metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2×Q8).7F5, (C4×D5).46D4, (Q8×C10).8C4, D5⋊(C4.10D4), Dic5.9(C2×D4), D5⋊M4(2).6C2, C4.22(C22⋊F5), C20.22(C22⋊C4), (C2×Dic10).14C4, Dic5.D4⋊4C2, D10.47(C22⋊C4), C22.16(C22×F5), C22.F5.4C22, (C2×Dic5).175C23, (C2×Dic10).146C22, (C2×C4×D5).6C4, (C2×C4).7(C2×F5), (C2×Q8×D5).11C2, C5⋊2(C2×C4.10D4), (C2×C20).28(C2×C4), C2.32(C2×C22⋊F5), C10.31(C2×C22⋊C4), (C2×Dic5).8(C2×C4), (C2×C4×D5).209C22, (C2×C10).88(C22×C4), (C22×D5).129(C2×C4), SmallGroup(320,1127)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C2×Q8).7F5
G = < a,b,c,d,e | a2=b4=d5=1, c2=e4=b2, ab=ba, ac=ca, ad=da, eae-1=ab2, cbc-1=b-1, bd=db, ebe-1=ab-1, cd=dc, ece-1=b2c, ede-1=d3 >
Subgroups: 554 in 146 conjugacy classes, 50 normal (18 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, Q8, C23, D5, D5, C10, C10, C2×C8, M4(2), C22×C4, C2×Q8, C2×Q8, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C4.10D4, C2×M4(2), C22×Q8, C5⋊C8, Dic10, C4×D5, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×Q8, C22×D5, C2×C4.10D4, D5⋊C8, C4.F5, C22.F5, C2×Dic10, C2×Dic10, C2×C4×D5, C2×C4×D5, Q8×D5, Q8×C10, Dic5.D4, D5⋊M4(2), C2×Q8×D5, (C2×Q8).7F5
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, F5, C4.10D4, C2×C22⋊C4, C2×F5, C2×C4.10D4, C22⋊F5, C22×F5, C2×C22⋊F5, (C2×Q8).7F5
►On 80 points
Generators in S80
(1 5)(3 7)(9 13)(11 15)(18 22)(20 24)(26 30)(28 32)(33 37)(35 39)(41 45)(43 47)(49 53)(51 55)(57 61)(59 63)(66 70)(68 72)(73 77)(75 79)
(1 68 5 72)(2 69 6 65)(3 66 7 70)(4 67 8 71)(9 61 13 57)(10 62 14 58)(11 59 15 63)(12 60 16 64)(17 46 21 42)(18 43 22 47)(19 44 23 48)(20 41 24 45)(25 40 29 36)(26 37 30 33)(27 38 31 34)(28 35 32 39)(49 79 53 75)(50 80 54 76)(51 77 55 73)(52 78 56 74)
(1 3 5 7)(2 8 6 4)(9 55 13 51)(10 52 14 56)(11 49 15 53)(12 54 16 50)(17 34 21 38)(18 39 22 35)(19 36 23 40)(20 33 24 37)(25 44 29 48)(26 41 30 45)(27 46 31 42)(28 43 32 47)(57 73 61 77)(58 78 62 74)(59 75 63 79)(60 80 64 76)(65 71 69 67)(66 68 70 72)
(1 63 37 18 77)(2 19 64 78 38)(3 79 20 39 57)(4 40 80 58 21)(5 59 33 22 73)(6 23 60 74 34)(7 75 24 35 61)(8 36 76 62 17)(9 66 53 41 28)(10 42 67 29 54)(11 30 43 55 68)(12 56 31 69 44)(13 70 49 45 32)(14 46 71 25 50)(15 26 47 51 72)(16 52 27 65 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)
G:=sub<Sym(80)| (1,5)(3,7)(9,13)(11,15)(18,22)(20,24)(26,30)(28,32)(33,37)(35,39)(41,45)(43,47)(49,53)(51,55)(57,61)(59,63)(66,70)(68,72)(73,77)(75,79), (1,68,5,72)(2,69,6,65)(3,66,7,70)(4,67,8,71)(9,61,13,57)(10,62,14,58)(11,59,15,63)(12,60,16,64)(17,46,21,42)(18,43,22,47)(19,44,23,48)(20,41,24,45)(25,40,29,36)(26,37,30,33)(27,38,31,34)(28,35,32,39)(49,79,53,75)(50,80,54,76)(51,77,55,73)(52,78,56,74), (1,3,5,7)(2,8,6,4)(9,55,13,51)(10,52,14,56)(11,49,15,53)(12,54,16,50)(17,34,21,38)(18,39,22,35)(19,36,23,40)(20,33,24,37)(25,44,29,48)(26,41,30,45)(27,46,31,42)(28,43,32,47)(57,73,61,77)(58,78,62,74)(59,75,63,79)(60,80,64,76)(65,71,69,67)(66,68,70,72), (1,63,37,18,77)(2,19,64,78,38)(3,79,20,39,57)(4,40,80,58,21)(5,59,33,22,73)(6,23,60,74,34)(7,75,24,35,61)(8,36,76,62,17)(9,66,53,41,28)(10,42,67,29,54)(11,30,43,55,68)(12,56,31,69,44)(13,70,49,45,32)(14,46,71,25,50)(15,26,47,51,72)(16,52,27,65,48), (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)>;
G:=Group( (1,5)(3,7)(9,13)(11,15)(18,22)(20,24)(26,30)(28,32)(33,37)(35,39)(41,45)(43,47)(49,53)(51,55)(57,61)(59,63)(66,70)(68,72)(73,77)(75,79), (1,68,5,72)(2,69,6,65)(3,66,7,70)(4,67,8,71)(9,61,13,57)(10,62,14,58)(11,59,15,63)(12,60,16,64)(17,46,21,42)(18,43,22,47)(19,44,23,48)(20,41,24,45)(25,40,29,36)(26,37,30,33)(27,38,31,34)(28,35,32,39)(49,79,53,75)(50,80,54,76)(51,77,55,73)(52,78,56,74), (1,3,5,7)(2,8,6,4)(9,55,13,51)(10,52,14,56)(11,49,15,53)(12,54,16,50)(17,34,21,38)(18,39,22,35)(19,36,23,40)(20,33,24,37)(25,44,29,48)(26,41,30,45)(27,46,31,42)(28,43,32,47)(57,73,61,77)(58,78,62,74)(59,75,63,79)(60,80,64,76)(65,71,69,67)(66,68,70,72), (1,63,37,18,77)(2,19,64,78,38)(3,79,20,39,57)(4,40,80,58,21)(5,59,33,22,73)(6,23,60,74,34)(7,75,24,35,61)(8,36,76,62,17)(9,66,53,41,28)(10,42,67,29,54)(11,30,43,55,68)(12,56,31,69,44)(13,70,49,45,32)(14,46,71,25,50)(15,26,47,51,72)(16,52,27,65,48), (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) );
G=PermutationGroup([[(1,5),(3,7),(9,13),(11,15),(18,22),(20,24),(26,30),(28,32),(33,37),(35,39),(41,45),(43,47),(49,53),(51,55),(57,61),(59,63),(66,70),(68,72),(73,77),(75,79)], [(1,68,5,72),(2,69,6,65),(3,66,7,70),(4,67,8,71),(9,61,13,57),(10,62,14,58),(11,59,15,63),(12,60,16,64),(17,46,21,42),(18,43,22,47),(19,44,23,48),(20,41,24,45),(25,40,29,36),(26,37,30,33),(27,38,31,34),(28,35,32,39),(49,79,53,75),(50,80,54,76),(51,77,55,73),(52,78,56,74)], [(1,3,5,7),(2,8,6,4),(9,55,13,51),(10,52,14,56),(11,49,15,53),(12,54,16,50),(17,34,21,38),(18,39,22,35),(19,36,23,40),(20,33,24,37),(25,44,29,48),(26,41,30,45),(27,46,31,42),(28,43,32,47),(57,73,61,77),(58,78,62,74),(59,75,63,79),(60,80,64,76),(65,71,69,67),(66,68,70,72)], [(1,63,37,18,77),(2,19,64,78,38),(3,79,20,39,57),(4,40,80,58,21),(5,59,33,22,73),(6,23,60,74,34),(7,75,24,35,61),(8,36,76,62,17),(9,66,53,41,28),(10,42,67,29,54),(11,30,43,55,68),(12,56,31,69,44),(13,70,49,45,32),(14,46,71,25,50),(15,26,47,51,72),(16,52,27,65,48)], [(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)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5 | 8A | ··· | 8H | 10A | 10B | 10C | 20A | ··· | 20F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 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 | 4 | 4 | 4 | 8 | ··· | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | - | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | F5 | C4.10D4 | C2×F5 | C22⋊F5 | (C2×Q8).7F5 |
| kernel | (C2×Q8).7F5 | Dic5.D4 | D5⋊M4(2) | C2×Q8×D5 | C2×Dic10 | C2×C4×D5 | Q8×C10 | C4×D5 | C2×Q8 | D5 | C2×C4 | C4 | C1 |
| # reps | 1 | 4 | 2 | 1 | 2 | 4 | 2 | 4 | 1 | 2 | 3 | 4 | 2 |
Matrix representation of (C2×Q8).7F5 ►in GL8(𝔽41)
| 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 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 23 | 19 | 1 | 0 |
| 0 | 0 | 0 | 0 | 30 | 18 | 0 | 1 |
| 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 | 36 | 19 | 0 | 0 |
| 0 | 0 | 0 | 0 | 31 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 34 | 26 | 15 |
| 0 | 0 | 0 | 0 | 9 | 9 | 15 | 15 |
| 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 | 15 | 37 | 0 | 0 |
| 0 | 0 | 0 | 0 | 36 | 26 | 0 | 0 |
| 0 | 0 | 0 | 0 | 24 | 36 | 0 | 40 |
| 0 | 0 | 0 | 0 | 16 | 20 | 1 | 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 |
| 40 | 40 | 40 | 40 | 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 | 1 |
| 2 | 0 | 38 | 38 | 0 | 0 | 0 | 0 |
| 38 | 38 | 0 | 2 | 0 | 0 | 0 | 0 |
| 3 | 5 | 3 | 0 | 0 | 0 | 0 | 0 |
| 39 | 36 | 36 | 39 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 37 | 27 | 5 | 0 |
| 0 | 0 | 0 | 0 | 38 | 8 | 29 | 9 |
| 0 | 0 | 0 | 0 | 16 | 35 | 36 | 17 |
| 0 | 0 | 0 | 0 | 13 | 15 | 33 | 1 |
G:=sub<GL(8,GF(41))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,23,30,0,0,0,0,0,40,19,18,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,36,31,0,9,0,0,0,0,19,5,34,9,0,0,0,0,0,0,26,15,0,0,0,0,0,0,15,15],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,15,36,24,16,0,0,0,0,37,26,36,20,0,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0],[0,0,0,40,0,0,0,0,1,0,0,40,0,0,0,0,0,1,0,40,0,0,0,0,0,0,1,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[2,38,3,39,0,0,0,0,0,38,5,36,0,0,0,0,38,0,3,36,0,0,0,0,38,2,0,39,0,0,0,0,0,0,0,0,37,38,16,13,0,0,0,0,27,8,35,15,0,0,0,0,5,29,36,33,0,0,0,0,0,9,17,1] >;
(C2×Q8).7F5 in GAP, Magma, Sage, TeX
(C_2\times Q_8)._7F_5
% in TeX
G:=Group("(C2xQ8).7F5"); // GroupNames label
G:=SmallGroup(320,1127);
// by ID
G=gap.SmallGroup(320,1127);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,232,422,387,184,297,136,1684,6278,1595]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=d^5=1,c^2=e^4=b^2,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=a*b^2,c*b*c^-1=b^-1,b*d=d*b,e*b*e^-1=a*b^-1,c*d=d*c,e*c*e^-1=b^2*c,e*d*e^-1=d^3>;
// generators/relations