direct product, non-abelian, soluble
Aliases: C2×Q8⋊D15, Q8⋊D30, C10⋊GL2(𝔽3), SL2(𝔽3)⋊3D10, (C5×Q8)⋊2D6, (C2×C10).6S4, (C2×Q8)⋊1D15, (Q8×C10)⋊1S3, C10.20(C2×S4), C22.5(C5⋊S4), C5⋊2(C2×GL2(𝔽3)), (C2×SL2(𝔽3))⋊2D5, (C10×SL2(𝔽3))⋊2C2, (C5×SL2(𝔽3))⋊3C22, C2.6(C2×C5⋊S4), SmallGroup(480,1028)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2 — Q8 — C5×SL2(𝔽3) — C2×Q8⋊D15 |
| C5×SL2(𝔽3) — C2×Q8⋊D15 |
Subgroups: 994 in 102 conjugacy classes, 21 normal (15 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×2], C22, C22 [×4], C5, S3 [×4], C6 [×3], C8 [×2], C2×C4, D4 [×3], Q8, Q8, C23, D5 [×2], C10, C10 [×2], D6 [×6], C2×C6, C15, C2×C8, SD16 [×4], C2×D4, C2×Q8, C20 [×2], D10 [×4], C2×C10, SL2(𝔽3), C22×S3, D15 [×4], C30 [×3], C2×SD16, C5⋊2C8 [×2], D20 [×3], C2×C20, C5×Q8, C5×Q8, C22×D5, GL2(𝔽3) [×2], C2×SL2(𝔽3), D30 [×6], C2×C30, C2×C5⋊2C8, Q8⋊D5 [×4], C2×D20, Q8×C10, C2×GL2(𝔽3), C5×SL2(𝔽3), C22×D15, C2×Q8⋊D5, Q8⋊D15 [×2], C10×SL2(𝔽3), C2×Q8⋊D15
Quotients:
C1, C2 [×3], C22, S3, D5, D6, D10, S4, D15, GL2(𝔽3) [×2], C2×S4, D30, C2×GL2(𝔽3), C5⋊S4, Q8⋊D15 [×2], C2×C5⋊S4, C2×Q8⋊D15
Generators and relations
G = < a,b,c,d,e | a2=b4=d15=e2=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, dbd-1=c, ebe=b-1c, dcd-1=bc, ece=b2c, ede=d-1 >
►On 80 points
Generators in S80
(1 15)(2 11)(3 12)(4 13)(5 14)(6 18)(7 19)(8 20)(9 16)(10 17)(21 69)(22 70)(23 71)(24 72)(25 73)(26 74)(27 75)(28 76)(29 77)(30 78)(31 79)(32 80)(33 66)(34 67)(35 68)(36 64)(37 65)(38 51)(39 52)(40 53)(41 54)(42 55)(43 56)(44 57)(45 58)(46 59)(47 60)(48 61)(49 62)(50 63)
(1 33 17 55)(2 24 18 61)(3 30 19 52)(4 21 20 58)(5 27 16 64)(6 48 11 72)(7 39 12 78)(8 45 13 69)(9 36 14 75)(10 42 15 66)(22 32 59 54)(23 65 60 28)(25 35 62 57)(26 53 63 31)(29 56 51 34)(37 47 76 71)(38 67 77 43)(40 50 79 74)(41 70 80 46)(44 73 68 49)
(1 23 17 60)(2 29 18 51)(3 35 19 57)(4 26 20 63)(5 32 16 54)(6 38 11 77)(7 44 12 68)(8 50 13 74)(9 41 14 80)(10 47 15 71)(21 31 58 53)(22 64 59 27)(24 34 61 56)(25 52 62 30)(28 55 65 33)(36 46 75 70)(37 66 76 42)(39 49 78 73)(40 69 79 45)(43 72 67 48)
(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)
(1 11)(2 15)(3 14)(4 13)(5 12)(6 17)(7 16)(8 20)(9 19)(10 18)(21 40)(22 39)(23 38)(24 37)(25 36)(26 50)(27 49)(28 48)(29 47)(30 46)(31 45)(32 44)(33 43)(34 42)(35 41)(51 71)(52 70)(53 69)(54 68)(55 67)(56 66)(57 80)(58 79)(59 78)(60 77)(61 76)(62 75)(63 74)(64 73)(65 72)
G:=sub<Sym(80)| (1,15)(2,11)(3,12)(4,13)(5,14)(6,18)(7,19)(8,20)(9,16)(10,17)(21,69)(22,70)(23,71)(24,72)(25,73)(26,74)(27,75)(28,76)(29,77)(30,78)(31,79)(32,80)(33,66)(34,67)(35,68)(36,64)(37,65)(38,51)(39,52)(40,53)(41,54)(42,55)(43,56)(44,57)(45,58)(46,59)(47,60)(48,61)(49,62)(50,63), (1,33,17,55)(2,24,18,61)(3,30,19,52)(4,21,20,58)(5,27,16,64)(6,48,11,72)(7,39,12,78)(8,45,13,69)(9,36,14,75)(10,42,15,66)(22,32,59,54)(23,65,60,28)(25,35,62,57)(26,53,63,31)(29,56,51,34)(37,47,76,71)(38,67,77,43)(40,50,79,74)(41,70,80,46)(44,73,68,49), (1,23,17,60)(2,29,18,51)(3,35,19,57)(4,26,20,63)(5,32,16,54)(6,38,11,77)(7,44,12,68)(8,50,13,74)(9,41,14,80)(10,47,15,71)(21,31,58,53)(22,64,59,27)(24,34,61,56)(25,52,62,30)(28,55,65,33)(36,46,75,70)(37,66,76,42)(39,49,78,73)(40,69,79,45)(43,72,67,48), (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), (1,11)(2,15)(3,14)(4,13)(5,12)(6,17)(7,16)(8,20)(9,19)(10,18)(21,40)(22,39)(23,38)(24,37)(25,36)(26,50)(27,49)(28,48)(29,47)(30,46)(31,45)(32,44)(33,43)(34,42)(35,41)(51,71)(52,70)(53,69)(54,68)(55,67)(56,66)(57,80)(58,79)(59,78)(60,77)(61,76)(62,75)(63,74)(64,73)(65,72)>;
G:=Group( (1,15)(2,11)(3,12)(4,13)(5,14)(6,18)(7,19)(8,20)(9,16)(10,17)(21,69)(22,70)(23,71)(24,72)(25,73)(26,74)(27,75)(28,76)(29,77)(30,78)(31,79)(32,80)(33,66)(34,67)(35,68)(36,64)(37,65)(38,51)(39,52)(40,53)(41,54)(42,55)(43,56)(44,57)(45,58)(46,59)(47,60)(48,61)(49,62)(50,63), (1,33,17,55)(2,24,18,61)(3,30,19,52)(4,21,20,58)(5,27,16,64)(6,48,11,72)(7,39,12,78)(8,45,13,69)(9,36,14,75)(10,42,15,66)(22,32,59,54)(23,65,60,28)(25,35,62,57)(26,53,63,31)(29,56,51,34)(37,47,76,71)(38,67,77,43)(40,50,79,74)(41,70,80,46)(44,73,68,49), (1,23,17,60)(2,29,18,51)(3,35,19,57)(4,26,20,63)(5,32,16,54)(6,38,11,77)(7,44,12,68)(8,50,13,74)(9,41,14,80)(10,47,15,71)(21,31,58,53)(22,64,59,27)(24,34,61,56)(25,52,62,30)(28,55,65,33)(36,46,75,70)(37,66,76,42)(39,49,78,73)(40,69,79,45)(43,72,67,48), (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), (1,11)(2,15)(3,14)(4,13)(5,12)(6,17)(7,16)(8,20)(9,19)(10,18)(21,40)(22,39)(23,38)(24,37)(25,36)(26,50)(27,49)(28,48)(29,47)(30,46)(31,45)(32,44)(33,43)(34,42)(35,41)(51,71)(52,70)(53,69)(54,68)(55,67)(56,66)(57,80)(58,79)(59,78)(60,77)(61,76)(62,75)(63,74)(64,73)(65,72) );
G=PermutationGroup([(1,15),(2,11),(3,12),(4,13),(5,14),(6,18),(7,19),(8,20),(9,16),(10,17),(21,69),(22,70),(23,71),(24,72),(25,73),(26,74),(27,75),(28,76),(29,77),(30,78),(31,79),(32,80),(33,66),(34,67),(35,68),(36,64),(37,65),(38,51),(39,52),(40,53),(41,54),(42,55),(43,56),(44,57),(45,58),(46,59),(47,60),(48,61),(49,62),(50,63)], [(1,33,17,55),(2,24,18,61),(3,30,19,52),(4,21,20,58),(5,27,16,64),(6,48,11,72),(7,39,12,78),(8,45,13,69),(9,36,14,75),(10,42,15,66),(22,32,59,54),(23,65,60,28),(25,35,62,57),(26,53,63,31),(29,56,51,34),(37,47,76,71),(38,67,77,43),(40,50,79,74),(41,70,80,46),(44,73,68,49)], [(1,23,17,60),(2,29,18,51),(3,35,19,57),(4,26,20,63),(5,32,16,54),(6,38,11,77),(7,44,12,68),(8,50,13,74),(9,41,14,80),(10,47,15,71),(21,31,58,53),(22,64,59,27),(24,34,61,56),(25,52,62,30),(28,55,65,33),(36,46,75,70),(37,66,76,42),(39,49,78,73),(40,69,79,45),(43,72,67,48)], [(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)], [(1,11),(2,15),(3,14),(4,13),(5,12),(6,17),(7,16),(8,20),(9,19),(10,18),(21,40),(22,39),(23,38),(24,37),(25,36),(26,50),(27,49),(28,48),(29,47),(30,46),(31,45),(32,44),(33,43),(34,42),(35,41),(51,71),(52,70),(53,69),(54,68),(55,67),(56,66),(57,80),(58,79),(59,78),(60,77),(61,76),(62,75),(63,74),(64,73),(65,72)])
Matrix representation ►G ⊆ GL4(𝔽241) generated by
| 240 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 106 | 67 |
| 0 | 0 | 174 | 135 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 68 | 105 |
| 0 | 0 | 174 | 173 |
| 8 | 11 | 0 | 0 |
| 197 | 30 | 0 | 0 |
| 0 | 0 | 67 | 68 |
| 0 | 0 | 135 | 173 |
| 30 | 237 | 0 | 0 |
| 44 | 211 | 0 | 0 |
| 0 | 0 | 68 | 105 |
| 0 | 0 | 135 | 173 |
G:=sub<GL(4,GF(241))| [240,0,0,0,0,240,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,106,174,0,0,67,135],[1,0,0,0,0,1,0,0,0,0,68,174,0,0,105,173],[8,197,0,0,11,30,0,0,0,0,67,135,0,0,68,173],[30,44,0,0,237,211,0,0,0,0,68,135,0,0,105,173] >;
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 5A | 5B | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 15A | 15B | 15C | 15D | 20A | 20B | 20C | 20D | 30A | ··· | 30L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 15 | 15 | 15 | 15 | 20 | 20 | 20 | 20 | 30 | ··· | 30 |
| size | 1 | 1 | 1 | 1 | 60 | 60 | 8 | 6 | 6 | 2 | 2 | 8 | 8 | 8 | 30 | 30 | 30 | 30 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 8 | ··· | 8 |
44 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 6 | 6 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | S3 | D5 | D6 | D10 | D15 | GL2(𝔽3) | D30 | S4 | C2×S4 | GL2(𝔽3) | Q8⋊D15 | C5⋊S4 | C2×C5⋊S4 |
| kernel | C2×Q8⋊D15 | Q8⋊D15 | C10×SL2(𝔽3) | Q8×C10 | C2×SL2(𝔽3) | C5×Q8 | SL2(𝔽3) | C2×Q8 | C10 | Q8 | C2×C10 | C10 | C10 | C2 | C22 | C2 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 2 | 4 | 4 | 4 | 2 | 2 | 2 | 12 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_2\times Q_8\rtimes D_{15} % in TeX
G:=Group("C2xQ8:D15"); // GroupNames label
G:=SmallGroup(480,1028);
// by ID
G=gap.SmallGroup(480,1028);
# by ID
G:=PCGroup([7,-2,-2,-3,-5,-2,2,-2,170,1347,4204,3168,172,2525,1909,285,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=d^15=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=b^-1,d*b*d^-1=c,e*b*e=b^-1*c,d*c*d^-1=b*c,e*c*e=b^2*c,e*d*e=d^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀