Copied to
clipboard

## G = C2×D5×SD16order 320 = 26·5

### Direct product of C2, D5 and SD16

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C2×D5×SD16
 Chief series C1 — C5 — C10 — C20 — C4×D5 — C2×C4×D5 — C2×D4×D5 — C2×D5×SD16
 Lower central C5 — C10 — C20 — C2×D5×SD16
 Upper central C1 — C22 — C2×C4 — C2×SD16

Generators and relations for C2×D5×SD16
G = < a,b,c,d,e | a2=b5=c2=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d3 >

Subgroups: 1278 in 298 conjugacy classes, 111 normal (33 characteristic)
C1, C2, C2 [×2], C2 [×8], C4 [×2], C4 [×6], C22, C22 [×22], C5, C8 [×2], C8 [×2], C2×C4, C2×C4 [×11], D4 [×2], D4 [×8], Q8 [×2], Q8 [×8], C23 [×11], D5 [×4], D5 [×2], C10, C10 [×2], C10 [×2], C2×C8, C2×C8 [×5], SD16 [×4], SD16 [×12], C22×C4 [×2], C2×D4, C2×D4 [×8], C2×Q8, C2×Q8 [×8], C24, Dic5 [×2], Dic5 [×2], C20 [×2], C20 [×2], D10 [×6], D10 [×12], C2×C10, C2×C10 [×4], C22×C8, C2×SD16, C2×SD16 [×11], C22×D4, C22×Q8, C52C8 [×2], C40 [×2], Dic10 [×2], Dic10 [×5], C4×D5 [×4], C4×D5 [×4], D20 [×2], D20, C2×Dic5, C2×Dic5, C5⋊D4 [×4], C2×C20, C2×C20, C5×D4 [×2], C5×D4, C5×Q8 [×2], C5×Q8, C22×D5, C22×D5 [×9], C22×C10, C22×SD16, C8×D5 [×4], C40⋊C2 [×4], C2×C52C8, D4.D5 [×4], Q8⋊D5 [×4], C2×C40, C5×SD16 [×4], C2×Dic10, C2×Dic10, C2×C4×D5, C2×C4×D5, C2×D20, D4×D5 [×4], D4×D5 [×2], Q8×D5 [×4], Q8×D5 [×2], C2×C5⋊D4, D4×C10, Q8×C10, C23×D5, D5×C2×C8, C2×C40⋊C2, D5×SD16 [×8], C2×D4.D5, C2×Q8⋊D5, C10×SD16, C2×D4×D5, C2×Q8×D5, C2×D5×SD16
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, SD16 [×4], C2×D4 [×6], C24, D10 [×7], C2×SD16 [×6], C22×D4, C22×D5 [×7], C22×SD16, D4×D5 [×2], C23×D5, D5×SD16 [×2], C2×D4×D5, C2×D5×SD16

Smallest permutation representation of C2×D5×SD16
On 80 points
Generators in S80
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 17)(8 18)(9 65)(10 66)(11 67)(12 68)(13 69)(14 70)(15 71)(16 72)(25 45)(26 46)(27 47)(28 48)(29 41)(30 42)(31 43)(32 44)(33 49)(34 50)(35 51)(36 52)(37 53)(38 54)(39 55)(40 56)(57 79)(58 80)(59 73)(60 74)(61 75)(62 76)(63 77)(64 78)
(1 49 46 62 68)(2 50 47 63 69)(3 51 48 64 70)(4 52 41 57 71)(5 53 42 58 72)(6 54 43 59 65)(7 55 44 60 66)(8 56 45 61 67)(9 24 38 31 73)(10 17 39 32 74)(11 18 40 25 75)(12 19 33 26 76)(13 20 34 27 77)(14 21 35 28 78)(15 22 36 29 79)(16 23 37 30 80)
(1 12)(2 13)(3 14)(4 15)(5 16)(6 9)(7 10)(8 11)(17 66)(18 67)(19 68)(20 69)(21 70)(22 71)(23 72)(24 65)(25 45)(26 46)(27 47)(28 48)(29 41)(30 42)(31 43)(32 44)(33 62)(34 63)(35 64)(36 57)(37 58)(38 59)(39 60)(40 61)(49 76)(50 77)(51 78)(52 79)(53 80)(54 73)(55 74)(56 75)
(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 19)(2 22)(3 17)(4 20)(5 23)(6 18)(7 21)(8 24)(9 67)(10 70)(11 65)(12 68)(13 71)(14 66)(15 69)(16 72)(25 43)(26 46)(27 41)(28 44)(29 47)(30 42)(31 45)(32 48)(33 49)(34 52)(35 55)(36 50)(37 53)(38 56)(39 51)(40 54)(57 77)(58 80)(59 75)(60 78)(61 73)(62 76)(63 79)(64 74)

G:=sub<Sym(80)| (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,17)(8,18)(9,65)(10,66)(11,67)(12,68)(13,69)(14,70)(15,71)(16,72)(25,45)(26,46)(27,47)(28,48)(29,41)(30,42)(31,43)(32,44)(33,49)(34,50)(35,51)(36,52)(37,53)(38,54)(39,55)(40,56)(57,79)(58,80)(59,73)(60,74)(61,75)(62,76)(63,77)(64,78), (1,49,46,62,68)(2,50,47,63,69)(3,51,48,64,70)(4,52,41,57,71)(5,53,42,58,72)(6,54,43,59,65)(7,55,44,60,66)(8,56,45,61,67)(9,24,38,31,73)(10,17,39,32,74)(11,18,40,25,75)(12,19,33,26,76)(13,20,34,27,77)(14,21,35,28,78)(15,22,36,29,79)(16,23,37,30,80), (1,12)(2,13)(3,14)(4,15)(5,16)(6,9)(7,10)(8,11)(17,66)(18,67)(19,68)(20,69)(21,70)(22,71)(23,72)(24,65)(25,45)(26,46)(27,47)(28,48)(29,41)(30,42)(31,43)(32,44)(33,62)(34,63)(35,64)(36,57)(37,58)(38,59)(39,60)(40,61)(49,76)(50,77)(51,78)(52,79)(53,80)(54,73)(55,74)(56,75), (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), (1,19)(2,22)(3,17)(4,20)(5,23)(6,18)(7,21)(8,24)(9,67)(10,70)(11,65)(12,68)(13,71)(14,66)(15,69)(16,72)(25,43)(26,46)(27,41)(28,44)(29,47)(30,42)(31,45)(32,48)(33,49)(34,52)(35,55)(36,50)(37,53)(38,56)(39,51)(40,54)(57,77)(58,80)(59,75)(60,78)(61,73)(62,76)(63,79)(64,74)>;

G:=Group( (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,17)(8,18)(9,65)(10,66)(11,67)(12,68)(13,69)(14,70)(15,71)(16,72)(25,45)(26,46)(27,47)(28,48)(29,41)(30,42)(31,43)(32,44)(33,49)(34,50)(35,51)(36,52)(37,53)(38,54)(39,55)(40,56)(57,79)(58,80)(59,73)(60,74)(61,75)(62,76)(63,77)(64,78), (1,49,46,62,68)(2,50,47,63,69)(3,51,48,64,70)(4,52,41,57,71)(5,53,42,58,72)(6,54,43,59,65)(7,55,44,60,66)(8,56,45,61,67)(9,24,38,31,73)(10,17,39,32,74)(11,18,40,25,75)(12,19,33,26,76)(13,20,34,27,77)(14,21,35,28,78)(15,22,36,29,79)(16,23,37,30,80), (1,12)(2,13)(3,14)(4,15)(5,16)(6,9)(7,10)(8,11)(17,66)(18,67)(19,68)(20,69)(21,70)(22,71)(23,72)(24,65)(25,45)(26,46)(27,47)(28,48)(29,41)(30,42)(31,43)(32,44)(33,62)(34,63)(35,64)(36,57)(37,58)(38,59)(39,60)(40,61)(49,76)(50,77)(51,78)(52,79)(53,80)(54,73)(55,74)(56,75), (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), (1,19)(2,22)(3,17)(4,20)(5,23)(6,18)(7,21)(8,24)(9,67)(10,70)(11,65)(12,68)(13,71)(14,66)(15,69)(16,72)(25,43)(26,46)(27,41)(28,44)(29,47)(30,42)(31,45)(32,48)(33,49)(34,52)(35,55)(36,50)(37,53)(38,56)(39,51)(40,54)(57,77)(58,80)(59,75)(60,78)(61,73)(62,76)(63,79)(64,74) );

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

56 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 4A 4B 4C 4D 4E 4F 4G 4H 5A 5B 8A 8B 8C 8D 8E 8F 8G 8H 10A ··· 10F 10G 10H 10I 10J 20A 20B 20C 20D 20E 20F 20G 20H 40A ··· 40H order 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 5 5 8 8 8 8 8 8 8 8 10 ··· 10 10 10 10 10 20 20 20 20 20 20 20 20 40 ··· 40 size 1 1 1 1 4 4 5 5 5 5 20 20 2 2 4 4 10 10 20 20 2 2 2 2 2 2 10 10 10 10 2 ··· 2 8 8 8 8 4 4 4 4 8 8 8 8 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 D4 D4 D4 D5 SD16 D10 D10 D10 D10 D4×D5 D4×D5 D5×SD16 kernel C2×D5×SD16 D5×C2×C8 C2×C40⋊C2 D5×SD16 C2×D4.D5 C2×Q8⋊D5 C10×SD16 C2×D4×D5 C2×Q8×D5 C4×D5 C2×Dic5 C22×D5 C2×SD16 D10 C2×C8 SD16 C2×D4 C2×Q8 C4 C22 C2 # reps 1 1 1 8 1 1 1 1 1 2 1 1 2 8 2 8 2 2 2 2 8

Matrix representation of C2×D5×SD16 in GL4(𝔽41) generated by

 40 0 0 0 0 40 0 0 0 0 40 0 0 0 0 40
,
 0 1 0 0 40 34 0 0 0 0 1 0 0 0 0 1
,
 0 40 0 0 40 0 0 0 0 0 40 0 0 0 0 40
,
 1 0 0 0 0 1 0 0 0 0 30 12 0 0 24 0
,
 40 0 0 0 0 40 0 0 0 0 40 0 0 0 23 1
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[0,40,0,0,1,34,0,0,0,0,1,0,0,0,0,1],[0,40,0,0,40,0,0,0,0,0,40,0,0,0,0,40],[1,0,0,0,0,1,0,0,0,0,30,24,0,0,12,0],[40,0,0,0,0,40,0,0,0,0,40,23,0,0,0,1] >;

C2×D5×SD16 in GAP, Magma, Sage, TeX

C_2\times D_5\times {\rm SD}_{16}
% in TeX

G:=Group("C2xD5xSD16");
// GroupNames label

G:=SmallGroup(320,1430);
// by ID

G=gap.SmallGroup(320,1430);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,185,136,438,235,102,12550]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^5=c^2=d^8=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^3>;
// generators/relations

׿
×
𝔽