Copied to
clipboard

## G = C2×C22⋊D20order 320 = 26·5

### Direct product of C2 and C22⋊D20

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C2×C22⋊D20
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C23×D5 — D5×C24 — C2×C22⋊D20
 Lower central C5 — C2×C10 — C2×C22⋊D20
 Upper central C1 — C23 — C2×C22⋊C4

Generators and relations for C2×C22⋊D20
G = < a,b,c,d,e | a2=b2=c2=d20=e2=1, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=ebe=bc=cb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 3038 in 662 conjugacy classes, 143 normal (17 characteristic)
C1, C2, C2 [×6], C2 [×14], C4 [×6], C22, C22 [×10], C22 [×86], C5, C2×C4 [×4], C2×C4 [×8], D4 [×24], C23, C23 [×6], C23 [×88], D5 [×10], C10, C10 [×6], C10 [×4], C22⋊C4 [×4], C22⋊C4 [×8], C22×C4 [×2], C22×C4, C2×D4 [×24], C24, C24 [×19], Dic5 [×2], C20 [×4], D10 [×8], D10 [×66], C2×C10, C2×C10 [×10], C2×C10 [×12], C2×C22⋊C4, C2×C22⋊C4 [×2], C22≀C2 [×8], C22×D4 [×3], C25, D20 [×16], C2×Dic5 [×2], C2×Dic5 [×2], C5⋊D4 [×8], C2×C20 [×4], C2×C20 [×4], C22×D5 [×14], C22×D5 [×70], C22×C10, C22×C10 [×6], C22×C10 [×4], C2×C22≀C2, D10⋊C4 [×8], C5×C22⋊C4 [×4], C2×D20 [×8], C2×D20 [×8], C22×Dic5, C2×C5⋊D4 [×4], C2×C5⋊D4 [×4], C22×C20 [×2], C23×D5, C23×D5 [×6], C23×D5 [×12], C23×C10, C22⋊D20 [×8], C2×D10⋊C4 [×2], C10×C22⋊C4, C22×D20 [×2], C22×C5⋊D4, D5×C24, C2×C22⋊D20
Quotients: C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], D5, C2×D4 [×18], C24, D10 [×7], C22≀C2 [×4], C22×D4 [×3], D20 [×4], C22×D5 [×7], C2×C22≀C2, C2×D20 [×6], D4×D5 [×4], C23×D5, C22⋊D20 [×4], C22×D20, C2×D4×D5 [×2], C2×C22⋊D20

Smallest permutation representation of C2×C22⋊D20
On 80 points
Generators in S80
(1 50)(2 51)(3 52)(4 53)(5 54)(6 55)(7 56)(8 57)(9 58)(10 59)(11 60)(12 41)(13 42)(14 43)(15 44)(16 45)(17 46)(18 47)(19 48)(20 49)(21 64)(22 65)(23 66)(24 67)(25 68)(26 69)(27 70)(28 71)(29 72)(30 73)(31 74)(32 75)(33 76)(34 77)(35 78)(36 79)(37 80)(38 61)(39 62)(40 63)
(1 74)(2 51)(3 76)(4 53)(5 78)(6 55)(7 80)(8 57)(9 62)(10 59)(11 64)(12 41)(13 66)(14 43)(15 68)(16 45)(17 70)(18 47)(19 72)(20 49)(21 60)(22 65)(23 42)(24 67)(25 44)(26 69)(27 46)(28 71)(29 48)(30 73)(31 50)(32 75)(33 52)(34 77)(35 54)(36 79)(37 56)(38 61)(39 58)(40 63)
(1 31)(2 32)(3 33)(4 34)(5 35)(6 36)(7 37)(8 38)(9 39)(10 40)(11 21)(12 22)(13 23)(14 24)(15 25)(16 26)(17 27)(18 28)(19 29)(20 30)(41 65)(42 66)(43 67)(44 68)(45 69)(46 70)(47 71)(48 72)(49 73)(50 74)(51 75)(52 76)(53 77)(54 78)(55 79)(56 80)(57 61)(58 62)(59 63)(60 64)
(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 49)(2 48)(3 47)(4 46)(5 45)(6 44)(7 43)(8 42)(9 41)(10 60)(11 59)(12 58)(13 57)(14 56)(15 55)(16 54)(17 53)(18 52)(19 51)(20 50)(21 63)(22 62)(23 61)(24 80)(25 79)(26 78)(27 77)(28 76)(29 75)(30 74)(31 73)(32 72)(33 71)(34 70)(35 69)(36 68)(37 67)(38 66)(39 65)(40 64)

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

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

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

68 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 2L ··· 2S 2T 2U 4A 4B 4C 4D 4E 4F 5A 5B 10A ··· 10N 10O ··· 10V 20A ··· 20P order 1 2 ··· 2 2 2 2 2 2 ··· 2 2 2 4 4 4 4 4 4 5 5 10 ··· 10 10 ··· 10 20 ··· 20 size 1 1 ··· 1 2 2 2 2 10 ··· 10 20 20 4 4 4 4 20 20 2 2 2 ··· 2 4 ··· 4 4 ··· 4

68 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 type + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 D4 D4 D5 D10 D10 D10 D20 D4×D5 kernel C2×C22⋊D20 C22⋊D20 C2×D10⋊C4 C10×C22⋊C4 C22×D20 C22×C5⋊D4 D5×C24 C22×D5 C22×C10 C2×C22⋊C4 C22⋊C4 C22×C4 C24 C23 C22 # reps 1 8 2 1 2 1 1 8 4 2 8 4 2 16 8

Matrix representation of C2×C22⋊D20 in GL6(𝔽41)

 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 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 9 40
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 25 39 0 0 0 0 2 13 0 0 0 0 0 0 26 4 0 0 0 0 5 15 0 0 0 0 0 0 21 9 0 0 0 0 1 20
,
 25 39 0 0 0 0 25 16 0 0 0 0 0 0 15 37 0 0 0 0 15 26 0 0 0 0 0 0 20 32 0 0 0 0 17 21

G:=sub<GL(6,GF(41))| [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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,9,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[25,2,0,0,0,0,39,13,0,0,0,0,0,0,26,5,0,0,0,0,4,15,0,0,0,0,0,0,21,1,0,0,0,0,9,20],[25,25,0,0,0,0,39,16,0,0,0,0,0,0,15,15,0,0,0,0,37,26,0,0,0,0,0,0,20,17,0,0,0,0,32,21] >;

C2×C22⋊D20 in GAP, Magma, Sage, TeX

C_2\times C_2^2\rtimes D_{20}
% in TeX

G:=Group("C2xC2^2:D20");
// GroupNames label

G:=SmallGroup(320,1159);
// by ID

G=gap.SmallGroup(320,1159);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,675,297,80,12550]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^20=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,d*b*d^-1=e*b*e=b*c=c*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations

׿
×
𝔽