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## G = C5×C23.31D4order 320 = 26·5

### Direct product of C5 and C23.31D4

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C4 — C5×C23.31D4
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C22×C20 — C5×C2.C42 — C5×C23.31D4
 Lower central C1 — C22 — C2×C4 — C5×C23.31D4
 Upper central C1 — C2×C10 — C22×C20 — C5×C23.31D4

Generators and relations for C5×C23.31D4
G = < a,b,c,d,e,f | a5=b2=c2=d2=1, e4=d, f2=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, ebe-1=bcd, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=bcde3 >

Subgroups: 162 in 80 conjugacy classes, 34 normal (all characteristic)
C1, C2, C2, C4, C22, C22, C5, C8, C2×C4, C2×C4, Q8, C23, C10, C10, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×Q8, C20, C2×C10, C2×C10, C2.C42, C22⋊C8, C22⋊Q8, C40, C2×C20, C2×C20, C5×Q8, C22×C10, C23.31D4, C5×C22⋊C4, C5×C4⋊C4, C5×C4⋊C4, C2×C40, C22×C20, C22×C20, Q8×C10, C5×C2.C42, C5×C22⋊C8, C5×C22⋊Q8, C5×C23.31D4
Quotients: C1, C2, C4, C22, C5, C2×C4, D4, C10, C22⋊C4, SD16, Q16, C20, C2×C10, C23⋊C4, Q8⋊C4, C4≀C2, C2×C20, C5×D4, C23.31D4, C5×C22⋊C4, C5×SD16, C5×Q16, C5×C23⋊C4, C5×Q8⋊C4, C5×C4≀C2, C5×C23.31D4

Smallest permutation representation of C5×C23.31D4
On 80 points
Generators in S80
(1 41 77 39 69)(2 42 78 40 70)(3 43 79 33 71)(4 44 80 34 72)(5 45 73 35 65)(6 46 74 36 66)(7 47 75 37 67)(8 48 76 38 68)(9 25 63 17 55)(10 26 64 18 56)(11 27 57 19 49)(12 28 58 20 50)(13 29 59 21 51)(14 30 60 22 52)(15 31 61 23 53)(16 32 62 24 54)
(2 16)(4 10)(6 12)(8 14)(18 34)(20 36)(22 38)(24 40)(26 44)(28 46)(30 48)(32 42)(50 66)(52 68)(54 70)(56 72)(58 74)(60 76)(62 78)(64 80)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 9)(8 10)(17 37)(18 38)(19 39)(20 40)(21 33)(22 34)(23 35)(24 36)(25 47)(26 48)(27 41)(28 42)(29 43)(30 44)(31 45)(32 46)(49 69)(50 70)(51 71)(52 72)(53 65)(54 66)(55 67)(56 68)(57 77)(58 78)(59 79)(60 80)(61 73)(62 74)(63 75)(64 76)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)(33 37)(34 38)(35 39)(36 40)(41 45)(42 46)(43 47)(44 48)(49 53)(50 54)(51 55)(52 56)(57 61)(58 62)(59 63)(60 64)(65 69)(66 70)(67 71)(68 72)(73 77)(74 78)(75 79)(76 80)
(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 5)(2 8 16 14)(3 9)(4 12 10 6)(7 13)(11 15)(17 33)(18 36 34 20)(19 23)(21 37)(22 40 38 24)(25 43)(26 46 44 28)(27 31)(29 47)(30 42 48 32)(35 39)(41 45)(49 53)(50 56 66 72)(51 67)(52 70 68 54)(55 71)(57 61)(58 64 74 80)(59 75)(60 78 76 62)(63 79)(65 69)(73 77)

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

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

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

95 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C ··· 4G 4H 4I 5A 5B 5C 5D 8A 8B 8C 8D 10A ··· 10L 10M ··· 10T 20A ··· 20H 20I ··· 20AB 20AC ··· 20AJ 40A ··· 40P order 1 2 2 2 2 2 4 4 4 ··· 4 4 4 5 5 5 5 8 8 8 8 10 ··· 10 10 ··· 10 20 ··· 20 20 ··· 20 20 ··· 20 40 ··· 40 size 1 1 1 1 2 2 2 2 4 ··· 4 8 8 1 1 1 1 4 4 4 4 1 ··· 1 2 ··· 2 2 ··· 2 4 ··· 4 8 ··· 8 4 ··· 4

95 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + - + image C1 C2 C2 C2 C4 C4 C5 C10 C10 C10 C20 C20 D4 D4 SD16 Q16 C4≀C2 C5×D4 C5×D4 C5×SD16 C5×Q16 C5×C4≀C2 C23⋊C4 C5×C23⋊C4 kernel C5×C23.31D4 C5×C2.C42 C5×C22⋊C8 C5×C22⋊Q8 C5×C4⋊C4 Q8×C10 C23.31D4 C2.C42 C22⋊C8 C22⋊Q8 C4⋊C4 C2×Q8 C2×C20 C22×C10 C2×C10 C2×C10 C10 C2×C4 C23 C22 C22 C2 C10 C2 # reps 1 1 1 1 2 2 4 4 4 4 8 8 1 1 2 2 4 4 4 8 8 16 1 4

Matrix representation of C5×C23.31D4 in GL4(𝔽41) generated by

 18 0 0 0 0 18 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 1 0 0 0 35 40
,
 40 0 0 0 0 40 0 0 0 0 1 0 0 0 0 1
,
 40 0 0 0 0 40 0 0 0 0 40 0 0 0 0 40
,
 26 15 0 0 26 26 0 0 0 0 27 9 0 0 2 14
,
 40 0 0 0 0 1 0 0 0 0 40 0 0 0 17 32
G:=sub<GL(4,GF(41))| [18,0,0,0,0,18,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,1,35,0,0,0,40],[40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[26,26,0,0,15,26,0,0,0,0,27,2,0,0,9,14],[40,0,0,0,0,1,0,0,0,0,40,17,0,0,0,32] >;

C5×C23.31D4 in GAP, Magma, Sage, TeX

C_5\times C_2^3._{31}D_4
% in TeX

G:=Group("C5xC2^3.31D4");
// GroupNames label

G:=SmallGroup(320,133);
// by ID

G=gap.SmallGroup(320,133);
# by ID

G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-2,280,309,568,2803,2530,248,4911]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^5=b^2=c^2=d^2=1,e^4=d,f^2=b,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,e*b*e^-1=b*c*d,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=b*c*d*e^3>;
// generators/relations

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