metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q8⋊3Dic15, D4⋊2Dic15, C60.213D4, C15⋊21C4≀C2, (D4×C15)⋊8C4, (Q8×C15)⋊8C4, (C2×C30).3D4, C60.83(C2×C4), (C3×Q8)⋊2Dic5, (C5×Q8)⋊8Dic3, (C3×D4)⋊2Dic5, C4○D4.3D15, (C5×D4)⋊5Dic3, (C2×C20).76D6, (C2×C4).40D30, (C4×Dic15)⋊2C2, (C2×C12).77D10, C60.7C4⋊16C2, C12.9(C2×Dic5), C4.3(C2×Dic15), C5⋊5(Q8⋊3Dic3), C3⋊3(D4⋊2Dic5), C4.31(C15⋊7D4), (C2×C60).62C22, C20.30(C2×Dic3), C12.110(C5⋊D4), C20.110(C3⋊D4), C22.3(C15⋊7D4), C6.19(C23.D5), C30.107(C22⋊C4), C2.8(C30.38D4), C10.30(C6.D4), (C5×C4○D4).5S3, (C3×C4○D4).1D5, (C15×C4○D4).1C2, (C2×C6).8(C5⋊D4), (C2×C10).7(C3⋊D4), SmallGroup(480,197)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8⋊3Dic15
G = < a,b,c,d | a4=c30=1, b2=a2, d2=c15, bab-1=a-1, ac=ca, ad=da, cbc-1=a2b, dbd-1=a-1b, dcd-1=c-1 >
Subgroups: 356 in 88 conjugacy classes, 39 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, C10, C10, Dic3, C12, C12, C2×C6, C2×C6, C15, C42, M4(2), C4○D4, Dic5, C20, C20, C2×C10, C2×C10, C3⋊C8, C2×Dic3, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C30, C30, C4≀C2, C5⋊2C8, C2×Dic5, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C4.Dic3, C4×Dic3, C3×C4○D4, Dic15, C60, C60, C2×C30, C2×C30, C4.Dic5, C4×Dic5, C5×C4○D4, Q8⋊3Dic3, C15⋊3C8, C2×Dic15, C2×C60, C2×C60, D4×C15, D4×C15, Q8×C15, D4⋊2Dic5, C60.7C4, C4×Dic15, C15×C4○D4, Q8⋊3Dic15
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D5, Dic3, D6, C22⋊C4, Dic5, D10, C2×Dic3, C3⋊D4, D15, C4≀C2, C2×Dic5, C5⋊D4, C6.D4, Dic15, D30, C23.D5, Q8⋊3Dic3, C2×Dic15, C15⋊7D4, D4⋊2Dic5, C30.38D4, Q8⋊3Dic15
►On 120 points
Generators in S120
(1 47 16 31)(2 48 17 32)(3 49 18 33)(4 50 19 34)(5 51 20 35)(6 52 21 36)(7 53 22 37)(8 54 23 38)(9 55 24 39)(10 56 25 40)(11 57 26 41)(12 58 27 42)(13 59 28 43)(14 60 29 44)(15 46 30 45)(61 99 76 114)(62 100 77 115)(63 101 78 116)(64 102 79 117)(65 103 80 118)(66 104 81 119)(67 105 82 120)(68 106 83 91)(69 107 84 92)(70 108 85 93)(71 109 86 94)(72 110 87 95)(73 111 88 96)(74 112 89 97)(75 113 90 98)
(1 114 16 99)(2 100 17 115)(3 116 18 101)(4 102 19 117)(5 118 20 103)(6 104 21 119)(7 120 22 105)(8 106 23 91)(9 92 24 107)(10 108 25 93)(11 94 26 109)(12 110 27 95)(13 96 28 111)(14 112 29 97)(15 98 30 113)(31 61 47 76)(32 77 48 62)(33 63 49 78)(34 79 50 64)(35 65 51 80)(36 81 52 66)(37 67 53 82)(38 83 54 68)(39 69 55 84)(40 85 56 70)(41 71 57 86)(42 87 58 72)(43 73 59 88)(44 89 60 74)(45 75 46 90)
(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 29)(2 28)(3 27)(4 26)(5 25)(6 24)(7 23)(8 22)(9 21)(10 20)(11 19)(12 18)(13 17)(14 16)(15 30)(31 60)(32 59)(33 58)(34 57)(35 56)(36 55)(37 54)(38 53)(39 52)(40 51)(41 50)(42 49)(43 48)(44 47)(45 46)(61 97 76 112)(62 96 77 111)(63 95 78 110)(64 94 79 109)(65 93 80 108)(66 92 81 107)(67 91 82 106)(68 120 83 105)(69 119 84 104)(70 118 85 103)(71 117 86 102)(72 116 87 101)(73 115 88 100)(74 114 89 99)(75 113 90 98)
G:=sub<Sym(120)| (1,47,16,31)(2,48,17,32)(3,49,18,33)(4,50,19,34)(5,51,20,35)(6,52,21,36)(7,53,22,37)(8,54,23,38)(9,55,24,39)(10,56,25,40)(11,57,26,41)(12,58,27,42)(13,59,28,43)(14,60,29,44)(15,46,30,45)(61,99,76,114)(62,100,77,115)(63,101,78,116)(64,102,79,117)(65,103,80,118)(66,104,81,119)(67,105,82,120)(68,106,83,91)(69,107,84,92)(70,108,85,93)(71,109,86,94)(72,110,87,95)(73,111,88,96)(74,112,89,97)(75,113,90,98), (1,114,16,99)(2,100,17,115)(3,116,18,101)(4,102,19,117)(5,118,20,103)(6,104,21,119)(7,120,22,105)(8,106,23,91)(9,92,24,107)(10,108,25,93)(11,94,26,109)(12,110,27,95)(13,96,28,111)(14,112,29,97)(15,98,30,113)(31,61,47,76)(32,77,48,62)(33,63,49,78)(34,79,50,64)(35,65,51,80)(36,81,52,66)(37,67,53,82)(38,83,54,68)(39,69,55,84)(40,85,56,70)(41,71,57,86)(42,87,58,72)(43,73,59,88)(44,89,60,74)(45,75,46,90), (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), (1,29)(2,28)(3,27)(4,26)(5,25)(6,24)(7,23)(8,22)(9,21)(10,20)(11,19)(12,18)(13,17)(14,16)(15,30)(31,60)(32,59)(33,58)(34,57)(35,56)(36,55)(37,54)(38,53)(39,52)(40,51)(41,50)(42,49)(43,48)(44,47)(45,46)(61,97,76,112)(62,96,77,111)(63,95,78,110)(64,94,79,109)(65,93,80,108)(66,92,81,107)(67,91,82,106)(68,120,83,105)(69,119,84,104)(70,118,85,103)(71,117,86,102)(72,116,87,101)(73,115,88,100)(74,114,89,99)(75,113,90,98)>;
G:=Group( (1,47,16,31)(2,48,17,32)(3,49,18,33)(4,50,19,34)(5,51,20,35)(6,52,21,36)(7,53,22,37)(8,54,23,38)(9,55,24,39)(10,56,25,40)(11,57,26,41)(12,58,27,42)(13,59,28,43)(14,60,29,44)(15,46,30,45)(61,99,76,114)(62,100,77,115)(63,101,78,116)(64,102,79,117)(65,103,80,118)(66,104,81,119)(67,105,82,120)(68,106,83,91)(69,107,84,92)(70,108,85,93)(71,109,86,94)(72,110,87,95)(73,111,88,96)(74,112,89,97)(75,113,90,98), (1,114,16,99)(2,100,17,115)(3,116,18,101)(4,102,19,117)(5,118,20,103)(6,104,21,119)(7,120,22,105)(8,106,23,91)(9,92,24,107)(10,108,25,93)(11,94,26,109)(12,110,27,95)(13,96,28,111)(14,112,29,97)(15,98,30,113)(31,61,47,76)(32,77,48,62)(33,63,49,78)(34,79,50,64)(35,65,51,80)(36,81,52,66)(37,67,53,82)(38,83,54,68)(39,69,55,84)(40,85,56,70)(41,71,57,86)(42,87,58,72)(43,73,59,88)(44,89,60,74)(45,75,46,90), (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), (1,29)(2,28)(3,27)(4,26)(5,25)(6,24)(7,23)(8,22)(9,21)(10,20)(11,19)(12,18)(13,17)(14,16)(15,30)(31,60)(32,59)(33,58)(34,57)(35,56)(36,55)(37,54)(38,53)(39,52)(40,51)(41,50)(42,49)(43,48)(44,47)(45,46)(61,97,76,112)(62,96,77,111)(63,95,78,110)(64,94,79,109)(65,93,80,108)(66,92,81,107)(67,91,82,106)(68,120,83,105)(69,119,84,104)(70,118,85,103)(71,117,86,102)(72,116,87,101)(73,115,88,100)(74,114,89,99)(75,113,90,98) );
G=PermutationGroup([[(1,47,16,31),(2,48,17,32),(3,49,18,33),(4,50,19,34),(5,51,20,35),(6,52,21,36),(7,53,22,37),(8,54,23,38),(9,55,24,39),(10,56,25,40),(11,57,26,41),(12,58,27,42),(13,59,28,43),(14,60,29,44),(15,46,30,45),(61,99,76,114),(62,100,77,115),(63,101,78,116),(64,102,79,117),(65,103,80,118),(66,104,81,119),(67,105,82,120),(68,106,83,91),(69,107,84,92),(70,108,85,93),(71,109,86,94),(72,110,87,95),(73,111,88,96),(74,112,89,97),(75,113,90,98)], [(1,114,16,99),(2,100,17,115),(3,116,18,101),(4,102,19,117),(5,118,20,103),(6,104,21,119),(7,120,22,105),(8,106,23,91),(9,92,24,107),(10,108,25,93),(11,94,26,109),(12,110,27,95),(13,96,28,111),(14,112,29,97),(15,98,30,113),(31,61,47,76),(32,77,48,62),(33,63,49,78),(34,79,50,64),(35,65,51,80),(36,81,52,66),(37,67,53,82),(38,83,54,68),(39,69,55,84),(40,85,56,70),(41,71,57,86),(42,87,58,72),(43,73,59,88),(44,89,60,74),(45,75,46,90)], [(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)], [(1,29),(2,28),(3,27),(4,26),(5,25),(6,24),(7,23),(8,22),(9,21),(10,20),(11,19),(12,18),(13,17),(14,16),(15,30),(31,60),(32,59),(33,58),(34,57),(35,56),(36,55),(37,54),(38,53),(39,52),(40,51),(41,50),(42,49),(43,48),(44,47),(45,46),(61,97,76,112),(62,96,77,111),(63,95,78,110),(64,94,79,109),(65,93,80,108),(66,92,81,107),(67,91,82,106),(68,120,83,105),(69,119,84,104),(70,118,85,103),(71,117,86,102),(72,116,87,101),(73,115,88,100),(74,114,89,99),(75,113,90,98)]])
84 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 6A | 6B | 6C | 6D | 8A | 8B | 10A | 10B | 10C | ··· | 10H | 12A | 12B | 12C | 12D | 12E | 15A | 15B | 15C | 15D | 20A | 20B | 20C | 20D | 20E | ··· | 20J | 30A | 30B | 30C | 30D | 30E | ··· | 30P | 60A | ··· | 60H | 60I | ··· | 60T |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 8 | 8 | 10 | 10 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 30 | 30 | 30 | 30 | 30 | ··· | 30 | 60 | ··· | 60 | 60 | ··· | 60 |
| size | 1 | 1 | 2 | 4 | 2 | 1 | 1 | 2 | 4 | 30 | 30 | 30 | 30 | 2 | 2 | 2 | 4 | 4 | 4 | 60 | 60 | 2 | 2 | 4 | ··· | 4 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | - | + | - | - | + | + | - | - | ||||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | S3 | D4 | D4 | D5 | D6 | Dic3 | Dic3 | D10 | Dic5 | Dic5 | C3⋊D4 | C3⋊D4 | D15 | C4≀C2 | C5⋊D4 | C5⋊D4 | D30 | Dic15 | Dic15 | C15⋊7D4 | C15⋊7D4 | Q8⋊3Dic3 | D4⋊2Dic5 | Q8⋊3Dic15 |
| kernel | Q8⋊3Dic15 | C60.7C4 | C4×Dic15 | C15×C4○D4 | D4×C15 | Q8×C15 | C5×C4○D4 | C60 | C2×C30 | C3×C4○D4 | C2×C20 | C5×D4 | C5×Q8 | C2×C12 | C3×D4 | C3×Q8 | C20 | C2×C10 | C4○D4 | C15 | C12 | C2×C6 | C2×C4 | D4 | Q8 | C4 | C22 | C5 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 2 | 4 | 8 |
Matrix representation of Q8⋊3Dic15 ►in GL4(𝔽241) generated by
| 240 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 |
| 0 | 0 | 64 | 0 |
| 0 | 0 | 0 | 177 |
| 165 | 49 | 0 | 0 |
| 192 | 76 | 0 | 0 |
| 0 | 0 | 0 | 177 |
| 0 | 0 | 177 | 0 |
| 110 | 147 | 0 | 0 |
| 94 | 84 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 240 |
| 51 | 190 | 0 | 0 |
| 240 | 190 | 0 | 0 |
| 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 177 |
G:=sub<GL(4,GF(241))| [240,0,0,0,0,240,0,0,0,0,64,0,0,0,0,177],[165,192,0,0,49,76,0,0,0,0,0,177,0,0,177,0],[110,94,0,0,147,84,0,0,0,0,1,0,0,0,0,240],[51,240,0,0,190,190,0,0,0,0,240,0,0,0,0,177] >;
Q8⋊3Dic15 in GAP, Magma, Sage, TeX
Q_8\rtimes_3{\rm Dic}_{15} % in TeX
G:=Group("Q8:3Dic15"); // GroupNames label
G:=SmallGroup(480,197);
// by ID
G=gap.SmallGroup(480,197);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,100,675,346,80,2693,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^30=1,b^2=a^2,d^2=c^15,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^2*b,d*b*d^-1=a^-1*b,d*c*d^-1=c^-1>;
// generators/relations