direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C23.F5, C24.2F5, C10⋊(C4.D4), (C23×C10).8C4, (C23×D5).6C4, C23.22(C2×F5), Dic5.12(C2×D4), C22.F5⋊2C22, (C2×Dic5).127D4, C22.18(C22×F5), C22.54(C22⋊F5), Dic5.17(C22⋊C4), (C2×Dic5).176C23, (C22×Dic5).190C22, C5⋊2(C2×C4.D4), (C2×C5⋊D4).26C4, (C2×C22.F5)⋊9C2, C2.40(C2×C22⋊F5), C10.40(C2×C22⋊C4), (C2×C10).93(C22×C4), (C22×C10).75(C2×C4), (C2×Dic5).79(C2×C4), (C22×D5).11(C2×C4), (C22×C5⋊D4).15C2, (C2×C10).64(C22⋊C4), (C2×C5⋊D4).156C22, SmallGroup(320,1137)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 842 in 186 conjugacy classes, 52 normal (20 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×4], C22 [×3], C22 [×18], C5, C8 [×4], C2×C4 [×6], D4 [×8], C23, C23 [×2], C23 [×10], D5 [×2], C10, C10 [×2], C10 [×4], C2×C8 [×2], M4(2) [×6], C22×C4, C2×D4 [×8], C24, C24, Dic5 [×4], D10 [×8], C2×C10 [×3], C2×C10 [×10], C4.D4 [×4], C2×M4(2) [×2], C22×D4, C5⋊C8 [×4], C2×Dic5 [×2], C2×Dic5 [×4], C5⋊D4 [×8], C22×D5 [×2], C22×D5 [×4], C22×C10, C22×C10 [×2], C22×C10 [×4], C2×C4.D4, C2×C5⋊C8 [×2], C22.F5 [×4], C22.F5 [×2], C22×Dic5, C2×C5⋊D4 [×4], C2×C5⋊D4 [×4], C23×D5, C23×C10, C23.F5 [×4], C2×C22.F5 [×2], C22×C5⋊D4, C2×C23.F5
Quotients:
C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], F5, C4.D4 [×2], C2×C22⋊C4, C2×F5 [×3], C2×C4.D4, C22⋊F5 [×2], C22×F5, C23.F5 [×2], C2×C22⋊F5, C2×C23.F5
Generators and relations
G = < a,b,c,d,e,f | a2=b2=c2=d2=e5=1, f4=d, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf-1=bcd, fcf-1=cd=dc, ce=ec, de=ed, df=fd, fef-1=e3 >
►On 80 points
Generators in S80
(1 68)(2 69)(3 70)(4 71)(5 72)(6 65)(7 66)(8 67)(9 22)(10 23)(11 24)(12 17)(13 18)(14 19)(15 20)(16 21)(25 74)(26 75)(27 76)(28 77)(29 78)(30 79)(31 80)(32 73)(33 48)(34 41)(35 42)(36 43)(37 44)(38 45)(39 46)(40 47)(49 59)(50 60)(51 61)(52 62)(53 63)(54 64)(55 57)(56 58)
(1 7)(2 4)(3 5)(6 8)(9 56)(10 49)(11 54)(12 55)(13 52)(14 53)(15 50)(16 51)(17 57)(18 62)(19 63)(20 60)(21 61)(22 58)(23 59)(24 64)(25 47)(26 48)(27 45)(28 46)(29 43)(30 44)(31 41)(32 42)(33 75)(34 80)(35 73)(36 78)(37 79)(38 76)(39 77)(40 74)(65 67)(66 68)(69 71)(70 72)
(2 6)(4 8)(9 13)(11 15)(18 22)(20 24)(25 29)(27 31)(34 38)(36 40)(41 45)(43 47)(50 54)(52 56)(58 62)(60 64)(65 69)(67 71)(74 78)(76 80)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)(33 37)(34 38)(35 39)(36 40)(41 45)(42 46)(43 47)(44 48)(49 53)(50 54)(51 55)(52 56)(57 61)(58 62)(59 63)(60 64)(65 69)(66 70)(67 71)(68 72)(73 77)(74 78)(75 79)(76 80)
(1 57 35 75 19)(2 76 58 20 36)(3 21 77 37 59)(4 38 22 60 78)(5 61 39 79 23)(6 80 62 24 40)(7 17 73 33 63)(8 34 18 64 74)(9 50 29 71 45)(10 72 51 46 30)(11 47 65 31 52)(12 32 48 53 66)(13 54 25 67 41)(14 68 55 42 26)(15 43 69 27 56)(16 28 44 49 70)
(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)| (1,68)(2,69)(3,70)(4,71)(5,72)(6,65)(7,66)(8,67)(9,22)(10,23)(11,24)(12,17)(13,18)(14,19)(15,20)(16,21)(25,74)(26,75)(27,76)(28,77)(29,78)(30,79)(31,80)(32,73)(33,48)(34,41)(35,42)(36,43)(37,44)(38,45)(39,46)(40,47)(49,59)(50,60)(51,61)(52,62)(53,63)(54,64)(55,57)(56,58), (1,7)(2,4)(3,5)(6,8)(9,56)(10,49)(11,54)(12,55)(13,52)(14,53)(15,50)(16,51)(17,57)(18,62)(19,63)(20,60)(21,61)(22,58)(23,59)(24,64)(25,47)(26,48)(27,45)(28,46)(29,43)(30,44)(31,41)(32,42)(33,75)(34,80)(35,73)(36,78)(37,79)(38,76)(39,77)(40,74)(65,67)(66,68)(69,71)(70,72), (2,6)(4,8)(9,13)(11,15)(18,22)(20,24)(25,29)(27,31)(34,38)(36,40)(41,45)(43,47)(50,54)(52,56)(58,62)(60,64)(65,69)(67,71)(74,78)(76,80), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(41,45)(42,46)(43,47)(44,48)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64)(65,69)(66,70)(67,71)(68,72)(73,77)(74,78)(75,79)(76,80), (1,57,35,75,19)(2,76,58,20,36)(3,21,77,37,59)(4,38,22,60,78)(5,61,39,79,23)(6,80,62,24,40)(7,17,73,33,63)(8,34,18,64,74)(9,50,29,71,45)(10,72,51,46,30)(11,47,65,31,52)(12,32,48,53,66)(13,54,25,67,41)(14,68,55,42,26)(15,43,69,27,56)(16,28,44,49,70), (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( (1,68)(2,69)(3,70)(4,71)(5,72)(6,65)(7,66)(8,67)(9,22)(10,23)(11,24)(12,17)(13,18)(14,19)(15,20)(16,21)(25,74)(26,75)(27,76)(28,77)(29,78)(30,79)(31,80)(32,73)(33,48)(34,41)(35,42)(36,43)(37,44)(38,45)(39,46)(40,47)(49,59)(50,60)(51,61)(52,62)(53,63)(54,64)(55,57)(56,58), (1,7)(2,4)(3,5)(6,8)(9,56)(10,49)(11,54)(12,55)(13,52)(14,53)(15,50)(16,51)(17,57)(18,62)(19,63)(20,60)(21,61)(22,58)(23,59)(24,64)(25,47)(26,48)(27,45)(28,46)(29,43)(30,44)(31,41)(32,42)(33,75)(34,80)(35,73)(36,78)(37,79)(38,76)(39,77)(40,74)(65,67)(66,68)(69,71)(70,72), (2,6)(4,8)(9,13)(11,15)(18,22)(20,24)(25,29)(27,31)(34,38)(36,40)(41,45)(43,47)(50,54)(52,56)(58,62)(60,64)(65,69)(67,71)(74,78)(76,80), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(41,45)(42,46)(43,47)(44,48)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64)(65,69)(66,70)(67,71)(68,72)(73,77)(74,78)(75,79)(76,80), (1,57,35,75,19)(2,76,58,20,36)(3,21,77,37,59)(4,38,22,60,78)(5,61,39,79,23)(6,80,62,24,40)(7,17,73,33,63)(8,34,18,64,74)(9,50,29,71,45)(10,72,51,46,30)(11,47,65,31,52)(12,32,48,53,66)(13,54,25,67,41)(14,68,55,42,26)(15,43,69,27,56)(16,28,44,49,70), (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([(1,68),(2,69),(3,70),(4,71),(5,72),(6,65),(7,66),(8,67),(9,22),(10,23),(11,24),(12,17),(13,18),(14,19),(15,20),(16,21),(25,74),(26,75),(27,76),(28,77),(29,78),(30,79),(31,80),(32,73),(33,48),(34,41),(35,42),(36,43),(37,44),(38,45),(39,46),(40,47),(49,59),(50,60),(51,61),(52,62),(53,63),(54,64),(55,57),(56,58)], [(1,7),(2,4),(3,5),(6,8),(9,56),(10,49),(11,54),(12,55),(13,52),(14,53),(15,50),(16,51),(17,57),(18,62),(19,63),(20,60),(21,61),(22,58),(23,59),(24,64),(25,47),(26,48),(27,45),(28,46),(29,43),(30,44),(31,41),(32,42),(33,75),(34,80),(35,73),(36,78),(37,79),(38,76),(39,77),(40,74),(65,67),(66,68),(69,71),(70,72)], [(2,6),(4,8),(9,13),(11,15),(18,22),(20,24),(25,29),(27,31),(34,38),(36,40),(41,45),(43,47),(50,54),(52,56),(58,62),(60,64),(65,69),(67,71),(74,78),(76,80)], [(1,5),(2,6),(3,7),(4,8),(9,13),(10,14),(11,15),(12,16),(17,21),(18,22),(19,23),(20,24),(25,29),(26,30),(27,31),(28,32),(33,37),(34,38),(35,39),(36,40),(41,45),(42,46),(43,47),(44,48),(49,53),(50,54),(51,55),(52,56),(57,61),(58,62),(59,63),(60,64),(65,69),(66,70),(67,71),(68,72),(73,77),(74,78),(75,79),(76,80)], [(1,57,35,75,19),(2,76,58,20,36),(3,21,77,37,59),(4,38,22,60,78),(5,61,39,79,23),(6,80,62,24,40),(7,17,73,33,63),(8,34,18,64,74),(9,50,29,71,45),(10,72,51,46,30),(11,47,65,31,52),(12,32,48,53,66),(13,54,25,67,41),(14,68,55,42,26),(15,43,69,27,56),(16,28,44,49,70)], [(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 ⊆ GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 18 | 40 | 0 | 0 |
| 0 | 0 | 36 | 23 | 0 | 0 |
| 0 | 0 | 9 | 0 | 17 | 40 |
| 0 | 0 | 28 | 32 | 1 | 24 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 14 | 2 | 40 | 0 |
| 0 | 0 | 14 | 2 | 0 | 40 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 1 | 0 | 0 |
| 0 | 0 | 5 | 35 | 0 | 0 |
| 0 | 0 | 13 | 9 | 40 | 34 |
| 0 | 0 | 23 | 28 | 7 | 7 |
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 1 |
| 0 | 0 | 22 | 9 | 39 | 34 |
| 0 | 0 | 4 | 30 | 32 | 0 |
| 0 | 0 | 25 | 6 | 32 | 0 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[0,40,0,0,0,0,40,0,0,0,0,0,0,0,18,36,9,28,0,0,40,23,0,32,0,0,0,0,17,1,0,0,0,0,40,24],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,14,14,0,0,0,1,2,2,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,5,13,23,0,0,1,35,9,28,0,0,0,0,40,7,0,0,0,0,34,7],[32,0,0,0,0,0,0,9,0,0,0,0,0,0,0,22,4,25,0,0,0,9,30,6,0,0,40,39,32,32,0,0,1,34,0,0] >;
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 5 | 8A | ··· | 8H | 10A | ··· | 10O |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 8 | ··· | 8 | 10 | ··· | 10 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 20 | 20 | 10 | 10 | 10 | 10 | 4 | 20 | ··· | 20 | 4 | ··· | 4 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | F5 | C4.D4 | C2×F5 | C22⋊F5 | C23.F5 |
| kernel | C2×C23.F5 | C23.F5 | C2×C22.F5 | C22×C5⋊D4 | C2×C5⋊D4 | C23×D5 | C23×C10 | C2×Dic5 | C24 | C10 | C23 | C22 | C2 |
| # reps | 1 | 4 | 2 | 1 | 4 | 2 | 2 | 4 | 1 | 2 | 3 | 4 | 8 |
In GAP, Magma, Sage, TeX
C_2\times C_2^3.F_5
% in TeX
G:=Group("C2xC2^3.F5"); // GroupNames label
G:=SmallGroup(320,1137);
// by ID
G=gap.SmallGroup(320,1137);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,422,297,136,1684,6278,1595]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^5=1,f^4=d,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,f*b*f^-1=b*c*d,f*c*f^-1=c*d=d*c,c*e=e*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^3>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀