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## G = C5×C23.D4order 320 = 26·5

### Direct product of C5 and C23.D4

direct product, metabelian, nilpotent (class 4), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

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

Generators and relations for C5×C23.D4
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, ece-1=fcf-1=cd=dc, de=ed, df=fd, fef-1=be3 >

Subgroups: 162 in 68 conjugacy classes, 26 normal (all characteristic)
C1, C2, C2 [×3], C4 [×4], C22, C22 [×4], C5, C8, C2×C4, C2×C4 [×4], D4, C23 [×2], C10, C10 [×3], C22⋊C4, C22⋊C4 [×2], C4⋊C4, M4(2), C22×C4, C2×D4, C20 [×4], C2×C10, C2×C10 [×4], C23⋊C4, C4.D4, C22.D4, C40, C2×C20, C2×C20 [×4], C5×D4, C22×C10 [×2], C23.D4, C5×C22⋊C4, C5×C22⋊C4 [×2], C5×C4⋊C4, C5×M4(2), C22×C20, D4×C10, C5×C23⋊C4, C5×C4.D4, C5×C22.D4, C5×C23.D4
Quotients: C1, C2 [×3], C4 [×2], C22, C5, C2×C4, D4 [×2], C10 [×3], C22⋊C4, C20 [×2], C2×C10, C23⋊C4, C2×C20, C5×D4 [×2], C23.D4, C5×C22⋊C4, C5×C23⋊C4, C5×C23.D4

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

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

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

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

65 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 4F 5A 5B 5C 5D 8A 8B 10A 10B 10C 10D 10E 10F 10G 10H 10I ··· 10P 20A ··· 20L 20M ··· 20X 40A ··· 40H order 1 2 2 2 2 4 4 4 4 4 4 5 5 5 5 8 8 10 10 10 10 10 10 10 10 10 ··· 10 20 ··· 20 20 ··· 20 40 ··· 40 size 1 1 2 4 4 4 4 4 8 8 8 1 1 1 1 8 8 1 1 1 1 2 2 2 2 4 ··· 4 4 ··· 4 8 ··· 8 8 ··· 8

65 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 4 4 4 4 type + + + + + + + image C1 C2 C2 C2 C4 C4 C5 C10 C10 C10 C20 C20 D4 D4 C5×D4 C5×D4 C23⋊C4 C23.D4 C5×C23⋊C4 C5×C23.D4 kernel C5×C23.D4 C5×C23⋊C4 C5×C4.D4 C5×C22.D4 C5×C22⋊C4 C22×C20 C23.D4 C23⋊C4 C4.D4 C22.D4 C22⋊C4 C22×C4 C2×C20 C22×C10 C2×C4 C23 C10 C5 C2 C1 # reps 1 1 1 1 2 2 4 4 4 4 8 8 1 1 4 4 1 2 4 8

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

 16 0 0 0 0 16 0 0 0 0 16 0 0 0 0 16
,
 0 1 0 0 1 0 0 0 0 0 0 40 0 0 40 0
,
 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0
,
 40 0 0 0 0 40 0 0 0 0 40 0 0 0 0 40
,
 16 16 16 25 25 25 16 25 16 25 25 25 16 25 16 16
,
 0 0 0 40 0 0 1 0 1 0 0 0 0 40 0 0
G:=sub<GL(4,GF(41))| [16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[0,1,0,0,1,0,0,0,0,0,0,40,0,0,40,0],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0],[40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[16,25,16,16,16,25,25,25,16,16,25,16,25,25,25,16],[0,0,1,0,0,0,0,40,0,1,0,0,40,0,0,0] >;

C5×C23.D4 in GAP, Magma, Sage, TeX

C_5\times C_2^3.D_4
% in TeX

G:=Group("C5xC2^3.D4");
// GroupNames label

G:=SmallGroup(320,157);
// by ID

G=gap.SmallGroup(320,157);
# by ID

G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-2,280,309,1128,2803,2111,375,10085]);
// 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,e*c*e^-1=f*c*f^-1=c*d=d*c,d*e=e*d,d*f=f*d,f*e*f^-1=b*e^3>;
// generators/relations

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