non-abelian, soluble, monomial
Aliases: C24⋊2D15, C23.5D30, (C2×C10)⋊4S4, (C5×A4)⋊7D4, C22⋊2(C5⋊S4), C5⋊3(A4⋊D4), A4⋊3(C5⋊D4), C10.26(C2×S4), A4⋊Dic5⋊1C2, (C23×C10)⋊4S3, C22⋊(C15⋊7D4), (C22×A4)⋊2D5, (C2×A4).12D10, (C22×C10).17D6, (C10×A4).12C22, (C2×C5⋊S4)⋊2C2, (A4×C2×C10)⋊2C2, C2.11(C2×C5⋊S4), (C2×C10)⋊4(C3⋊D4), SmallGroup(480,1034)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1036 in 124 conjugacy classes, 21 normal (all characteristic)
C1, C2, C2 [×5], C3, C4 [×3], C22 [×2], C22 [×11], C5, S3, C6 [×2], C2×C4 [×3], D4 [×6], C23, C23 [×5], D5, C10, C10 [×4], Dic3, A4, D6, C2×C6, C15, C22⋊C4 [×3], C2×D4 [×3], C24, Dic5 [×3], D10 [×3], C2×C10 [×2], C2×C10 [×8], C3⋊D4, S4, C2×A4, C2×A4, D15, C30 [×2], C22≀C2, C2×Dic5 [×3], C5⋊D4 [×6], C22×D5, C22×C10, C22×C10 [×4], A4⋊C4, C2×S4, C22×A4, Dic15, C5×A4, D30, C2×C30, C23.D5 [×3], C2×C5⋊D4 [×3], C23×C10, A4⋊D4, C15⋊7D4, C5⋊S4, C10×A4, C10×A4, C24⋊2D5, A4⋊Dic5, C2×C5⋊S4, A4×C2×C10, C24⋊2D15
Quotients:
C1, C2 [×3], C22, S3, D4, D5, D6, D10, C3⋊D4, S4, D15, C5⋊D4, C2×S4, D30, A4⋊D4, C15⋊7D4, C5⋊S4, C2×C5⋊S4, C24⋊2D15
Generators and relations
G = < a,b,c,d,e,f | a2=b2=c2=d2=e15=f2=1, faf=ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, bf=fb, ece-1=fdf=cd=dc, cf=fc, ede-1=c, fef=e-1 >
►On 60 points
(16 38)(17 39)(18 40)(19 41)(20 42)(21 43)(22 44)(23 45)(24 31)(25 32)(26 33)(27 34)(28 35)(29 36)(30 37)
(1 52)(2 53)(3 54)(4 55)(5 56)(6 57)(7 58)(8 59)(9 60)(10 46)(11 47)(12 48)(13 49)(14 50)(15 51)(16 38)(17 39)(18 40)(19 41)(20 42)(21 43)(22 44)(23 45)(24 31)(25 32)(26 33)(27 34)(28 35)(29 36)(30 37)
(2 53)(3 54)(5 56)(6 57)(8 59)(9 60)(11 47)(12 48)(14 50)(15 51)(16 38)(17 39)(19 41)(20 42)(22 44)(23 45)(25 32)(26 33)(28 35)(29 36)
(1 52)(3 54)(4 55)(6 57)(7 58)(9 60)(10 46)(12 48)(13 49)(15 51)(17 39)(18 40)(20 42)(21 43)(23 45)(24 31)(26 33)(27 34)(29 36)(30 37)
(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)
(1 27)(2 26)(3 25)(4 24)(5 23)(6 22)(7 21)(8 20)(9 19)(10 18)(11 17)(12 16)(13 30)(14 29)(15 28)(31 55)(32 54)(33 53)(34 52)(35 51)(36 50)(37 49)(38 48)(39 47)(40 46)(41 60)(42 59)(43 58)(44 57)(45 56)
G:=sub<Sym(60)| (16,38)(17,39)(18,40)(19,41)(20,42)(21,43)(22,44)(23,45)(24,31)(25,32)(26,33)(27,34)(28,35)(29,36)(30,37), (1,52)(2,53)(3,54)(4,55)(5,56)(6,57)(7,58)(8,59)(9,60)(10,46)(11,47)(12,48)(13,49)(14,50)(15,51)(16,38)(17,39)(18,40)(19,41)(20,42)(21,43)(22,44)(23,45)(24,31)(25,32)(26,33)(27,34)(28,35)(29,36)(30,37), (2,53)(3,54)(5,56)(6,57)(8,59)(9,60)(11,47)(12,48)(14,50)(15,51)(16,38)(17,39)(19,41)(20,42)(22,44)(23,45)(25,32)(26,33)(28,35)(29,36), (1,52)(3,54)(4,55)(6,57)(7,58)(9,60)(10,46)(12,48)(13,49)(15,51)(17,39)(18,40)(20,42)(21,43)(23,45)(24,31)(26,33)(27,34)(29,36)(30,37), (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), (1,27)(2,26)(3,25)(4,24)(5,23)(6,22)(7,21)(8,20)(9,19)(10,18)(11,17)(12,16)(13,30)(14,29)(15,28)(31,55)(32,54)(33,53)(34,52)(35,51)(36,50)(37,49)(38,48)(39,47)(40,46)(41,60)(42,59)(43,58)(44,57)(45,56)>;
G:=Group( (16,38)(17,39)(18,40)(19,41)(20,42)(21,43)(22,44)(23,45)(24,31)(25,32)(26,33)(27,34)(28,35)(29,36)(30,37), (1,52)(2,53)(3,54)(4,55)(5,56)(6,57)(7,58)(8,59)(9,60)(10,46)(11,47)(12,48)(13,49)(14,50)(15,51)(16,38)(17,39)(18,40)(19,41)(20,42)(21,43)(22,44)(23,45)(24,31)(25,32)(26,33)(27,34)(28,35)(29,36)(30,37), (2,53)(3,54)(5,56)(6,57)(8,59)(9,60)(11,47)(12,48)(14,50)(15,51)(16,38)(17,39)(19,41)(20,42)(22,44)(23,45)(25,32)(26,33)(28,35)(29,36), (1,52)(3,54)(4,55)(6,57)(7,58)(9,60)(10,46)(12,48)(13,49)(15,51)(17,39)(18,40)(20,42)(21,43)(23,45)(24,31)(26,33)(27,34)(29,36)(30,37), (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), (1,27)(2,26)(3,25)(4,24)(5,23)(6,22)(7,21)(8,20)(9,19)(10,18)(11,17)(12,16)(13,30)(14,29)(15,28)(31,55)(32,54)(33,53)(34,52)(35,51)(36,50)(37,49)(38,48)(39,47)(40,46)(41,60)(42,59)(43,58)(44,57)(45,56) );
G=PermutationGroup([(16,38),(17,39),(18,40),(19,41),(20,42),(21,43),(22,44),(23,45),(24,31),(25,32),(26,33),(27,34),(28,35),(29,36),(30,37)], [(1,52),(2,53),(3,54),(4,55),(5,56),(6,57),(7,58),(8,59),(9,60),(10,46),(11,47),(12,48),(13,49),(14,50),(15,51),(16,38),(17,39),(18,40),(19,41),(20,42),(21,43),(22,44),(23,45),(24,31),(25,32),(26,33),(27,34),(28,35),(29,36),(30,37)], [(2,53),(3,54),(5,56),(6,57),(8,59),(9,60),(11,47),(12,48),(14,50),(15,51),(16,38),(17,39),(19,41),(20,42),(22,44),(23,45),(25,32),(26,33),(28,35),(29,36)], [(1,52),(3,54),(4,55),(6,57),(7,58),(9,60),(10,46),(12,48),(13,49),(15,51),(17,39),(18,40),(20,42),(21,43),(23,45),(24,31),(26,33),(27,34),(29,36),(30,37)], [(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)], [(1,27),(2,26),(3,25),(4,24),(5,23),(6,22),(7,21),(8,20),(9,19),(10,18),(11,17),(12,16),(13,30),(14,29),(15,28),(31,55),(32,54),(33,53),(34,52),(35,51),(36,50),(37,49),(38,48),(39,47),(40,46),(41,60),(42,59),(43,58),(44,57),(45,56)])
Matrix representation ►G ⊆ GL5(𝔽61)
| 0 | 1 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 60 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 60 | 0 | 1 |
| 0 | 0 | 60 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 1 |
| 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 1 | 60 | 0 |
| 39 | 19 | 0 | 0 | 0 |
| 19 | 39 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 22 | 19 | 0 | 0 | 0 |
| 42 | 39 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(5,GF(61))| [0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[60,0,0,0,0,0,60,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,60,60,60,0,0,0,0,1,0,0,0,1,0],[1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,60,60,60,0,0,1,0,0],[39,19,0,0,0,19,39,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0],[22,42,0,0,0,19,39,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0] >;
46 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 5A | 5B | 6A | 6B | 6C | 10A | ··· | 10F | 10G | ··· | 10N | 15A | 15B | 15C | 15D | 30A | ··· | 30L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | ··· | 10 | 15 | 15 | 15 | 15 | 30 | ··· | 30 |
| size | 1 | 1 | 2 | 3 | 3 | 6 | 60 | 8 | 60 | 60 | 60 | 2 | 2 | 8 | 8 | 8 | 2 | ··· | 2 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 8 | ··· | 8 |
46 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 6 | 6 | 6 | 6 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | S3 | D4 | D5 | D6 | D10 | C3⋊D4 | D15 | C5⋊D4 | D30 | C15⋊7D4 | S4 | C2×S4 | A4⋊D4 | C5⋊S4 | C2×C5⋊S4 | C24⋊2D15 |
| kernel | C24⋊2D15 | A4⋊Dic5 | C2×C5⋊S4 | A4×C2×C10 | C23×C10 | C5×A4 | C22×A4 | C22×C10 | C2×A4 | C2×C10 | C24 | A4 | C23 | C22 | C2×C10 | C10 | C5 | C22 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 4 | 4 | 4 | 8 | 2 | 2 | 1 | 2 | 2 | 4 |
In GAP, Magma, Sage, TeX
C_2^4\rtimes_2D_{15} % in TeX
G:=Group("C2^4:2D15"); // GroupNames label
G:=SmallGroup(480,1034);
// by ID
G=gap.SmallGroup(480,1034);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-5,-2,2,85,451,3364,10085,1286,5886,2232]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^15=f^2=1,f*a*f=a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,e*c*e^-1=f*d*f=c*d=d*c,c*f=f*c,e*d*e^-1=c,f*e*f=e^-1>;
// generators/relations