direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C6×C5⋊D4, C30⋊8D4, C30.47C23, C5⋊3(C6×D4), C10⋊2(C3×D4), (C2×C6)⋊7D10, C15⋊17(C2×D4), D10⋊3(C2×C6), C23⋊2(C3×D5), (C22×C6)⋊1D5, C22⋊3(C6×D5), (C22×C10)⋊4C6, (C22×C30)⋊4C2, Dic5⋊2(C2×C6), (C2×Dic5)⋊4C6, (C22×D5)⋊4C6, (C2×C30)⋊11C22, (C6×Dic5)⋊10C2, (C6×D5)⋊11C22, C6.47(C22×D5), C10.10(C22×C6), (C3×Dic5)⋊9C22, (D5×C2×C6)⋊7C2, C2.10(D5×C2×C6), (C2×C10)⋊5(C2×C6), SmallGroup(240,164)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6×C5⋊D4
G = < a,b,c,d | a6=b5=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >
Subgroups: 292 in 108 conjugacy classes, 54 normal (22 characteristic)
C1, C2, C2 [×2], C2 [×4], C3, C4 [×2], C22, C22 [×2], C22 [×6], C5, C6, C6 [×2], C6 [×4], C2×C4, D4 [×4], C23, C23, D5 [×2], C10, C10 [×2], C10 [×2], C12 [×2], C2×C6, C2×C6 [×2], C2×C6 [×6], C15, C2×D4, Dic5 [×2], D10 [×2], D10 [×2], C2×C10, C2×C10 [×2], C2×C10 [×2], C2×C12, C3×D4 [×4], C22×C6, C22×C6, C3×D5 [×2], C30, C30 [×2], C30 [×2], C2×Dic5, C5⋊D4 [×4], C22×D5, C22×C10, C6×D4, C3×Dic5 [×2], C6×D5 [×2], C6×D5 [×2], C2×C30, C2×C30 [×2], C2×C30 [×2], C2×C5⋊D4, C6×Dic5, C3×C5⋊D4 [×4], D5×C2×C6, C22×C30, C6×C5⋊D4
Quotients: C1, C2 [×7], C3, C22 [×7], C6 [×7], D4 [×2], C23, D5, C2×C6 [×7], C2×D4, D10 [×3], C3×D4 [×2], C22×C6, C3×D5, C5⋊D4 [×2], C22×D5, C6×D4, C6×D5 [×3], C2×C5⋊D4, C3×C5⋊D4 [×2], D5×C2×C6, C6×C5⋊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 27 99 69 115)(2 28 100 70 116)(3 29 101 71 117)(4 30 102 72 118)(5 25 97 67 119)(6 26 98 68 120)(7 47 39 86 91)(8 48 40 87 92)(9 43 41 88 93)(10 44 42 89 94)(11 45 37 90 95)(12 46 38 85 96)(13 32 52 81 111)(14 33 53 82 112)(15 34 54 83 113)(16 35 49 84 114)(17 36 50 79 109)(18 31 51 80 110)(19 59 76 62 105)(20 60 77 63 106)(21 55 78 64 107)(22 56 73 65 108)(23 57 74 66 103)(24 58 75 61 104)
(1 63 79 38)(2 64 80 39)(3 65 81 40)(4 66 82 41)(5 61 83 42)(6 62 84 37)(7 100 55 18)(8 101 56 13)(9 102 57 14)(10 97 58 15)(11 98 59 16)(12 99 60 17)(19 35 95 68)(20 36 96 69)(21 31 91 70)(22 32 92 71)(23 33 93 72)(24 34 94 67)(25 75 113 44)(26 76 114 45)(27 77 109 46)(28 78 110 47)(29 73 111 48)(30 74 112 43)(49 90 120 105)(50 85 115 106)(51 86 116 107)(52 87 117 108)(53 88 118 103)(54 89 119 104)
(7 21)(8 22)(9 23)(10 24)(11 19)(12 20)(13 32)(14 33)(15 34)(16 35)(17 36)(18 31)(25 119)(26 120)(27 115)(28 116)(29 117)(30 118)(37 62)(38 63)(39 64)(40 65)(41 66)(42 61)(43 103)(44 104)(45 105)(46 106)(47 107)(48 108)(49 114)(50 109)(51 110)(52 111)(53 112)(54 113)(55 91)(56 92)(57 93)(58 94)(59 95)(60 96)(67 97)(68 98)(69 99)(70 100)(71 101)(72 102)(73 87)(74 88)(75 89)(76 90)(77 85)(78 86)
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), (1,27,99,69,115)(2,28,100,70,116)(3,29,101,71,117)(4,30,102,72,118)(5,25,97,67,119)(6,26,98,68,120)(7,47,39,86,91)(8,48,40,87,92)(9,43,41,88,93)(10,44,42,89,94)(11,45,37,90,95)(12,46,38,85,96)(13,32,52,81,111)(14,33,53,82,112)(15,34,54,83,113)(16,35,49,84,114)(17,36,50,79,109)(18,31,51,80,110)(19,59,76,62,105)(20,60,77,63,106)(21,55,78,64,107)(22,56,73,65,108)(23,57,74,66,103)(24,58,75,61,104), (1,63,79,38)(2,64,80,39)(3,65,81,40)(4,66,82,41)(5,61,83,42)(6,62,84,37)(7,100,55,18)(8,101,56,13)(9,102,57,14)(10,97,58,15)(11,98,59,16)(12,99,60,17)(19,35,95,68)(20,36,96,69)(21,31,91,70)(22,32,92,71)(23,33,93,72)(24,34,94,67)(25,75,113,44)(26,76,114,45)(27,77,109,46)(28,78,110,47)(29,73,111,48)(30,74,112,43)(49,90,120,105)(50,85,115,106)(51,86,116,107)(52,87,117,108)(53,88,118,103)(54,89,119,104), (7,21)(8,22)(9,23)(10,24)(11,19)(12,20)(13,32)(14,33)(15,34)(16,35)(17,36)(18,31)(25,119)(26,120)(27,115)(28,116)(29,117)(30,118)(37,62)(38,63)(39,64)(40,65)(41,66)(42,61)(43,103)(44,104)(45,105)(46,106)(47,107)(48,108)(49,114)(50,109)(51,110)(52,111)(53,112)(54,113)(55,91)(56,92)(57,93)(58,94)(59,95)(60,96)(67,97)(68,98)(69,99)(70,100)(71,101)(72,102)(73,87)(74,88)(75,89)(76,90)(77,85)(78,86)>;
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), (1,27,99,69,115)(2,28,100,70,116)(3,29,101,71,117)(4,30,102,72,118)(5,25,97,67,119)(6,26,98,68,120)(7,47,39,86,91)(8,48,40,87,92)(9,43,41,88,93)(10,44,42,89,94)(11,45,37,90,95)(12,46,38,85,96)(13,32,52,81,111)(14,33,53,82,112)(15,34,54,83,113)(16,35,49,84,114)(17,36,50,79,109)(18,31,51,80,110)(19,59,76,62,105)(20,60,77,63,106)(21,55,78,64,107)(22,56,73,65,108)(23,57,74,66,103)(24,58,75,61,104), (1,63,79,38)(2,64,80,39)(3,65,81,40)(4,66,82,41)(5,61,83,42)(6,62,84,37)(7,100,55,18)(8,101,56,13)(9,102,57,14)(10,97,58,15)(11,98,59,16)(12,99,60,17)(19,35,95,68)(20,36,96,69)(21,31,91,70)(22,32,92,71)(23,33,93,72)(24,34,94,67)(25,75,113,44)(26,76,114,45)(27,77,109,46)(28,78,110,47)(29,73,111,48)(30,74,112,43)(49,90,120,105)(50,85,115,106)(51,86,116,107)(52,87,117,108)(53,88,118,103)(54,89,119,104), (7,21)(8,22)(9,23)(10,24)(11,19)(12,20)(13,32)(14,33)(15,34)(16,35)(17,36)(18,31)(25,119)(26,120)(27,115)(28,116)(29,117)(30,118)(37,62)(38,63)(39,64)(40,65)(41,66)(42,61)(43,103)(44,104)(45,105)(46,106)(47,107)(48,108)(49,114)(50,109)(51,110)(52,111)(53,112)(54,113)(55,91)(56,92)(57,93)(58,94)(59,95)(60,96)(67,97)(68,98)(69,99)(70,100)(71,101)(72,102)(73,87)(74,88)(75,89)(76,90)(77,85)(78,86) );
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)], [(1,27,99,69,115),(2,28,100,70,116),(3,29,101,71,117),(4,30,102,72,118),(5,25,97,67,119),(6,26,98,68,120),(7,47,39,86,91),(8,48,40,87,92),(9,43,41,88,93),(10,44,42,89,94),(11,45,37,90,95),(12,46,38,85,96),(13,32,52,81,111),(14,33,53,82,112),(15,34,54,83,113),(16,35,49,84,114),(17,36,50,79,109),(18,31,51,80,110),(19,59,76,62,105),(20,60,77,63,106),(21,55,78,64,107),(22,56,73,65,108),(23,57,74,66,103),(24,58,75,61,104)], [(1,63,79,38),(2,64,80,39),(3,65,81,40),(4,66,82,41),(5,61,83,42),(6,62,84,37),(7,100,55,18),(8,101,56,13),(9,102,57,14),(10,97,58,15),(11,98,59,16),(12,99,60,17),(19,35,95,68),(20,36,96,69),(21,31,91,70),(22,32,92,71),(23,33,93,72),(24,34,94,67),(25,75,113,44),(26,76,114,45),(27,77,109,46),(28,78,110,47),(29,73,111,48),(30,74,112,43),(49,90,120,105),(50,85,115,106),(51,86,116,107),(52,87,117,108),(53,88,118,103),(54,89,119,104)], [(7,21),(8,22),(9,23),(10,24),(11,19),(12,20),(13,32),(14,33),(15,34),(16,35),(17,36),(18,31),(25,119),(26,120),(27,115),(28,116),(29,117),(30,118),(37,62),(38,63),(39,64),(40,65),(41,66),(42,61),(43,103),(44,104),(45,105),(46,106),(47,107),(48,108),(49,114),(50,109),(51,110),(52,111),(53,112),(54,113),(55,91),(56,92),(57,93),(58,94),(59,95),(60,96),(67,97),(68,98),(69,99),(70,100),(71,101),(72,102),(73,87),(74,88),(75,89),(76,90),(77,85),(78,86)])
C6×C5⋊D4 is a maximal subgroup of
(C2×C6).D20 C3⋊(C23⋊F5) C5⋊(C12.D4) D30⋊6D4 C6.(D4×D5) (C2×C30).D4 C6.(C2×D20) C23.17(S3×D5) (C6×D5)⋊D4 Dic15⋊3D4 Dic15⋊16D4 (S3×C10)⋊D4 Dic15⋊5D4 C15⋊C22≀C2 (C2×C6)⋊D20 Dic15⋊18D4 D30⋊18D4 D30⋊8D4 C15⋊2+ 1+4 C6×D4×D5
78 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 4A | 4B | 5A | 5B | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 6K | 6L | 6M | 6N | 10A | ··· | 10N | 12A | 12B | 12C | 12D | 15A | 15B | 15C | 15D | 30A | ··· | 30AB |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 5 | 5 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 30 | ··· | 30 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 10 | 10 | 1 | 1 | 10 | 10 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 10 | 10 | 10 | 10 | 2 | 2 | 2 | 2 | 2 | ··· | 2 |
78 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 | C3 | C6 | C6 | C6 | C6 | D4 | D5 | D10 | C3×D4 | C3×D5 | C5⋊D4 | C6×D5 | C3×C5⋊D4 |
| kernel | C6×C5⋊D4 | C6×Dic5 | C3×C5⋊D4 | D5×C2×C6 | C22×C30 | C2×C5⋊D4 | C2×Dic5 | C5⋊D4 | C22×D5 | C22×C10 | C30 | C22×C6 | C2×C6 | C10 | C23 | C6 | C22 | C2 |
| # reps | 1 | 1 | 4 | 1 | 1 | 2 | 2 | 8 | 2 | 2 | 2 | 2 | 6 | 4 | 4 | 8 | 12 | 16 |
Matrix representation of C6×C5⋊D4 ►in GL3(𝔽61) generated by
| 48 | 0 | 0 |
| 0 | 60 | 0 |
| 0 | 0 | 60 |
| 1 | 0 | 0 |
| 0 | 0 | 60 |
| 0 | 1 | 43 |
| 60 | 0 | 0 |
| 0 | 31 | 8 |
| 0 | 17 | 30 |
| 60 | 0 | 0 |
| 0 | 1 | 43 |
| 0 | 0 | 60 |
G:=sub<GL(3,GF(61))| [48,0,0,0,60,0,0,0,60],[1,0,0,0,0,1,0,60,43],[60,0,0,0,31,17,0,8,30],[60,0,0,0,1,0,0,43,60] >;
C6×C5⋊D4 in GAP, Magma, Sage, TeX
C_6\times C_5\rtimes D_4
% in TeX
G:=Group("C6xC5:D4"); // GroupNames label
G:=SmallGroup(240,164);
// by ID
G=gap.SmallGroup(240,164);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-2,-5,506,6917]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^5=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