direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5×D4⋊2S3, C20.40D6, Dic6⋊3C10, C60.47C22, C30.53C23, D4⋊2(C5×S3), (C5×D4)⋊5S3, (C4×S3)⋊2C10, (S3×C20)⋊7C2, (D4×C15)⋊9C2, (C3×D4)⋊3C10, C3⋊D4⋊2C10, C4.5(S3×C10), (C2×C10).4D6, C15⋊16(C4○D4), C12.5(C2×C10), (C5×Dic6)⋊9C2, D6.2(C2×C10), (C10×Dic3)⋊9C2, (C2×Dic3)⋊3C10, C6.6(C22×C10), C22.1(S3×C10), (C2×C30).20C22, C10.43(C22×S3), Dic3.3(C2×C10), (S3×C10).13C22, (C5×Dic3).16C22, C3⋊2(C5×C4○D4), (C2×C6).(C2×C10), C2.7(S3×C2×C10), (C5×C3⋊D4)⋊6C2, (C5×D4)○(C5×Dic3), SmallGroup(240,170)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×D4⋊2S3
G = < a,b,c,d,e | a5=b4=c2=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece=b2c, ede=d-1 >
Subgroups: 144 in 80 conjugacy classes, 46 normal (26 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C2×C4, D4, D4, Q8, C10, C10, Dic3, Dic3, C12, D6, C2×C6, C15, C4○D4, C20, C20, C2×C10, C2×C10, Dic6, C4×S3, C2×Dic3, C3⋊D4, C3×D4, C5×S3, C30, C30, C2×C20, C5×D4, C5×D4, C5×Q8, D4⋊2S3, C5×Dic3, C5×Dic3, C60, S3×C10, C2×C30, C5×C4○D4, C5×Dic6, S3×C20, C10×Dic3, C5×C3⋊D4, D4×C15, C5×D4⋊2S3
Quotients: C1, C2, C22, C5, S3, C23, C10, D6, C4○D4, C2×C10, C22×S3, C5×S3, C22×C10, D4⋊2S3, S3×C10, C5×C4○D4, S3×C2×C10, C5×D4⋊2S3
►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 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 6)(2 7)(3 8)(4 9)(5 10)(11 34)(12 35)(13 31)(14 32)(15 33)(16 44)(17 45)(18 41)(19 42)(20 43)(21 37)(22 38)(23 39)(24 40)(25 36)(26 116)(27 117)(28 118)(29 119)(30 120)(46 113)(47 114)(48 115)(49 111)(50 112)(51 81)(52 82)(53 83)(54 84)(55 85)(56 86)(57 87)(58 88)(59 89)(60 90)(61 91)(62 92)(63 93)(64 94)(65 95)(66 96)(67 97)(68 98)(69 99)(70 100)(71 101)(72 102)(73 103)(74 104)(75 105)(76 106)(77 107)(78 108)(79 109)(80 110)
(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), (1,86,26,107)(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), (1,6)(2,7)(3,8)(4,9)(5,10)(11,34)(12,35)(13,31)(14,32)(15,33)(16,44)(17,45)(18,41)(19,42)(20,43)(21,37)(22,38)(23,39)(24,40)(25,36)(26,116)(27,117)(28,118)(29,119)(30,120)(46,113)(47,114)(48,115)(49,111)(50,112)(51,81)(52,82)(53,83)(54,84)(55,85)(56,86)(57,87)(58,88)(59,89)(60,90)(61,91)(62,92)(63,93)(64,94)(65,95)(66,96)(67,97)(68,98)(69,99)(70,100)(71,101)(72,102)(73,103)(74,104)(75,105)(76,106)(77,107)(78,108)(79,109)(80,110), (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), (1,86,26,107)(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), (1,6)(2,7)(3,8)(4,9)(5,10)(11,34)(12,35)(13,31)(14,32)(15,33)(16,44)(17,45)(18,41)(19,42)(20,43)(21,37)(22,38)(23,39)(24,40)(25,36)(26,116)(27,117)(28,118)(29,119)(30,120)(46,113)(47,114)(48,115)(49,111)(50,112)(51,81)(52,82)(53,83)(54,84)(55,85)(56,86)(57,87)(58,88)(59,89)(60,90)(61,91)(62,92)(63,93)(64,94)(65,95)(66,96)(67,97)(68,98)(69,99)(70,100)(71,101)(72,102)(73,103)(74,104)(75,105)(76,106)(77,107)(78,108)(79,109)(80,110), (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)], [(1,86,26,107),(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)], [(1,6),(2,7),(3,8),(4,9),(5,10),(11,34),(12,35),(13,31),(14,32),(15,33),(16,44),(17,45),(18,41),(19,42),(20,43),(21,37),(22,38),(23,39),(24,40),(25,36),(26,116),(27,117),(28,118),(29,119),(30,120),(46,113),(47,114),(48,115),(49,111),(50,112),(51,81),(52,82),(53,83),(54,84),(55,85),(56,86),(57,87),(58,88),(59,89),(60,90),(61,91),(62,92),(63,93),(64,94),(65,95),(66,96),(67,97),(68,98),(69,99),(70,100),(71,101),(72,102),(73,103),(74,104),(75,105),(76,106),(77,107),(78,108),(79,109),(80,110)], [(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×D4⋊2S3 is a maximal subgroup of
D60.C22 C60.10C23 D20.24D6 C60.19C23 C15⋊2- 1+4 D30.C23 D20⋊14D6 C5×S3×C4○D4
C5×D4⋊2S3 is a maximal quotient of
C5×D4×Dic3
75 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 5C | 5D | 6A | 6B | 6C | 10A | 10B | 10C | 10D | 10E | ··· | 10L | 10M | 10N | 10O | 10P | 12 | 15A | 15B | 15C | 15D | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 20M | ··· | 20T | 30A | 30B | 30C | 30D | 30E | ··· | 30L | 60A | 60B | 60C | 60D |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 12 | 15 | 15 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 20 | ··· | 20 | 30 | 30 | 30 | 30 | 30 | ··· | 30 | 60 | 60 | 60 | 60 |
| size | 1 | 1 | 2 | 2 | 6 | 2 | 2 | 3 | 3 | 6 | 6 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 6 | 6 | 6 | 6 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | ··· | 3 | 6 | ··· | 6 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 |
75 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | ||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | C10 | S3 | D6 | D6 | C4○D4 | C5×S3 | S3×C10 | S3×C10 | C5×C4○D4 | D4⋊2S3 | C5×D4⋊2S3 |
| kernel | C5×D4⋊2S3 | C5×Dic6 | S3×C20 | C10×Dic3 | C5×C3⋊D4 | D4×C15 | D4⋊2S3 | Dic6 | C4×S3 | C2×Dic3 | C3⋊D4 | C3×D4 | C5×D4 | C20 | C2×C10 | C15 | D4 | C4 | C22 | C3 | C5 | C1 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 4 | 4 | 4 | 8 | 8 | 4 | 1 | 1 | 2 | 2 | 4 | 4 | 8 | 8 | 1 | 4 |
Matrix representation of C5×D4⋊2S3 ►in GL4(𝔽61) generated by
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 50 | 0 |
| 0 | 0 | 50 | 11 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 11 | 39 |
| 0 | 0 | 11 | 50 |
| 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 | 1 | 60 |
G:=sub<GL(4,GF(61))| [9,0,0,0,0,9,0,0,0,0,1,0,0,0,0,1],[60,0,0,0,0,60,0,0,0,0,50,50,0,0,0,11],[1,0,0,0,0,1,0,0,0,0,11,11,0,0,39,50],[60,60,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,1,0,0,0,60] >;
C5×D4⋊2S3 in GAP, Magma, Sage, TeX
C_5\times D_4\rtimes_2S_3
% in TeX
G:=Group("C5xD4:2S3"); // GroupNames label
G:=SmallGroup(240,170);
// by ID
G=gap.SmallGroup(240,170);
# by ID
G:=PCGroup([6,-2,-2,-2,-5,-2,-3,247,794,404,5765]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^4=c^2=d^3=e^2=1,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,e*c*e=b^2*c,e*d*e=d^-1>;
// generators/relations