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G = C2×C23⋊Dic5order 320 = 26·5

Direct product of C2 and C23⋊Dic5

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2×C23⋊Dic5, C243Dic5, C24.17D10, (D4×C10)⋊25C4, (C23×C10)⋊4C4, (C2×D4)⋊5Dic5, C104(C23⋊C4), (C22×C20)⋊17C4, (C22×D4).5D5, C231(C2×Dic5), (C22×C4)⋊3Dic5, (C2×D4).200D10, (C22×C10).108D4, C23.64(C5⋊D4), C23.D543C22, C23.76(C22×D5), (D4×C10).280C22, (C23×C10).45C22, C22.1(C23.D5), C22.6(C22×Dic5), (C22×C10).115C23, C56(C2×C23⋊C4), (C2×C20)⋊15(C2×C4), (D4×C2×C10).10C2, (C2×C4)⋊1(C2×Dic5), (C22×C10)⋊7(C2×C4), (C2×C10).38(C2×D4), (C2×C23.D5)⋊8C2, C22.10(C2×C5⋊D4), C2.10(C2×C23.D5), C10.115(C2×C22⋊C4), (C2×C10).296(C22×C4), (C2×C10).177(C22⋊C4), SmallGroup(320,846)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C2×C23⋊Dic5
C1C5C10C2×C10C22×C10C23.D5C2×C23.D5 — C2×C23⋊Dic5
C5C10C2×C10 — C2×C23⋊Dic5
C1C22C24C22×D4

Generators and relations for C2×C23⋊Dic5
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e10=1, f2=e5, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, ebe-1=bd=db, fbf-1=bcd, fcf-1=cd=dc, ce=ec, de=ed, df=fd, fef-1=e-1 >

Subgroups: 670 in 210 conjugacy classes, 71 normal (25 characteristic)
C1, C2, C2 [×2], C2 [×8], C4 [×6], C22 [×3], C22 [×4], C22 [×18], C5, C2×C4 [×2], C2×C4 [×10], D4 [×8], C23 [×3], C23 [×6], C23 [×6], C10, C10 [×2], C10 [×8], C22⋊C4 [×6], C22×C4, C22×C4 [×2], C2×D4 [×4], C2×D4 [×4], C24 [×2], Dic5 [×4], C20 [×2], C2×C10 [×3], C2×C10 [×4], C2×C10 [×18], C23⋊C4 [×4], C2×C22⋊C4 [×2], C22×D4, C2×Dic5 [×8], C2×C20 [×2], C2×C20 [×2], C5×D4 [×8], C22×C10 [×3], C22×C10 [×6], C22×C10 [×6], C2×C23⋊C4, C23.D5 [×4], C23.D5 [×2], C22×Dic5 [×2], C22×C20, D4×C10 [×4], D4×C10 [×4], C23×C10 [×2], C23⋊Dic5 [×4], C2×C23.D5 [×2], D4×C2×C10, C2×C23⋊Dic5
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, D5, C22⋊C4 [×4], C22×C4, C2×D4 [×2], Dic5 [×4], D10 [×3], C23⋊C4 [×2], C2×C22⋊C4, C2×Dic5 [×6], C5⋊D4 [×4], C22×D5, C2×C23⋊C4, C23.D5 [×4], C22×Dic5, C2×C5⋊D4 [×2], C23⋊Dic5 [×2], C2×C23.D5, C2×C23⋊Dic5

Smallest permutation representation of C2×C23⋊Dic5
On 80 points
Generators in S80
(1 52)(2 53)(3 54)(4 55)(5 56)(6 57)(7 58)(8 59)(9 60)(10 51)(11 45)(12 46)(13 47)(14 48)(15 49)(16 50)(17 41)(18 42)(19 43)(20 44)(21 61)(22 62)(23 63)(24 64)(25 65)(26 66)(27 67)(28 68)(29 69)(30 70)(31 71)(32 72)(33 73)(34 74)(35 75)(36 76)(37 77)(38 78)(39 79)(40 80)
(1 13)(2 77)(3 15)(4 79)(5 17)(6 71)(7 19)(8 73)(9 11)(10 75)(12 21)(14 23)(16 25)(18 27)(20 29)(22 76)(24 78)(26 80)(28 72)(30 74)(31 57)(32 68)(33 59)(34 70)(35 51)(36 62)(37 53)(38 64)(39 55)(40 66)(41 56)(42 67)(43 58)(44 69)(45 60)(46 61)(47 52)(48 63)(49 54)(50 65)
(1 6)(2 7)(3 8)(4 9)(5 10)(11 79)(12 80)(13 71)(14 72)(15 73)(16 74)(17 75)(18 76)(19 77)(20 78)(21 26)(22 27)(23 28)(24 29)(25 30)(31 47)(32 48)(33 49)(34 50)(35 41)(36 42)(37 43)(38 44)(39 45)(40 46)(51 56)(52 57)(53 58)(54 59)(55 60)(61 66)(62 67)(63 68)(64 69)(65 70)
(1 22)(2 23)(3 24)(4 25)(5 26)(6 27)(7 28)(8 29)(9 30)(10 21)(11 74)(12 75)(13 76)(14 77)(15 78)(16 79)(17 80)(18 71)(19 72)(20 73)(31 42)(32 43)(33 44)(34 45)(35 46)(36 47)(37 48)(38 49)(39 50)(40 41)(51 61)(52 62)(53 63)(54 64)(55 65)(56 66)(57 67)(58 68)(59 69)(60 70)
(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 47 6 42)(2 46 7 41)(3 45 8 50)(4 44 9 49)(5 43 10 48)(11 59 16 54)(12 58 17 53)(13 57 18 52)(14 56 19 51)(15 55 20 60)(21 37 26 32)(22 36 27 31)(23 35 28 40)(24 34 29 39)(25 33 30 38)(61 77 66 72)(62 76 67 71)(63 75 68 80)(64 74 69 79)(65 73 70 78)

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

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

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

62 conjugacy classes

class 1 2A2B2C2D···2I2J2K4A4B4C···4J5A5B10A···10N10O···10AD20A···20H
order12222···222444···45510···1010···1020···20
size11112···2444420···20222···24···44···4

62 irreducible representations

dim11111112222222244
type++++++--+-++
imageC1C2C2C2C4C4C4D4D5Dic5Dic5D10Dic5D10C5⋊D4C23⋊C4C23⋊Dic5
kernelC2×C23⋊Dic5C23⋊Dic5C2×C23.D5D4×C2×C10C22×C20D4×C10C23×C10C22×C10C22×D4C22×C4C2×D4C2×D4C24C24C23C10C2
# reps142124242244221628

Matrix representation of C2×C23⋊Dic5 in GL6(𝔽41)

4000000
0400000
001000
000100
000010
000001
,
120000
0400000
001000
000100
00390400
00039040
,
4000000
0400000
0018600
00352300
0000186
00003523
,
100000
010000
0040000
0004000
0000400
0000040
,
100000
010000
00618618
0023212321
00003523
00001820
,
9180000
32320000
00161234
00040329
0039294035
000201

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,2,40,0,0,0,0,0,0,1,0,39,0,0,0,0,1,0,39,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,18,35,0,0,0,0,6,23,0,0,0,0,0,0,18,35,0,0,0,0,6,23],[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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,6,23,0,0,0,0,18,21,0,0,0,0,6,23,35,18,0,0,18,21,23,20],[9,32,0,0,0,0,18,32,0,0,0,0,0,0,1,0,39,0,0,0,6,40,29,2,0,0,12,3,40,0,0,0,34,29,35,1] >;

C2×C23⋊Dic5 in GAP, Magma, Sage, TeX

C_2\times C_2^3\rtimes {\rm Dic}_5
% in TeX

G:=Group("C2xC2^3:Dic5");
// GroupNames label

G:=SmallGroup(320,846);
// by ID

G=gap.SmallGroup(320,846);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,422,297,1684,12550]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^10=1,f^2=e^5,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,e*b*e^-1=b*d=d*b,f*b*f^-1=b*c*d,f*c*f^-1=c*d=d*c,c*e=e*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^-1>;
// generators/relations

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