Copied to
clipboard

## G = (C2×D4).9F5order 320 = 26·5

### 6th non-split extension by C2×D4 of F5 acting via F5/D5=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — (C2×D4).9F5
 Chief series C1 — C5 — C10 — Dic5 — C2×Dic5 — C22.F5 — C2×C22.F5 — (C2×D4).9F5
 Lower central C5 — C10 — C2×C10 — (C2×D4).9F5
 Upper central C1 — C2 — C23 — C2×D4

Generators and relations for (C2×D4).9F5
G = < a,b,c,d,e | a2=b4=c2=d5=1, e4=b2, ebe-1=ab=ba, ac=ca, ad=da, eae-1=ab2, cbc=b-1, bd=db, cd=dc, ece-1=b2c, ede-1=d3 >

Subgroups: 586 in 150 conjugacy classes, 48 normal (28 characteristic)
C1, C2, C2 [×5], C4 [×6], C22, C22 [×2], C22 [×5], C5, C8 [×4], C2×C4, C2×C4 [×11], D4 [×6], Q8 [×2], C23 [×2], C23, D5, C10, C10 [×4], C2×C8 [×2], M4(2) [×6], C22×C4 [×3], C2×D4, C2×D4 [×2], C2×Q8, C4○D4 [×4], Dic5 [×2], Dic5 [×2], Dic5, C20, D10 [×2], C2×C10, C2×C10 [×2], C2×C10 [×3], C4.D4 [×2], C4.10D4 [×2], C2×M4(2) [×2], C2×C4○D4, C5⋊C8 [×4], Dic10 [×2], C4×D5 [×2], C2×Dic5 [×3], C2×Dic5 [×4], C2×Dic5 [×2], C5⋊D4 [×4], C2×C20, C5×D4 [×2], C22×D5, C22×C10 [×2], M4(2).8C22, C2×C5⋊C8 [×2], C22.F5 [×4], C22.F5 [×2], C2×Dic10, C2×C4×D5, D42D5 [×4], C22×Dic5 [×2], C2×C5⋊D4 [×2], D4×C10, Dic5.D4 [×2], C23.F5 [×2], C2×C22.F5 [×2], C2×D42D5, (C2×D4).9F5
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, C2×F5 [×3], M4(2).8C22, C22⋊F5 [×2], C22×F5, C2×C22⋊F5, (C2×D4).9F5

Smallest permutation representation of (C2×D4).9F5
On 80 points
Generators in S80
(1 5)(3 7)(9 13)(11 15)(17 21)(19 23)(25 29)(27 31)(33 37)(35 39)(41 45)(43 47)(49 53)(51 55)(57 61)(59 63)(65 69)(67 71)(73 77)(75 79)
(1 67 5 71)(2 72 6 68)(3 65 7 69)(4 70 8 66)(9 57 13 61)(10 62 14 58)(11 63 15 59)(12 60 16 64)(17 49 21 53)(18 54 22 50)(19 55 23 51)(20 52 24 56)(25 79 29 75)(26 76 30 80)(27 77 31 73)(28 74 32 78)(33 45 37 41)(34 42 38 46)(35 43 39 47)(36 48 40 44)
(1 65)(2 70)(3 67)(4 72)(5 69)(6 66)(7 71)(8 68)(9 23)(10 20)(11 17)(12 22)(13 19)(14 24)(15 21)(16 18)(25 39)(26 36)(27 33)(28 38)(29 35)(30 40)(31 37)(32 34)(41 77)(42 74)(43 79)(44 76)(45 73)(46 78)(47 75)(48 80)(49 59)(50 64)(51 61)(52 58)(53 63)(54 60)(55 57)(56 62)
(1 17 37 75 57)(2 76 18 58 38)(3 59 77 39 19)(4 40 60 20 78)(5 21 33 79 61)(6 80 22 62 34)(7 63 73 35 23)(8 36 64 24 74)(9 71 53 45 29)(10 46 72 30 54)(11 31 47 55 65)(12 56 32 66 48)(13 67 49 41 25)(14 42 68 26 50)(15 27 43 51 69)(16 52 28 70 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)(25,29)(27,31)(33,37)(35,39)(41,45)(43,47)(49,53)(51,55)(57,61)(59,63)(65,69)(67,71)(73,77)(75,79), (1,67,5,71)(2,72,6,68)(3,65,7,69)(4,70,8,66)(9,57,13,61)(10,62,14,58)(11,63,15,59)(12,60,16,64)(17,49,21,53)(18,54,22,50)(19,55,23,51)(20,52,24,56)(25,79,29,75)(26,76,30,80)(27,77,31,73)(28,74,32,78)(33,45,37,41)(34,42,38,46)(35,43,39,47)(36,48,40,44), (1,65)(2,70)(3,67)(4,72)(5,69)(6,66)(7,71)(8,68)(9,23)(10,20)(11,17)(12,22)(13,19)(14,24)(15,21)(16,18)(25,39)(26,36)(27,33)(28,38)(29,35)(30,40)(31,37)(32,34)(41,77)(42,74)(43,79)(44,76)(45,73)(46,78)(47,75)(48,80)(49,59)(50,64)(51,61)(52,58)(53,63)(54,60)(55,57)(56,62), (1,17,37,75,57)(2,76,18,58,38)(3,59,77,39,19)(4,40,60,20,78)(5,21,33,79,61)(6,80,22,62,34)(7,63,73,35,23)(8,36,64,24,74)(9,71,53,45,29)(10,46,72,30,54)(11,31,47,55,65)(12,56,32,66,48)(13,67,49,41,25)(14,42,68,26,50)(15,27,43,51,69)(16,52,28,70,44), (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)(25,29)(27,31)(33,37)(35,39)(41,45)(43,47)(49,53)(51,55)(57,61)(59,63)(65,69)(67,71)(73,77)(75,79), (1,67,5,71)(2,72,6,68)(3,65,7,69)(4,70,8,66)(9,57,13,61)(10,62,14,58)(11,63,15,59)(12,60,16,64)(17,49,21,53)(18,54,22,50)(19,55,23,51)(20,52,24,56)(25,79,29,75)(26,76,30,80)(27,77,31,73)(28,74,32,78)(33,45,37,41)(34,42,38,46)(35,43,39,47)(36,48,40,44), (1,65)(2,70)(3,67)(4,72)(5,69)(6,66)(7,71)(8,68)(9,23)(10,20)(11,17)(12,22)(13,19)(14,24)(15,21)(16,18)(25,39)(26,36)(27,33)(28,38)(29,35)(30,40)(31,37)(32,34)(41,77)(42,74)(43,79)(44,76)(45,73)(46,78)(47,75)(48,80)(49,59)(50,64)(51,61)(52,58)(53,63)(54,60)(55,57)(56,62), (1,17,37,75,57)(2,76,18,58,38)(3,59,77,39,19)(4,40,60,20,78)(5,21,33,79,61)(6,80,22,62,34)(7,63,73,35,23)(8,36,64,24,74)(9,71,53,45,29)(10,46,72,30,54)(11,31,47,55,65)(12,56,32,66,48)(13,67,49,41,25)(14,42,68,26,50)(15,27,43,51,69)(16,52,28,70,44), (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),(25,29),(27,31),(33,37),(35,39),(41,45),(43,47),(49,53),(51,55),(57,61),(59,63),(65,69),(67,71),(73,77),(75,79)], [(1,67,5,71),(2,72,6,68),(3,65,7,69),(4,70,8,66),(9,57,13,61),(10,62,14,58),(11,63,15,59),(12,60,16,64),(17,49,21,53),(18,54,22,50),(19,55,23,51),(20,52,24,56),(25,79,29,75),(26,76,30,80),(27,77,31,73),(28,74,32,78),(33,45,37,41),(34,42,38,46),(35,43,39,47),(36,48,40,44)], [(1,65),(2,70),(3,67),(4,72),(5,69),(6,66),(7,71),(8,68),(9,23),(10,20),(11,17),(12,22),(13,19),(14,24),(15,21),(16,18),(25,39),(26,36),(27,33),(28,38),(29,35),(30,40),(31,37),(32,34),(41,77),(42,74),(43,79),(44,76),(45,73),(46,78),(47,75),(48,80),(49,59),(50,64),(51,61),(52,58),(53,63),(54,60),(55,57),(56,62)], [(1,17,37,75,57),(2,76,18,58,38),(3,59,77,39,19),(4,40,60,20,78),(5,21,33,79,61),(6,80,22,62,34),(7,63,73,35,23),(8,36,64,24,74),(9,71,53,45,29),(10,46,72,30,54),(11,31,47,55,65),(12,56,32,66,48),(13,67,49,41,25),(14,42,68,26,50),(15,27,43,51,69),(16,52,28,70,44)], [(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)])

32 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 4E 4F 4G 5 8A ··· 8H 10A 10B 10C 10D 10E 10F 10G 20A 20B order 1 2 2 2 2 2 2 4 4 4 4 4 4 4 5 8 ··· 8 10 10 10 10 10 10 10 20 20 size 1 1 2 2 2 4 20 4 5 5 10 10 10 20 4 20 ··· 20 4 4 4 8 8 8 8 8 8

32 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 4 4 4 4 4 8 type + + + + + + + + + + - image C1 C2 C2 C2 C2 C4 C4 C4 C4 D4 F5 C2×F5 C2×F5 M4(2).8C22 C22⋊F5 (C2×D4).9F5 kernel (C2×D4).9F5 Dic5.D4 C23.F5 C2×C22.F5 C2×D4⋊2D5 C2×C4×D5 C22×Dic5 C2×C5⋊D4 D4×C10 C2×Dic5 C2×D4 C2×C4 C23 C5 C22 C1 # reps 1 2 2 2 1 2 2 2 2 4 1 1 2 2 4 2

Matrix representation of (C2×D4).9F5 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 4 0 1 0 0 0 0 0 7 0 0 1
,
 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 32 0 0 0 0 0 0 0 28 9 0 0 0 0 0 0 0 0 32 0 0 0 0 0 22 0 0 9
,
 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 26 5 0 0 0 0 0 0 29 15 0 0 0 0 0 0 19 31 0 32 0 0 0 0 9 3 9 0
,
 40 40 0 0 0 0 0 0 8 7 0 0 0 0 0 0 39 39 34 40 0 0 0 0 0 0 1 0 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
,
 0 15 36 36 0 0 0 0 33 18 35 5 0 0 0 0 32 32 8 0 0 0 0 0 0 29 15 15 0 0 0 0 0 0 0 0 33 0 37 0 0 0 0 0 0 0 29 1 0 0 0 0 13 1 8 0 0 0 0 0 18 0 14 0

G:=sub<GL(8,GF(41))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,4,7,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,32,28,0,22,0,0,0,0,0,9,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,9],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,26,29,19,9,0,0,0,0,5,15,31,3,0,0,0,0,0,0,0,9,0,0,0,0,0,0,32,0],[40,8,39,0,0,0,0,0,40,7,39,0,0,0,0,0,0,0,34,1,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,33,32,0,0,0,0,0,15,18,32,29,0,0,0,0,36,35,8,15,0,0,0,0,36,5,0,15,0,0,0,0,0,0,0,0,33,0,13,18,0,0,0,0,0,0,1,0,0,0,0,0,37,29,8,14,0,0,0,0,0,1,0,0] >;

(C2×D4).9F5 in GAP, Magma, Sage, TeX

(C_2\times D_4)._9F_5
% in TeX

G:=Group("(C2xD4).9F5");
// GroupNames label

G:=SmallGroup(320,1115);
// by ID

G=gap.SmallGroup(320,1115);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,422,387,297,136,1684,6278,1595]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^4=c^2=d^5=1,e^4=b^2,e*b*e^-1=a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=a*b^2,c*b*c=b^-1,b*d=d*b,c*d=d*c,e*c*e^-1=b^2*c,e*d*e^-1=d^3>;
// generators/relations

׿
×
𝔽