direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5×Q8⋊3S3, D12⋊4C10, C20.42D6, C60.49C22, C30.55C23, Q8⋊3(C5×S3), (C5×Q8)⋊7S3, (C4×S3)⋊3C10, (S3×C20)⋊8C2, C4.7(S3×C10), (Q8×C15)⋊9C2, (C3×Q8)⋊3C10, (C5×D12)⋊10C2, C15⋊19(C4○D4), C12.7(C2×C10), D6.3(C2×C10), C6.8(C22×C10), C10.45(C22×S3), Dic3.5(C2×C10), (S3×C10).14C22, (C5×Dic3).17C22, C3⋊3(C5×C4○D4), C2.9(S3×C2×C10), (C5×Q8)○(C5×Dic3), SmallGroup(240,172)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×Q8⋊3S3
G = < a,b,c,d,e | a5=b4=d3=e2=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe=b-1, bd=db, cd=dc, ce=ec, ede=d-1 >
Subgroups: 160 in 80 conjugacy classes, 46 normal (16 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C2×C4, D4, Q8, C10, C10, Dic3, C12, D6, C15, C4○D4, C20, C20, C2×C10, C4×S3, D12, C3×Q8, C5×S3, C30, C2×C20, C5×D4, C5×Q8, Q8⋊3S3, C5×Dic3, C60, S3×C10, C5×C4○D4, S3×C20, C5×D12, Q8×C15, C5×Q8⋊3S3
Quotients: C1, C2, C22, C5, S3, C23, C10, D6, C4○D4, C2×C10, C22×S3, C5×S3, C22×C10, Q8⋊3S3, S3×C10, C5×C4○D4, S3×C2×C10, C5×Q8⋊3S3
►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 56 26 77)(2 57 27 78)(3 58 28 79)(4 59 29 80)(5 60 30 76)(6 86 116 107)(7 87 117 108)(8 88 118 109)(9 89 119 110)(10 90 120 106)(11 96 25 94)(12 97 21 95)(13 98 22 91)(14 99 23 92)(15 100 24 93)(16 83 113 104)(17 84 114 105)(18 85 115 101)(19 81 111 102)(20 82 112 103)(31 68 38 61)(32 69 39 62)(33 70 40 63)(34 66 36 64)(35 67 37 65)(41 55 48 71)(42 51 49 72)(43 52 50 73)(44 53 46 74)(45 54 47 75)
(1 86 26 107)(2 87 27 108)(3 88 28 109)(4 89 29 110)(5 90 30 106)(6 77 116 56)(7 78 117 57)(8 79 118 58)(9 80 119 59)(10 76 120 60)(11 64 25 66)(12 65 21 67)(13 61 22 68)(14 62 23 69)(15 63 24 70)(16 74 113 53)(17 75 114 54)(18 71 115 55)(19 72 111 51)(20 73 112 52)(31 98 38 91)(32 99 39 92)(33 100 40 93)(34 96 36 94)(35 97 37 95)(41 85 48 101)(42 81 49 102)(43 82 50 103)(44 83 46 104)(45 84 47 105)
(1 33 45)(2 34 41)(3 35 42)(4 31 43)(5 32 44)(6 15 17)(7 11 18)(8 12 19)(9 13 20)(10 14 16)(21 111 118)(22 112 119)(23 113 120)(24 114 116)(25 115 117)(26 40 47)(27 36 48)(28 37 49)(29 38 50)(30 39 46)(51 58 67)(52 59 68)(53 60 69)(54 56 70)(55 57 66)(61 73 80)(62 74 76)(63 75 77)(64 71 78)(65 72 79)(81 88 97)(82 89 98)(83 90 99)(84 86 100)(85 87 96)(91 103 110)(92 104 106)(93 105 107)(94 101 108)(95 102 109)
(6 116)(7 117)(8 118)(9 119)(10 120)(11 115)(12 111)(13 112)(14 113)(15 114)(16 23)(17 24)(18 25)(19 21)(20 22)(31 43)(32 44)(33 45)(34 41)(35 42)(36 48)(37 49)(38 50)(39 46)(40 47)(51 65)(52 61)(53 62)(54 63)(55 64)(56 77)(57 78)(58 79)(59 80)(60 76)(66 71)(67 72)(68 73)(69 74)(70 75)(81 97)(82 98)(83 99)(84 100)(85 96)(91 103)(92 104)(93 105)(94 101)(95 102)
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), (1,56,26,77)(2,57,27,78)(3,58,28,79)(4,59,29,80)(5,60,30,76)(6,86,116,107)(7,87,117,108)(8,88,118,109)(9,89,119,110)(10,90,120,106)(11,96,25,94)(12,97,21,95)(13,98,22,91)(14,99,23,92)(15,100,24,93)(16,83,113,104)(17,84,114,105)(18,85,115,101)(19,81,111,102)(20,82,112,103)(31,68,38,61)(32,69,39,62)(33,70,40,63)(34,66,36,64)(35,67,37,65)(41,55,48,71)(42,51,49,72)(43,52,50,73)(44,53,46,74)(45,54,47,75), (1,86,26,107)(2,87,27,108)(3,88,28,109)(4,89,29,110)(5,90,30,106)(6,77,116,56)(7,78,117,57)(8,79,118,58)(9,80,119,59)(10,76,120,60)(11,64,25,66)(12,65,21,67)(13,61,22,68)(14,62,23,69)(15,63,24,70)(16,74,113,53)(17,75,114,54)(18,71,115,55)(19,72,111,51)(20,73,112,52)(31,98,38,91)(32,99,39,92)(33,100,40,93)(34,96,36,94)(35,97,37,95)(41,85,48,101)(42,81,49,102)(43,82,50,103)(44,83,46,104)(45,84,47,105), (1,33,45)(2,34,41)(3,35,42)(4,31,43)(5,32,44)(6,15,17)(7,11,18)(8,12,19)(9,13,20)(10,14,16)(21,111,118)(22,112,119)(23,113,120)(24,114,116)(25,115,117)(26,40,47)(27,36,48)(28,37,49)(29,38,50)(30,39,46)(51,58,67)(52,59,68)(53,60,69)(54,56,70)(55,57,66)(61,73,80)(62,74,76)(63,75,77)(64,71,78)(65,72,79)(81,88,97)(82,89,98)(83,90,99)(84,86,100)(85,87,96)(91,103,110)(92,104,106)(93,105,107)(94,101,108)(95,102,109), (6,116)(7,117)(8,118)(9,119)(10,120)(11,115)(12,111)(13,112)(14,113)(15,114)(16,23)(17,24)(18,25)(19,21)(20,22)(31,43)(32,44)(33,45)(34,41)(35,42)(36,48)(37,49)(38,50)(39,46)(40,47)(51,65)(52,61)(53,62)(54,63)(55,64)(56,77)(57,78)(58,79)(59,80)(60,76)(66,71)(67,72)(68,73)(69,74)(70,75)(81,97)(82,98)(83,99)(84,100)(85,96)(91,103)(92,104)(93,105)(94,101)(95,102)>;
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), (1,56,26,77)(2,57,27,78)(3,58,28,79)(4,59,29,80)(5,60,30,76)(6,86,116,107)(7,87,117,108)(8,88,118,109)(9,89,119,110)(10,90,120,106)(11,96,25,94)(12,97,21,95)(13,98,22,91)(14,99,23,92)(15,100,24,93)(16,83,113,104)(17,84,114,105)(18,85,115,101)(19,81,111,102)(20,82,112,103)(31,68,38,61)(32,69,39,62)(33,70,40,63)(34,66,36,64)(35,67,37,65)(41,55,48,71)(42,51,49,72)(43,52,50,73)(44,53,46,74)(45,54,47,75), (1,86,26,107)(2,87,27,108)(3,88,28,109)(4,89,29,110)(5,90,30,106)(6,77,116,56)(7,78,117,57)(8,79,118,58)(9,80,119,59)(10,76,120,60)(11,64,25,66)(12,65,21,67)(13,61,22,68)(14,62,23,69)(15,63,24,70)(16,74,113,53)(17,75,114,54)(18,71,115,55)(19,72,111,51)(20,73,112,52)(31,98,38,91)(32,99,39,92)(33,100,40,93)(34,96,36,94)(35,97,37,95)(41,85,48,101)(42,81,49,102)(43,82,50,103)(44,83,46,104)(45,84,47,105), (1,33,45)(2,34,41)(3,35,42)(4,31,43)(5,32,44)(6,15,17)(7,11,18)(8,12,19)(9,13,20)(10,14,16)(21,111,118)(22,112,119)(23,113,120)(24,114,116)(25,115,117)(26,40,47)(27,36,48)(28,37,49)(29,38,50)(30,39,46)(51,58,67)(52,59,68)(53,60,69)(54,56,70)(55,57,66)(61,73,80)(62,74,76)(63,75,77)(64,71,78)(65,72,79)(81,88,97)(82,89,98)(83,90,99)(84,86,100)(85,87,96)(91,103,110)(92,104,106)(93,105,107)(94,101,108)(95,102,109), (6,116)(7,117)(8,118)(9,119)(10,120)(11,115)(12,111)(13,112)(14,113)(15,114)(16,23)(17,24)(18,25)(19,21)(20,22)(31,43)(32,44)(33,45)(34,41)(35,42)(36,48)(37,49)(38,50)(39,46)(40,47)(51,65)(52,61)(53,62)(54,63)(55,64)(56,77)(57,78)(58,79)(59,80)(60,76)(66,71)(67,72)(68,73)(69,74)(70,75)(81,97)(82,98)(83,99)(84,100)(85,96)(91,103)(92,104)(93,105)(94,101)(95,102) );
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)], [(1,56,26,77),(2,57,27,78),(3,58,28,79),(4,59,29,80),(5,60,30,76),(6,86,116,107),(7,87,117,108),(8,88,118,109),(9,89,119,110),(10,90,120,106),(11,96,25,94),(12,97,21,95),(13,98,22,91),(14,99,23,92),(15,100,24,93),(16,83,113,104),(17,84,114,105),(18,85,115,101),(19,81,111,102),(20,82,112,103),(31,68,38,61),(32,69,39,62),(33,70,40,63),(34,66,36,64),(35,67,37,65),(41,55,48,71),(42,51,49,72),(43,52,50,73),(44,53,46,74),(45,54,47,75)], [(1,86,26,107),(2,87,27,108),(3,88,28,109),(4,89,29,110),(5,90,30,106),(6,77,116,56),(7,78,117,57),(8,79,118,58),(9,80,119,59),(10,76,120,60),(11,64,25,66),(12,65,21,67),(13,61,22,68),(14,62,23,69),(15,63,24,70),(16,74,113,53),(17,75,114,54),(18,71,115,55),(19,72,111,51),(20,73,112,52),(31,98,38,91),(32,99,39,92),(33,100,40,93),(34,96,36,94),(35,97,37,95),(41,85,48,101),(42,81,49,102),(43,82,50,103),(44,83,46,104),(45,84,47,105)], [(1,33,45),(2,34,41),(3,35,42),(4,31,43),(5,32,44),(6,15,17),(7,11,18),(8,12,19),(9,13,20),(10,14,16),(21,111,118),(22,112,119),(23,113,120),(24,114,116),(25,115,117),(26,40,47),(27,36,48),(28,37,49),(29,38,50),(30,39,46),(51,58,67),(52,59,68),(53,60,69),(54,56,70),(55,57,66),(61,73,80),(62,74,76),(63,75,77),(64,71,78),(65,72,79),(81,88,97),(82,89,98),(83,90,99),(84,86,100),(85,87,96),(91,103,110),(92,104,106),(93,105,107),(94,101,108),(95,102,109)], [(6,116),(7,117),(8,118),(9,119),(10,120),(11,115),(12,111),(13,112),(14,113),(15,114),(16,23),(17,24),(18,25),(19,21),(20,22),(31,43),(32,44),(33,45),(34,41),(35,42),(36,48),(37,49),(38,50),(39,46),(40,47),(51,65),(52,61),(53,62),(54,63),(55,64),(56,77),(57,78),(58,79),(59,80),(60,76),(66,71),(67,72),(68,73),(69,74),(70,75),(81,97),(82,98),(83,99),(84,100),(85,96),(91,103),(92,104),(93,105),(94,101),(95,102)]])
C5×Q8⋊3S3 is a maximal subgroup of
D12⋊D10 Dic10.26D6 D20.27D6 Dic10.27D6 D12.29D10 D20⋊16D6 D20⋊17D6 C5×S3×C4○D4
C5×Q8⋊3S3 is a maximal quotient of
C5×Q8×Dic3
75 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 5C | 5D | 6 | 10A | 10B | 10C | 10D | 10E | ··· | 10P | 12A | 12B | 12C | 15A | 15B | 15C | 15D | 20A | ··· | 20L | 20M | ··· | 20T | 30A | 30B | 30C | 30D | 60A | ··· | 60L |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 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 | 6 | 6 | 6 | 2 | 2 | 2 | 2 | 3 | 3 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 6 | ··· | 6 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 3 | ··· | 3 | 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 | D6 | C4○D4 | C5×S3 | S3×C10 | C5×C4○D4 | Q8⋊3S3 | C5×Q8⋊3S3 |
| kernel | C5×Q8⋊3S3 | S3×C20 | C5×D12 | Q8×C15 | Q8⋊3S3 | C4×S3 | D12 | C3×Q8 | C5×Q8 | C20 | C15 | Q8 | C4 | C3 | C5 | C1 |
| # reps | 1 | 3 | 3 | 1 | 4 | 12 | 12 | 4 | 1 | 3 | 2 | 4 | 12 | 8 | 1 | 4 |
Matrix representation of C5×Q8⋊3S3 ►in GL4(𝔽61) generated by
| 58 | 0 | 0 | 0 |
| 0 | 58 | 0 | 0 |
| 0 | 0 | 34 | 0 |
| 0 | 0 | 0 | 34 |
| 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 | 50 | 0 |
| 0 | 0 | 0 | 11 |
| 60 | 1 | 0 | 0 |
| 60 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 60 |
G:=sub<GL(4,GF(61))| [58,0,0,0,0,58,0,0,0,0,34,0,0,0,0,34],[60,0,0,0,0,60,0,0,0,0,0,60,0,0,1,0],[60,0,0,0,0,60,0,0,0,0,50,0,0,0,0,11],[60,60,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,60] >;
C5×Q8⋊3S3 in GAP, Magma, Sage, TeX
C_5\times Q_8\rtimes_3S_3
% in TeX
G:=Group("C5xQ8:3S3"); // GroupNames label
G:=SmallGroup(240,172);
// by ID
G=gap.SmallGroup(240,172);
# by ID
G:=PCGroup([6,-2,-2,-2,-5,-2,-3,247,794,404,194,5765]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^4=d^3=e^2=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=e*b*e=b^-1,b*d=d*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations