direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C10×C3⋊D4, C30⋊9D4, C30.57C23, C6⋊2(C5×D4), C3⋊3(D4×C10), C15⋊18(C2×D4), D6⋊3(C2×C10), (C2×C10)⋊10D6, C23⋊2(C5×S3), C22⋊3(S3×C10), (C22×C10)⋊3S3, (C22×C30)⋊6C2, (C22×C6)⋊2C10, (C22×S3)⋊3C10, (C2×C30)⋊13C22, (C2×Dic3)⋊4C10, Dic3⋊2(C2×C10), (S3×C10)⋊11C22, (C10×Dic3)⋊10C2, C10.47(C22×S3), C6.10(C22×C10), (C5×Dic3)⋊9C22, (S3×C2×C10)⋊7C2, (C2×C6)⋊3(C2×C10), C2.10(S3×C2×C10), SmallGroup(240,174)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C10×C3⋊D4
G = < a,b,c,d | a10=b3=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >
Subgroups: 216 in 108 conjugacy classes, 54 normal (22 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C5, S3, C6, C6, C6, C2×C4, D4, C23, C23, C10, C10, C10, Dic3, D6, D6, C2×C6, C2×C6, C2×C6, C15, C2×D4, C20, C2×C10, C2×C10, C2×C10, C2×Dic3, C3⋊D4, C22×S3, C22×C6, C5×S3, C30, C30, C30, C2×C20, C5×D4, C22×C10, C22×C10, C2×C3⋊D4, C5×Dic3, S3×C10, S3×C10, C2×C30, C2×C30, C2×C30, D4×C10, C10×Dic3, C5×C3⋊D4, S3×C2×C10, C22×C30, C10×C3⋊D4
Quotients: C1, C2, C22, C5, S3, D4, C23, C10, D6, C2×D4, C2×C10, C3⋊D4, C22×S3, C5×S3, C5×D4, C22×C10, C2×C3⋊D4, S3×C10, D4×C10, C5×C3⋊D4, S3×C2×C10, C10×C3⋊D4
►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 41 81)(2 42 82)(3 43 83)(4 44 84)(5 45 85)(6 46 86)(7 47 87)(8 48 88)(9 49 89)(10 50 90)(11 34 120)(12 35 111)(13 36 112)(14 37 113)(15 38 114)(16 39 115)(17 40 116)(18 31 117)(19 32 118)(20 33 119)(21 107 78)(22 108 79)(23 109 80)(24 110 71)(25 101 72)(26 102 73)(27 103 74)(28 104 75)(29 105 76)(30 106 77)(51 68 91)(52 69 92)(53 70 93)(54 61 94)(55 62 95)(56 63 96)(57 64 97)(58 65 98)(59 66 99)(60 67 100)
(1 22 54 113)(2 23 55 114)(3 24 56 115)(4 25 57 116)(5 26 58 117)(6 27 59 118)(7 28 60 119)(8 29 51 120)(9 30 52 111)(10 21 53 112)(11 88 105 91)(12 89 106 92)(13 90 107 93)(14 81 108 94)(15 82 109 95)(16 83 110 96)(17 84 101 97)(18 85 102 98)(19 86 103 99)(20 87 104 100)(31 45 73 65)(32 46 74 66)(33 47 75 67)(34 48 76 68)(35 49 77 69)(36 50 78 70)(37 41 79 61)(38 42 80 62)(39 43 71 63)(40 44 72 64)
(11 76)(12 77)(13 78)(14 79)(15 80)(16 71)(17 72)(18 73)(19 74)(20 75)(21 112)(22 113)(23 114)(24 115)(25 116)(26 117)(27 118)(28 119)(29 120)(30 111)(31 102)(32 103)(33 104)(34 105)(35 106)(36 107)(37 108)(38 109)(39 110)(40 101)(41 81)(42 82)(43 83)(44 84)(45 85)(46 86)(47 87)(48 88)(49 89)(50 90)(61 94)(62 95)(63 96)(64 97)(65 98)(66 99)(67 100)(68 91)(69 92)(70 93)
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), (1,41,81)(2,42,82)(3,43,83)(4,44,84)(5,45,85)(6,46,86)(7,47,87)(8,48,88)(9,49,89)(10,50,90)(11,34,120)(12,35,111)(13,36,112)(14,37,113)(15,38,114)(16,39,115)(17,40,116)(18,31,117)(19,32,118)(20,33,119)(21,107,78)(22,108,79)(23,109,80)(24,110,71)(25,101,72)(26,102,73)(27,103,74)(28,104,75)(29,105,76)(30,106,77)(51,68,91)(52,69,92)(53,70,93)(54,61,94)(55,62,95)(56,63,96)(57,64,97)(58,65,98)(59,66,99)(60,67,100), (1,22,54,113)(2,23,55,114)(3,24,56,115)(4,25,57,116)(5,26,58,117)(6,27,59,118)(7,28,60,119)(8,29,51,120)(9,30,52,111)(10,21,53,112)(11,88,105,91)(12,89,106,92)(13,90,107,93)(14,81,108,94)(15,82,109,95)(16,83,110,96)(17,84,101,97)(18,85,102,98)(19,86,103,99)(20,87,104,100)(31,45,73,65)(32,46,74,66)(33,47,75,67)(34,48,76,68)(35,49,77,69)(36,50,78,70)(37,41,79,61)(38,42,80,62)(39,43,71,63)(40,44,72,64), (11,76)(12,77)(13,78)(14,79)(15,80)(16,71)(17,72)(18,73)(19,74)(20,75)(21,112)(22,113)(23,114)(24,115)(25,116)(26,117)(27,118)(28,119)(29,120)(30,111)(31,102)(32,103)(33,104)(34,105)(35,106)(36,107)(37,108)(38,109)(39,110)(40,101)(41,81)(42,82)(43,83)(44,84)(45,85)(46,86)(47,87)(48,88)(49,89)(50,90)(61,94)(62,95)(63,96)(64,97)(65,98)(66,99)(67,100)(68,91)(69,92)(70,93)>;
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), (1,41,81)(2,42,82)(3,43,83)(4,44,84)(5,45,85)(6,46,86)(7,47,87)(8,48,88)(9,49,89)(10,50,90)(11,34,120)(12,35,111)(13,36,112)(14,37,113)(15,38,114)(16,39,115)(17,40,116)(18,31,117)(19,32,118)(20,33,119)(21,107,78)(22,108,79)(23,109,80)(24,110,71)(25,101,72)(26,102,73)(27,103,74)(28,104,75)(29,105,76)(30,106,77)(51,68,91)(52,69,92)(53,70,93)(54,61,94)(55,62,95)(56,63,96)(57,64,97)(58,65,98)(59,66,99)(60,67,100), (1,22,54,113)(2,23,55,114)(3,24,56,115)(4,25,57,116)(5,26,58,117)(6,27,59,118)(7,28,60,119)(8,29,51,120)(9,30,52,111)(10,21,53,112)(11,88,105,91)(12,89,106,92)(13,90,107,93)(14,81,108,94)(15,82,109,95)(16,83,110,96)(17,84,101,97)(18,85,102,98)(19,86,103,99)(20,87,104,100)(31,45,73,65)(32,46,74,66)(33,47,75,67)(34,48,76,68)(35,49,77,69)(36,50,78,70)(37,41,79,61)(38,42,80,62)(39,43,71,63)(40,44,72,64), (11,76)(12,77)(13,78)(14,79)(15,80)(16,71)(17,72)(18,73)(19,74)(20,75)(21,112)(22,113)(23,114)(24,115)(25,116)(26,117)(27,118)(28,119)(29,120)(30,111)(31,102)(32,103)(33,104)(34,105)(35,106)(36,107)(37,108)(38,109)(39,110)(40,101)(41,81)(42,82)(43,83)(44,84)(45,85)(46,86)(47,87)(48,88)(49,89)(50,90)(61,94)(62,95)(63,96)(64,97)(65,98)(66,99)(67,100)(68,91)(69,92)(70,93) );
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)], [(1,41,81),(2,42,82),(3,43,83),(4,44,84),(5,45,85),(6,46,86),(7,47,87),(8,48,88),(9,49,89),(10,50,90),(11,34,120),(12,35,111),(13,36,112),(14,37,113),(15,38,114),(16,39,115),(17,40,116),(18,31,117),(19,32,118),(20,33,119),(21,107,78),(22,108,79),(23,109,80),(24,110,71),(25,101,72),(26,102,73),(27,103,74),(28,104,75),(29,105,76),(30,106,77),(51,68,91),(52,69,92),(53,70,93),(54,61,94),(55,62,95),(56,63,96),(57,64,97),(58,65,98),(59,66,99),(60,67,100)], [(1,22,54,113),(2,23,55,114),(3,24,56,115),(4,25,57,116),(5,26,58,117),(6,27,59,118),(7,28,60,119),(8,29,51,120),(9,30,52,111),(10,21,53,112),(11,88,105,91),(12,89,106,92),(13,90,107,93),(14,81,108,94),(15,82,109,95),(16,83,110,96),(17,84,101,97),(18,85,102,98),(19,86,103,99),(20,87,104,100),(31,45,73,65),(32,46,74,66),(33,47,75,67),(34,48,76,68),(35,49,77,69),(36,50,78,70),(37,41,79,61),(38,42,80,62),(39,43,71,63),(40,44,72,64)], [(11,76),(12,77),(13,78),(14,79),(15,80),(16,71),(17,72),(18,73),(19,74),(20,75),(21,112),(22,113),(23,114),(24,115),(25,116),(26,117),(27,118),(28,119),(29,120),(30,111),(31,102),(32,103),(33,104),(34,105),(35,106),(36,107),(37,108),(38,109),(39,110),(40,101),(41,81),(42,82),(43,83),(44,84),(45,85),(46,86),(47,87),(48,88),(49,89),(50,90),(61,94),(62,95),(63,96),(64,97),(65,98),(66,99),(67,100),(68,91),(69,92),(70,93)]])
C10×C3⋊D4 is a maximal subgroup of
C15⋊8(C23⋊C4) C23.D5⋊S3 C30.(C2×D4) (C2×C10).D12 (C6×D5)⋊D4 (S3×C10).D4 D30⋊7D4 Dic15⋊4D4 Dic15⋊17D4 (C2×C30)⋊D4 (S3×C10)⋊D4 (C2×C10)⋊4D12 Dic15⋊5D4 Dic15⋊18D4 D30⋊19D4 D30⋊8D4 C15⋊2+ 1+4 S3×D4×C10
90 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 5A | 5B | 5C | 5D | 6A | ··· | 6G | 10A | ··· | 10L | 10M | ··· | 10T | 10U | ··· | 10AB | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 30A | ··· | 30AB |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | ··· | 6 | 10 | ··· | 10 | 10 | ··· | 10 | 10 | ··· | 10 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 6 | 6 | 2 | 6 | 6 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | 2 | 2 | 2 | 6 | ··· | 6 | 2 | ··· | 2 |
90 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | S3 | D4 | D6 | C3⋊D4 | C5×S3 | C5×D4 | S3×C10 | C5×C3⋊D4 |
| kernel | C10×C3⋊D4 | C10×Dic3 | C5×C3⋊D4 | S3×C2×C10 | C22×C30 | C2×C3⋊D4 | C2×Dic3 | C3⋊D4 | C22×S3 | C22×C6 | C22×C10 | C30 | C2×C10 | C10 | C23 | C6 | C22 | C2 |
| # reps | 1 | 1 | 4 | 1 | 1 | 4 | 4 | 16 | 4 | 4 | 1 | 2 | 3 | 4 | 4 | 8 | 12 | 16 |
Matrix representation of C10×C3⋊D4 ►in GL3(𝔽61) generated by
| 60 | 0 | 0 |
| 0 | 27 | 0 |
| 0 | 0 | 27 |
| 1 | 0 | 0 |
| 0 | 60 | 60 |
| 0 | 1 | 0 |
| 60 | 0 | 0 |
| 0 | 9 | 18 |
| 0 | 9 | 52 |
| 60 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 60 | 60 |
G:=sub<GL(3,GF(61))| [60,0,0,0,27,0,0,0,27],[1,0,0,0,60,1,0,60,0],[60,0,0,0,9,9,0,18,52],[60,0,0,0,1,60,0,0,60] >;
C10×C3⋊D4 in GAP, Magma, Sage, TeX
C_{10}\times C_3\rtimes D_4 % in TeX
G:=Group("C10xC3:D4"); // GroupNames label
G:=SmallGroup(240,174);
// by ID
G=gap.SmallGroup(240,174);
# by ID
G:=PCGroup([6,-2,-2,-2,-5,-2,-3,794,5765]);
// Polycyclic
G:=Group<a,b,c,d|a^10=b^3=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations