Copied to
clipboard

## G = (C2×D20)⋊25C4order 320 = 26·5

### 10th semidirect product of C2×D20 and C4 acting via C4/C2=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — (C2×D20)⋊25C4
 Chief series C1 — C5 — C10 — C2×C10 — C22×C10 — C2×C5⋊D4 — C2×C4○D20 — (C2×D20)⋊25C4
 Lower central C5 — C10 — C2×C10 — (C2×D20)⋊25C4
 Upper central C1 — C4 — C22×C4 — C42⋊C2

Generators and relations for (C2×D20)⋊25C4
G = < a,b,c,d | a2=b20=c2=d4=1, ab=ba, dcd-1=ac=ca, dad-1=ab10, cbc=b-1, bd=db >

Subgroups: 638 in 158 conjugacy classes, 59 normal (41 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, Dic5, C20, C20, D10, C2×C10, C2×C10, C23⋊C4, C42⋊C2, C42⋊C2, C2×C4○D4, Dic10, C4×D5, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×C10, C23.C23, C4×Dic5, C4⋊Dic5, C23.D5, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×Dic10, C2×C4×D5, C2×D20, C4○D20, C2×C5⋊D4, C22×C20, C23.1D10, C23.21D10, C5×C42⋊C2, C2×C4○D20, (C2×D20)⋊25C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C22⋊C4, C22×C4, C2×D4, D10, C2×C22⋊C4, C4×D5, D20, C5⋊D4, C22×D5, C23.C23, D10⋊C4, C2×C4×D5, C2×D20, C2×C5⋊D4, C2×D10⋊C4, (C2×D20)⋊25C4

Smallest permutation representation of (C2×D20)⋊25C4
On 80 points
Generators in S80
(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 70)(42 71)(43 72)(44 73)(45 74)(46 75)(47 76)(48 77)(49 78)(50 79)(51 80)(52 61)(53 62)(54 63)(55 64)(56 65)(57 66)(58 67)(59 68)(60 69)
(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 43)(2 42)(3 41)(4 60)(5 59)(6 58)(7 57)(8 56)(9 55)(10 54)(11 53)(12 52)(13 51)(14 50)(15 49)(16 48)(17 47)(18 46)(19 45)(20 44)(21 62)(22 61)(23 80)(24 79)(25 78)(26 77)(27 76)(28 75)(29 74)(30 73)(31 72)(32 71)(33 70)(34 69)(35 68)(36 67)(37 66)(38 65)(39 64)(40 63)
(1 49 21 78)(2 50 22 79)(3 51 23 80)(4 52 24 61)(5 53 25 62)(6 54 26 63)(7 55 27 64)(8 56 28 65)(9 57 29 66)(10 58 30 67)(11 59 31 68)(12 60 32 69)(13 41 33 70)(14 42 34 71)(15 43 35 72)(16 44 36 73)(17 45 37 74)(18 46 38 75)(19 47 39 76)(20 48 40 77)

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

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

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

62 conjugacy classes

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

62 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + image C1 C2 C2 C2 C2 C4 C4 C4 D4 D5 D10 D10 C4×D5 D20 C5⋊D4 C23.C23 (C2×D20)⋊25C4 kernel (C2×D20)⋊25C4 C23.1D10 C23.21D10 C5×C42⋊C2 C2×C4○D20 C2×Dic10 C2×C4×D5 C2×D20 C2×C20 C42⋊C2 C22⋊C4 C22×C4 C2×C4 C2×C4 C2×C4 C5 C1 # reps 1 4 1 1 1 2 4 2 4 2 4 2 8 8 8 2 8

Matrix representation of (C2×D20)⋊25C4 in GL4(𝔽41) generated by

 24 40 24 40 1 17 1 17 0 0 17 1 0 0 40 24
,
 9 30 0 0 11 14 0 0 0 0 9 30 0 0 11 14
,
 38 38 38 38 17 3 17 3 6 6 3 3 7 35 24 38
,
 17 1 8 21 40 24 20 32 7 39 24 40 2 34 1 17
G:=sub<GL(4,GF(41))| [24,1,0,0,40,17,0,0,24,1,17,40,40,17,1,24],[9,11,0,0,30,14,0,0,0,0,9,11,0,0,30,14],[38,17,6,7,38,3,6,35,38,17,3,24,38,3,3,38],[17,40,7,2,1,24,39,34,8,20,24,1,21,32,40,17] >;

(C2×D20)⋊25C4 in GAP, Magma, Sage, TeX

(C_2\times D_{20})\rtimes_{25}C_4
% in TeX

G:=Group("(C2xD20):25C4");
// GroupNames label

G:=SmallGroup(320,633);
// by ID

G=gap.SmallGroup(320,633);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,422,58,1123,438,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^20=c^2=d^4=1,a*b=b*a,d*c*d^-1=a*c=c*a,d*a*d^-1=a*b^10,c*b*c=b^-1,b*d=d*b>;
// generators/relations

׿
×
𝔽