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), Dic5.D4⋊4C2, (C2×Dic10).14C4, 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
Subgroups: 554 in 146 conjugacy classes, 50 normal (18 characteristic)
C1, C2, C2 [×4], C4 [×2], C4 [×6], C22, C22 [×4], C5, C8 [×4], C2×C4, C2×C4 [×2], C2×C4 [×11], Q8 [×8], C23, D5 [×2], D5, C10, C10, C2×C8 [×2], M4(2) [×6], C22×C4 [×3], C2×Q8, C2×Q8 [×7], Dic5 [×2], Dic5 [×2], C20 [×2], C20 [×2], D10 [×2], D10 [×2], C2×C10, C4.10D4 [×4], C2×M4(2) [×2], C22×Q8, C5⋊C8 [×4], Dic10 [×6], C4×D5 [×4], C4×D5 [×4], C2×Dic5, C2×Dic5 [×2], C2×C20, C2×C20 [×2], C5×Q8 [×2], C22×D5, C2×C4.10D4, D5⋊C8 [×2], C4.F5 [×2], C22.F5 [×4], C2×Dic10, C2×Dic10 [×2], C2×C4×D5, C2×C4×D5 [×2], Q8×D5 [×4], Q8×C10, Dic5.D4 [×4], D5⋊M4(2) [×2], C2×Q8×D5, (C2×Q8).7F5
Quotients:
C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], F5, C4.10D4 [×2], C2×C22⋊C4, C2×F5 [×3], C2×C4.10D4, C22⋊F5 [×2], C22×F5, C2×C22⋊F5, (C2×Q8).7F5
Generators and relations
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 >
►On 80 points
(1 5)(3 7)(9 13)(11 15)(17 21)(19 23)(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 28 5 32)(2 29 6 25)(3 26 7 30)(4 27 8 31)(9 23 13 19)(10 24 14 20)(11 21 15 17)(12 22 16 18)(33 55 37 51)(34 56 38 52)(35 53 39 49)(36 54 40 50)(41 61 45 57)(42 62 46 58)(43 59 47 63)(44 60 48 64)(65 78 69 74)(66 75 70 79)(67 76 71 80)(68 73 72 77)
(1 3 5 7)(2 8 6 4)(9 68 13 72)(10 65 14 69)(11 70 15 66)(12 67 16 71)(17 79 21 75)(18 76 22 80)(19 73 23 77)(20 78 24 74)(25 31 29 27)(26 28 30 32)(33 59 37 63)(34 64 38 60)(35 61 39 57)(36 58 40 62)(41 49 45 53)(42 54 46 50)(43 51 47 55)(44 56 48 52)
(1 57 75 23 33)(2 24 58 34 76)(3 35 17 77 59)(4 78 36 60 18)(5 61 79 19 37)(6 20 62 38 80)(7 39 21 73 63)(8 74 40 64 22)(9 51 32 45 66)(10 46 52 67 25)(11 68 47 26 53)(12 27 69 54 48)(13 55 28 41 70)(14 42 56 71 29)(15 72 43 30 49)(16 31 65 50 44)
(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)(17,21)(19,23)(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), (1,28,5,32)(2,29,6,25)(3,26,7,30)(4,27,8,31)(9,23,13,19)(10,24,14,20)(11,21,15,17)(12,22,16,18)(33,55,37,51)(34,56,38,52)(35,53,39,49)(36,54,40,50)(41,61,45,57)(42,62,46,58)(43,59,47,63)(44,60,48,64)(65,78,69,74)(66,75,70,79)(67,76,71,80)(68,73,72,77), (1,3,5,7)(2,8,6,4)(9,68,13,72)(10,65,14,69)(11,70,15,66)(12,67,16,71)(17,79,21,75)(18,76,22,80)(19,73,23,77)(20,78,24,74)(25,31,29,27)(26,28,30,32)(33,59,37,63)(34,64,38,60)(35,61,39,57)(36,58,40,62)(41,49,45,53)(42,54,46,50)(43,51,47,55)(44,56,48,52), (1,57,75,23,33)(2,24,58,34,76)(3,35,17,77,59)(4,78,36,60,18)(5,61,79,19,37)(6,20,62,38,80)(7,39,21,73,63)(8,74,40,64,22)(9,51,32,45,66)(10,46,52,67,25)(11,68,47,26,53)(12,27,69,54,48)(13,55,28,41,70)(14,42,56,71,29)(15,72,43,30,49)(16,31,65,50,44), (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)(17,21)(19,23)(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), (1,28,5,32)(2,29,6,25)(3,26,7,30)(4,27,8,31)(9,23,13,19)(10,24,14,20)(11,21,15,17)(12,22,16,18)(33,55,37,51)(34,56,38,52)(35,53,39,49)(36,54,40,50)(41,61,45,57)(42,62,46,58)(43,59,47,63)(44,60,48,64)(65,78,69,74)(66,75,70,79)(67,76,71,80)(68,73,72,77), (1,3,5,7)(2,8,6,4)(9,68,13,72)(10,65,14,69)(11,70,15,66)(12,67,16,71)(17,79,21,75)(18,76,22,80)(19,73,23,77)(20,78,24,74)(25,31,29,27)(26,28,30,32)(33,59,37,63)(34,64,38,60)(35,61,39,57)(36,58,40,62)(41,49,45,53)(42,54,46,50)(43,51,47,55)(44,56,48,52), (1,57,75,23,33)(2,24,58,34,76)(3,35,17,77,59)(4,78,36,60,18)(5,61,79,19,37)(6,20,62,38,80)(7,39,21,73,63)(8,74,40,64,22)(9,51,32,45,66)(10,46,52,67,25)(11,68,47,26,53)(12,27,69,54,48)(13,55,28,41,70)(14,42,56,71,29)(15,72,43,30,49)(16,31,65,50,44), (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),(17,21),(19,23),(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)], [(1,28,5,32),(2,29,6,25),(3,26,7,30),(4,27,8,31),(9,23,13,19),(10,24,14,20),(11,21,15,17),(12,22,16,18),(33,55,37,51),(34,56,38,52),(35,53,39,49),(36,54,40,50),(41,61,45,57),(42,62,46,58),(43,59,47,63),(44,60,48,64),(65,78,69,74),(66,75,70,79),(67,76,71,80),(68,73,72,77)], [(1,3,5,7),(2,8,6,4),(9,68,13,72),(10,65,14,69),(11,70,15,66),(12,67,16,71),(17,79,21,75),(18,76,22,80),(19,73,23,77),(20,78,24,74),(25,31,29,27),(26,28,30,32),(33,59,37,63),(34,64,38,60),(35,61,39,57),(36,58,40,62),(41,49,45,53),(42,54,46,50),(43,51,47,55),(44,56,48,52)], [(1,57,75,23,33),(2,24,58,34,76),(3,35,17,77,59),(4,78,36,60,18),(5,61,79,19,37),(6,20,62,38,80),(7,39,21,73,63),(8,74,40,64,22),(9,51,32,45,66),(10,46,52,67,25),(11,68,47,26,53),(12,27,69,54,48),(13,55,28,41,70),(14,42,56,71,29),(15,72,43,30,49),(16,31,65,50,44)], [(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)])
Matrix representation ►G ⊆ 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] >;
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 |
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