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G = C23⋊2Dic10  order 320 = 26·5

2nd semidirect product of C23 and Dic10 acting via Dic10/C10=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C23⋊2Dic10, C24.23D10, C10.12+ (1+4), (C22×C10)⋊5Q8, C4⋊Dic5⋊3C22, C5⋊1(C23⋊2Q8), C20.48D4⋊3C2, C10.6(C22×Q8), (C2×C10).27C24, C22⋊C4.86D10, C2.6(D4⋊6D10), (C2×C20).127C23, (C2×Dic10)⋊2C22, (C22×C4).169D10, C10.D4⋊1C22, (C2×Dic5).8C23, C2.8(C22×Dic10), C22.5(C2×Dic10), C22.69(C23×D5), Dic5.14D4⋊1C2, (C22×C20).71C22, (C23×C10).53C22, C23.144(C22×D5), C23.D5.85C22, (C22×C10).119C23, (C22×Dic5).76C22, (C2×C10).49(C2×Q8), (C2×C22⋊C4).18D5, (C10×C22⋊C4).18C2, (C2×C4).133(C22×D5), (C2×C23.D5).22C2, (C5×C22⋊C4).97C22, SmallGroup(320,1155)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C10 — C23⋊2Dic10
Chief series
C1 — C5 — C10 — C2×C10 — C2×Dic5 — C22×Dic5 — C2×C23.D5 — C23⋊2Dic10
Lower central
C5 — C2×C10 — C23⋊2Dic10

Subgroups: 782 in 242 conjugacy classes, 111 normal (13 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×12], C22, C22 [×6], C22 [×10], C5, C2×C4 [×4], C2×C4 [×14], Q8 [×4], C23, C23 [×6], C23 [×2], C10, C10 [×2], C10 [×6], C22⋊C4 [×4], C22⋊C4 [×8], C4⋊C4 [×12], C22×C4 [×2], C22×C4 [×4], C2×Q8 [×4], C24, Dic5 [×8], C20 [×4], C2×C10, C2×C10 [×6], C2×C10 [×10], C2×C22⋊C4, C2×C22⋊C4 [×2], C22⋊Q8 [×12], Dic10 [×4], C2×Dic5 [×8], C2×Dic5 [×4], C2×C20 [×4], C2×C20 [×2], C22×C10, C22×C10 [×6], C22×C10 [×2], C23⋊2Q8, C10.D4 [×8], C4⋊Dic5 [×4], C23.D5 [×8], C5×C22⋊C4 [×4], C2×Dic10 [×4], C22×Dic5 [×4], C22×C20 [×2], C23×C10, Dic5.14D4 [×8], C20.48D4 [×4], C2×C23.D5 [×2], C10×C22⋊C4, C23⋊2Dic10

Quotients:
C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], D5, C2×Q8 [×6], C24, D10 [×7], C22×Q8, 2+ (1+4) [×2], Dic10 [×4], C22×D5 [×7], C23⋊2Q8, C2×Dic10 [×6], C23×D5, C22×Dic10, D4⋊6D10 [×2], C23⋊2Dic10

Generators and relations
 G = < a,b,c,d,e | a2=b2=c2=d20=1, e2=d10, ab=ba, dad-1=ac=ca, ae=ea, ebe-1=bc=cb, bd=db, cd=dc, ce=ec, ede-1=d-1 >

Smallest permutation representation
►On 80 points
Generators in S80
(2 80)(4 62)(6 64)(8 66)(10 68)(12 70)(14 72)(16 74)(18 76)(20 78)(22 43)(24 45)(26 47)(28 49)(30 51)(32 53)(34 55)(36 57)(38 59)(40 41)

(21 42)(22 43)(23 44)(24 45)(25 46)(26 47)(27 48)(28 49)(29 50)(30 51)(31 52)(32 53)(33 54)(34 55)(35 56)(36 57)(37 58)(38 59)(39 60)(40 41)

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

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

Matrix representation ►G ⊆ GL6(𝔽41)

4000000
0400000
001000
0004000
000010
0000040
,
100000
010000
001000
000100
0000400
0000040
,
100000
010000
0040000
0004000
0000400
0000040
,
0400000
100000
0001800
0023000
0000025
0000160
,
1300000
30400000
000010
000001
0040000
0004000

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,40],[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],[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],[0,1,0,0,0,0,40,0,0,0,0,0,0,0,0,23,0,0,0,0,18,0,0,0,0,0,0,0,0,16,0,0,0,0,25,0],[1,30,0,0,0,0,30,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,1,0,0,0,0,0,0,1,0,0] >;
 

62 conjugacy classes

class 1 2A2B2C2D···2I4A4B4C4D4E···4L5A5B10A···10N10O···10V20A···20P
order12222···244444···45510···1010···1020···20
size11112···2444420···20222···24···44···4

62 irreducible representations

dim1111122222244
type+++++-++++-+
imageC1C2C2C2C2Q8D5D10D10D10Dic102+ (1+4)D4⋊6D10
kernelC23⋊2Dic10Dic5.14D4C20.48D4C2×C23.D5C10×C22⋊C4C22×C10C2×C22⋊C4C22⋊C4C22×C4C24C23C10C2
# reps18421428421628

In GAP, Magma, Sage, TeX 

C_2^3\rtimes_2Dic_{10}
 
% in TeX
 
G:=Group("C2^3:2Dic10");
 
// GroupNames label
 
G:=SmallGroup(320,1155);
 
// by ID
 
G=gap.SmallGroup(320,1155);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,758,675,570,80,12550]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^20=1,e^2=d^10,a*b=b*a,d*a*d^-1=a*c=c*a,a*e=e*a,e*b*e^-1=b*c=c*b,b*d=d*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
 
// generators/relations
 

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