metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (Q8×D5)⋊7C4, (C2×Q8)⋊4F5, (Q8×C10)⋊4C4, Q8.7(C2×F5), Q8⋊F5⋊3C2, (C4×D5).41D4, C4⋊F5.8C22, D10.98(C2×D4), D5⋊C8.7C22, Dic5.8(C2×D4), C5⋊(C23.38D4), C4.18(C22×F5), (C2×Dic10)⋊12C4, C20.18(C22×C4), Dic10.8(C2×C4), D5⋊M4(2).5C2, (C4×D5).40C23, C4.17(C22⋊F5), (Q8×D5).11C22, C20.17(C22⋊C4), (C2×Dic5).122D4, (C22×D5).148D4, D5.4(C8.C22), D10.46(C22⋊C4), C22.28(C22⋊F5), Dic5.13(C22⋊C4), D10.C23.5C2, (C2×Q8×D5).10C2, (C2×C4).39(C2×F5), (C5×Q8).7(C2×C4), (C2×C20).60(C2×C4), (C4×D5).24(C2×C4), C2.27(C2×C22⋊F5), C10.26(C2×C22⋊C4), (C2×C4×D5).205C22, (C2×C10).58(C22⋊C4), SmallGroup(320,1120)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (Q8×C10)⋊C4
G = < a,b,c,d | a10=b4=d4=1, c2=b2, ab=ba, ac=ca, dad-1=a3b2, cbc-1=b-1, bd=db, dcd-1=a5b-1c >
Subgroups: 602 in 150 conjugacy classes, 50 normal (30 characteristic)
C1, C2, C2 [×4], C4 [×2], C4 [×8], C22, C22 [×4], C5, C8 [×2], C2×C4, C2×C4 [×13], Q8 [×2], Q8 [×8], C23, D5 [×2], D5, C10, C10, C42, C22⋊C4, C4⋊C4 [×2], C2×C8 [×2], M4(2) [×2], C22×C4 [×2], C2×Q8, C2×Q8 [×8], Dic5 [×2], Dic5 [×2], C20 [×2], C20 [×2], F5 [×2], D10 [×2], D10 [×2], C2×C10, Q8⋊C4 [×4], C42⋊C2, C2×M4(2), C22×Q8, C5⋊C8 [×2], Dic10 [×2], Dic10 [×5], C4×D5 [×4], C4×D5 [×4], C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×Q8 [×2], C5×Q8, C2×F5 [×2], C22×D5, C23.38D4, D5⋊C8 [×2], C4.F5, C4×F5, C4⋊F5 [×2], C22.F5, C22⋊F5, C2×Dic10, C2×Dic10, C2×C4×D5, C2×C4×D5, Q8×D5 [×4], Q8×D5 [×2], Q8×C10, Q8⋊F5 [×4], D5⋊M4(2), D10.C23, C2×Q8×D5, (Q8×C10)⋊C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], F5, C2×C22⋊C4, C8.C22 [×2], C2×F5 [×3], C23.38D4, C22⋊F5 [×2], C22×F5, C2×C22⋊F5, (Q8×C10)⋊C4
►On 80 points
(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 35 15 62)(2 36 16 63)(3 37 17 64)(4 38 18 65)(5 39 19 66)(6 40 20 67)(7 31 11 68)(8 32 12 69)(9 33 13 70)(10 34 14 61)(21 55 71 47)(22 56 72 48)(23 57 73 49)(24 58 74 50)(25 59 75 41)(26 60 76 42)(27 51 77 43)(28 52 78 44)(29 53 79 45)(30 54 80 46)
(1 43 15 51)(2 44 16 52)(3 45 17 53)(4 46 18 54)(5 47 19 55)(6 48 20 56)(7 49 11 57)(8 50 12 58)(9 41 13 59)(10 42 14 60)(21 39 71 66)(22 40 72 67)(23 31 73 68)(24 32 74 69)(25 33 75 70)(26 34 76 61)(27 35 77 62)(28 36 78 63)(29 37 79 64)(30 38 80 65)
(2 12 10 18)(3 5 9 7)(4 16 8 14)(6 20)(11 17 19 13)(21 54 23 58)(22 43)(24 47 30 49)(25 52 29 60)(26 41 28 45)(27 56)(31 37 39 33)(32 61 38 63)(34 65 36 69)(40 67)(42 75 44 79)(46 73 50 71)(48 77)(51 72)(53 76 59 78)(55 80 57 74)(64 66 70 68)
G:=sub<Sym(80)| (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), (1,35,15,62)(2,36,16,63)(3,37,17,64)(4,38,18,65)(5,39,19,66)(6,40,20,67)(7,31,11,68)(8,32,12,69)(9,33,13,70)(10,34,14,61)(21,55,71,47)(22,56,72,48)(23,57,73,49)(24,58,74,50)(25,59,75,41)(26,60,76,42)(27,51,77,43)(28,52,78,44)(29,53,79,45)(30,54,80,46), (1,43,15,51)(2,44,16,52)(3,45,17,53)(4,46,18,54)(5,47,19,55)(6,48,20,56)(7,49,11,57)(8,50,12,58)(9,41,13,59)(10,42,14,60)(21,39,71,66)(22,40,72,67)(23,31,73,68)(24,32,74,69)(25,33,75,70)(26,34,76,61)(27,35,77,62)(28,36,78,63)(29,37,79,64)(30,38,80,65), (2,12,10,18)(3,5,9,7)(4,16,8,14)(6,20)(11,17,19,13)(21,54,23,58)(22,43)(24,47,30,49)(25,52,29,60)(26,41,28,45)(27,56)(31,37,39,33)(32,61,38,63)(34,65,36,69)(40,67)(42,75,44,79)(46,73,50,71)(48,77)(51,72)(53,76,59,78)(55,80,57,74)(64,66,70,68)>;
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), (1,35,15,62)(2,36,16,63)(3,37,17,64)(4,38,18,65)(5,39,19,66)(6,40,20,67)(7,31,11,68)(8,32,12,69)(9,33,13,70)(10,34,14,61)(21,55,71,47)(22,56,72,48)(23,57,73,49)(24,58,74,50)(25,59,75,41)(26,60,76,42)(27,51,77,43)(28,52,78,44)(29,53,79,45)(30,54,80,46), (1,43,15,51)(2,44,16,52)(3,45,17,53)(4,46,18,54)(5,47,19,55)(6,48,20,56)(7,49,11,57)(8,50,12,58)(9,41,13,59)(10,42,14,60)(21,39,71,66)(22,40,72,67)(23,31,73,68)(24,32,74,69)(25,33,75,70)(26,34,76,61)(27,35,77,62)(28,36,78,63)(29,37,79,64)(30,38,80,65), (2,12,10,18)(3,5,9,7)(4,16,8,14)(6,20)(11,17,19,13)(21,54,23,58)(22,43)(24,47,30,49)(25,52,29,60)(26,41,28,45)(27,56)(31,37,39,33)(32,61,38,63)(34,65,36,69)(40,67)(42,75,44,79)(46,73,50,71)(48,77)(51,72)(53,76,59,78)(55,80,57,74)(64,66,70,68) );
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)], [(1,35,15,62),(2,36,16,63),(3,37,17,64),(4,38,18,65),(5,39,19,66),(6,40,20,67),(7,31,11,68),(8,32,12,69),(9,33,13,70),(10,34,14,61),(21,55,71,47),(22,56,72,48),(23,57,73,49),(24,58,74,50),(25,59,75,41),(26,60,76,42),(27,51,77,43),(28,52,78,44),(29,53,79,45),(30,54,80,46)], [(1,43,15,51),(2,44,16,52),(3,45,17,53),(4,46,18,54),(5,47,19,55),(6,48,20,56),(7,49,11,57),(8,50,12,58),(9,41,13,59),(10,42,14,60),(21,39,71,66),(22,40,72,67),(23,31,73,68),(24,32,74,69),(25,33,75,70),(26,34,76,61),(27,35,77,62),(28,36,78,63),(29,37,79,64),(30,38,80,65)], [(2,12,10,18),(3,5,9,7),(4,16,8,14),(6,20),(11,17,19,13),(21,54,23,58),(22,43),(24,47,30,49),(25,52,29,60),(26,41,28,45),(27,56),(31,37,39,33),(32,61,38,63),(34,65,36,69),(40,67),(42,75,44,79),(46,73,50,71),(48,77),(51,72),(53,76,59,78),(55,80,57,74),(64,66,70,68)])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4L | 5 | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 20A | ··· | 20F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 2 | 5 | 5 | 10 | 2 | 2 | 4 | 4 | 10 | 10 | 20 | ··· | 20 | 4 | 20 | 20 | 20 | 20 | 4 | 4 | 4 | 8 | ··· | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | - | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | D4 | D4 | F5 | C8.C22 | C2×F5 | C2×F5 | C22⋊F5 | C22⋊F5 | (Q8×C10)⋊C4 |
| kernel | (Q8×C10)⋊C4 | Q8⋊F5 | D5⋊M4(2) | D10.C23 | C2×Q8×D5 | C2×Dic10 | Q8×D5 | Q8×C10 | C4×D5 | C2×Dic5 | C22×D5 | C2×Q8 | D5 | C2×C4 | Q8 | C4 | C22 | C1 |
| # reps | 1 | 4 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 2 |
Matrix representation of (Q8×C10)⋊C4 ►in GL8(𝔽41)
| 0 | 40 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 34 | 4 | 37 |
| 0 | 0 | 0 | 0 | 7 | 0 | 37 | 37 |
| 0 | 0 | 0 | 0 | 37 | 4 | 0 | 34 |
| 0 | 0 | 0 | 0 | 4 | 4 | 7 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 34 | 0 | 4 | 4 |
| 0 | 0 | 0 | 0 | 0 | 34 | 4 | 37 |
| 0 | 0 | 0 | 0 | 4 | 4 | 7 | 0 |
| 0 | 0 | 0 | 0 | 4 | 37 | 0 | 7 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 37 | 4 | 0 | 34 |
| 0 | 0 | 0 | 0 | 4 | 4 | 7 | 0 |
| 0 | 0 | 0 | 0 | 0 | 7 | 37 | 4 |
| 0 | 0 | 0 | 0 | 34 | 0 | 4 | 4 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 32 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 9 |
| 0 | 0 | 0 | 0 | 0 | 0 | 9 | 0 |
G:=sub<GL(8,GF(41))| [0,0,0,1,0,0,0,0,40,40,40,40,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,7,37,4,0,0,0,0,34,0,4,4,0,0,0,0,4,37,0,7,0,0,0,0,37,37,34,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,34,0,4,4,0,0,0,0,0,34,4,37,0,0,0,0,4,4,7,0,0,0,0,0,4,37,0,7],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,37,4,0,34,0,0,0,0,4,4,7,0,0,0,0,0,0,7,37,4,0,0,0,0,34,0,4,4],[0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0] >;
(Q8×C10)⋊C4 in GAP, Magma, Sage, TeX
(Q_8\times C_{10})\rtimes C_4 % in TeX
G:=Group("(Q8xC10):C4"); // GroupNames label
G:=SmallGroup(320,1120);
// by ID
G=gap.SmallGroup(320,1120);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,232,422,387,184,1684,438,102,6278,1595]);
// Polycyclic
G:=Group<a,b,c,d|a^10=b^4=d^4=1,c^2=b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^3*b^2,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=a^5*b^-1*c>;
// generators/relations