,106,23,92)(9,107,24,93)(10,108,25,94)(11,84,114,99)(12,85,115,100)(13,81,111,96)(14,82,112,97)(15,83,113,98)(16,89,119,104)(17,90,120,105)(18,86,116,101)(19,87,117,102)(20,88,118,103)(31,80,50,61)(32,76,46,62)(33,77,47,63)(34,78,48,64)(35,79,49,65)(36,51,43,66)(37,52,44,67)(38,53,45,68)(39,54,41,69)(40,55,42,70), (1,86,26,101)(2,87,27,102)(3,88,28,103)(4,89,29,104)(5,90,30,105)(6,65,21,79)(7,61,22,80)(8,62,23,76)(9,63,24,77)(10,64,25,78)(11,69,114,54)(12,70,115,55)(13,66,111,51)(14,67,112,52)(15,68,113,53)(16,74,119,59)(17,75,120,60)(18,71,116,56)(19,72,117,57)(20,73,118,58)(31,110,50,91)(32,106,46,92)(33,107,47,93)(34,108,48,94)(35,109,49,95)(36,81,43,96)(37,82,44,97)(38,83,45,98)(39,84,41,99)(40,85,42,100) );
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)], [(1,39,33),(2,40,34),(3,36,35),(4,37,31),(5,38,32),(6,20,13),(7,16,14),(8,17,15),(9,18,11),(10,19,12),(21,118,111),(22,119,112),(23,120,113),(24,116,114),(25,117,115),(26,41,47),(27,42,48),(28,43,49),(29,44,50),(30,45,46),(51,79,58),(52,80,59),(53,76,60),(54,77,56),(55,78,57),(61,74,67),(62,75,68),(63,71,69),(64,72,70),(65,73,66),(81,109,88),(82,110,89),(83,106,90),(84,107,86),(85,108,87),(91,104,97),(92,105,98),(93,101,99),(94,102,100),(95,103,96)], [(1,26),(2,27),(3,28),(4,29),(5,30),(6,111),(7,112),(8,113),(9,114),(10,115),(11,24),(12,25),(13,21),(14,22),(15,23),(16,119),(17,120),(18,116),(19,117),(20,118),(31,44),(32,45),(33,41),(34,42),(35,43),(36,49),(37,50),(38,46),(39,47),(40,48),(51,65),(52,61),(53,62),(54,63),(55,64),(56,71),(57,72),(58,73),(59,74),(60,75),(66,79),(67,80),(68,76),(69,77),(70,78),(81,95),(82,91),(83,92),(84,93),(85,94),(86,101),(87,102),(88,103),(89,104),(90,105),(96,109),(97,110),(98,106),(99,107),(100,108)], [(1,56,26,71),(2,57,27,72),(3,58,28,73),(4,59,29,74),(5,60,30,75),(6,109,21,95),(7,110,22,91),(8,106,23,92),(9,107,24,93),(10,108,25,94),(11,84,114,99),(12,85,115,100),(13,81,111,96),(14,82,112,97),(15,83,113,98),(16,89,119,104),(17,90,120,105),(18,86,116,101),(19,87,117,102),(20,88,118,103),(31,80,50,61),(32,76,46,62),(33,77,47,63),(34,78,48,64),(35,79,49,65),(36,51,43,66),(37,52,44,67),(38,53,45,68),(39,54,41,69),(40,55,42,70)], [(1,86,26,101),(2,87,27,102),(3,88,28,103),(4,89,29,104),(5,90,30,105),(6,65,21,79),(7,61,22,80),(8,62,23,76),(9,63,24,77),(10,64,25,78),(11,69,114,54),(12,70,115,55),(13,66,111,51),(14,67,112,52),(15,68,113,53),(16,74,119,59),(17,75,120,60),(18,71,116,56),(19,72,117,57),(20,73,118,58),(31,110,50,91),(32,106,46,92),(33,107,47,93),(34,108,48,94),(35,109,49,95),(36,81,43,96),(37,82,44,97),(38,83,45,98),(39,84,41,99),(40,85,42,100)]])
C5×S3×Q8 is a maximal subgroup of
D20.28D6 C60.44C23 D20.29D6 C30.33C24
75 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 5C | 5D | 6 | 10A | 10B | 10C | 10D | 10E | ··· | 10L | 12A | 12B | 12C | 15A | 15B | 15C | 15D | 20A | ··· | 20L | 20M | ··· | 20X | 30A | 30B | 30C | 30D | 60A | ··· | 60L |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 30 | 30 | 30 | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 3 | 3 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 3 | ··· | 3 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
75 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | - | + | - | ||||||||
| image | C1 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | S3 | Q8 | D6 | C5×S3 | C5×Q8 | S3×C10 | S3×Q8 | C5×S3×Q8 |
| kernel | C5×S3×Q8 | C5×Dic6 | S3×C20 | Q8×C15 | S3×Q8 | Dic6 | C4×S3 | C3×Q8 | C5×Q8 | C5×S3 | C20 | Q8 | S3 | C4 | C5 | C1 |
| # reps | 1 | 3 | 3 | 1 | 4 | 12 | 12 | 4 | 1 | 2 | 3 | 4 | 8 | 12 | 1 | 4 |
Matrix representation of C5×S3×Q8 ►in GL4(𝔽61) generated by
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 58 | 0 |
| 0 | 0 | 0 | 58 |
| 0 | 60 | 0 | 0 |
| 1 | 60 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 60 |
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 60 | 0 |
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 13 | 47 |
| 0 | 0 | 47 | 48 |
G:=sub<GL(4,GF(61))| [9,0,0,0,0,9,0,0,0,0,58,0,0,0,0,58],[0,1,0,0,60,60,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,60,0,0,0,0,60],[60,0,0,0,0,60,0,0,0,0,0,60,0,0,1,0],[60,0,0,0,0,60,0,0,0,0,13,47,0,0,47,48] >;
C5×S3×Q8 in GAP, Magma, Sage, TeX
C_5\times S_3\times Q_8
% in TeX
G:=Group("C5xS3xQ8"); // GroupNames label
G:=SmallGroup(240,171);
// by ID
G=gap.SmallGroup(240,171);
# by ID
G:=PCGroup([6,-2,-2,-2,-5,-2,-3,247,404,194,5765]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^3=c^2=d^4=1,e^2=d^2,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,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations