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## G = C2×C23⋊Dic5order 320 = 26·5

### Direct product of C2 and C23⋊Dic5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C2×C23⋊Dic5
 Chief series C1 — C5 — C10 — C2×C10 — C22×C10 — C23.D5 — C2×C23.D5 — C2×C23⋊Dic5
 Lower central C5 — C10 — C2×C10 — C2×C23⋊Dic5
 Upper central C1 — C22 — C24 — C22×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, C2, C4, C22, C22, C22, C5, C2×C4, C2×C4, D4, C23, C23, C23, C10, C10, C10, C22⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, Dic5, C20, C2×C10, C2×C10, C2×C10, C23⋊C4, C2×C22⋊C4, C22×D4, C2×Dic5, C2×C20, C2×C20, C5×D4, C22×C10, C22×C10, C22×C10, C2×C23⋊C4, C23.D5, C23.D5, C22×Dic5, C22×C20, D4×C10, D4×C10, C23×C10, C23⋊Dic5, C2×C23.D5, D4×C2×C10, C2×C23⋊Dic5
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C22⋊C4, C22×C4, C2×D4, Dic5, D10, C23⋊C4, C2×C22⋊C4, C2×Dic5, C5⋊D4, C22×D5, C2×C23⋊C4, C23.D5, C22×Dic5, C2×C5⋊D4, C23⋊Dic5, C2×C23.D5, C2×C23⋊Dic5

Smallest permutation representation of C2×C23⋊Dic5
On 80 points
Generators in S80
(1 60)(2 51)(3 52)(4 53)(5 54)(6 55)(7 56)(8 57)(9 58)(10 59)(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 55)(32 68)(33 57)(34 70)(35 59)(36 62)(37 51)(38 64)(39 53)(40 66)(41 54)(42 67)(43 56)(44 69)(45 58)(46 61)(47 60)(48 63)(49 52)(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 63)(52 64)(53 65)(54 66)(55 67)(56 68)(57 69)(58 70)(59 61)(60 62)
(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 57 16 52)(12 56 17 51)(13 55 18 60)(14 54 19 59)(15 53 20 58)(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)| (1,60)(2,51)(3,52)(4,53)(5,54)(6,55)(7,56)(8,57)(9,58)(10,59)(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)(31,55)(32,68)(33,57)(34,70)(35,59)(36,62)(37,51)(38,64)(39,53)(40,66)(41,54)(42,67)(43,56)(44,69)(45,58)(46,61)(47,60)(48,63)(49,52)(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)(51,63)(52,64)(53,65)(54,66)(55,67)(56,68)(57,69)(58,70)(59,61)(60,62), (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)(11,57,16,52)(12,56,17,51)(13,55,18,60)(14,54,19,59)(15,53,20,58)(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( (1,60)(2,51)(3,52)(4,53)(5,54)(6,55)(7,56)(8,57)(9,58)(10,59)(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)(31,55)(32,68)(33,57)(34,70)(35,59)(36,62)(37,51)(38,64)(39,53)(40,66)(41,54)(42,67)(43,56)(44,69)(45,58)(46,61)(47,60)(48,63)(49,52)(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)(51,63)(52,64)(53,65)(54,66)(55,67)(56,68)(57,69)(58,70)(59,61)(60,62), (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)(11,57,16,52)(12,56,17,51)(13,55,18,60)(14,54,19,59)(15,53,20,58)(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([[(1,60),(2,51),(3,52),(4,53),(5,54),(6,55),(7,56),(8,57),(9,58),(10,59),(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),(31,55),(32,68),(33,57),(34,70),(35,59),(36,62),(37,51),(38,64),(39,53),(40,66),(41,54),(42,67),(43,56),(44,69),(45,58),(46,61),(47,60),(48,63),(49,52),(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),(51,63),(52,64),(53,65),(54,66),(55,67),(56,68),(57,69),(58,70),(59,61),(60,62)], [(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),(11,57,16,52),(12,56,17,51),(13,55,18,60),(14,54,19,59),(15,53,20,58),(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 2A 2B 2C 2D ··· 2I 2J 2K 4A 4B 4C ··· 4J 5A 5B 10A ··· 10N 10O ··· 10AD 20A ··· 20H order 1 2 2 2 2 ··· 2 2 2 4 4 4 ··· 4 5 5 10 ··· 10 10 ··· 10 20 ··· 20 size 1 1 1 1 2 ··· 2 4 4 4 4 20 ··· 20 2 2 2 ··· 2 4 ··· 4 4 ··· 4

62 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + - - + - + + image C1 C2 C2 C2 C4 C4 C4 D4 D5 Dic5 Dic5 D10 Dic5 D10 C5⋊D4 C23⋊C4 C23⋊Dic5 kernel C2×C23⋊Dic5 C23⋊Dic5 C2×C23.D5 D4×C2×C10 C22×C20 D4×C10 C23×C10 C22×C10 C22×D4 C22×C4 C2×D4 C2×D4 C24 C24 C23 C10 C2 # reps 1 4 2 1 2 4 2 4 2 2 4 4 2 2 16 2 8

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

 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 2 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 39 0 40 0 0 0 0 39 0 40
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 18 6 0 0 0 0 35 23 0 0 0 0 0 0 18 6 0 0 0 0 35 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 18 6 18 0 0 23 21 23 21 0 0 0 0 35 23 0 0 0 0 18 20
,
 9 18 0 0 0 0 32 32 0 0 0 0 0 0 1 6 12 34 0 0 0 40 3 29 0 0 39 29 40 35 0 0 0 2 0 1

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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