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## G = C5×C23.7D4order 320 = 26·5

### Direct product of C5 and C23.7D4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C23 — C5×C23.7D4
 Chief series C1 — C2 — C22 — C23 — C22×C10 — C22×C20 — C5×C22.D4 — C5×C23.7D4
 Lower central C1 — C2 — C23 — C5×C23.7D4
 Upper central C1 — C10 — C22×C10 — C5×C23.7D4

Generators and relations for C5×C23.7D4
G = < a,b,c,d,e,f | a5=b2=c2=d2=e4=1, f2=d, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, ebe-1=fbf-1=bcd, ece-1=cd=dc, cf=fc, de=ed, df=fd, fef-1=de-1 >

Subgroups: 322 in 160 conjugacy classes, 54 normal (12 characteristic)
C1, C2, C2, C4, C22, C22, C5, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C10, C10, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, C4○D4, C20, C2×C10, C2×C10, C23⋊C4, C22.D4, 2+ 1+4, C2×C20, C2×C20, C5×D4, C5×Q8, C22×C10, C22×C10, C22×C10, C23.7D4, C5×C22⋊C4, C5×C22⋊C4, C5×C4⋊C4, C22×C20, D4×C10, D4×C10, C5×C4○D4, C5×C23⋊C4, C5×C22.D4, C5×2+ 1+4, C5×C23.7D4
Quotients: C1, C2, C22, C5, D4, C23, C10, C2×D4, C2×C10, C22≀C2, C5×D4, C22×C10, C23.7D4, D4×C10, C5×C22≀C2, C5×C23.7D4

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

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

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

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

80 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A ··· 4E 4F 4G 4H 5A 5B 5C 5D 10A 10B 10C 10D 10E ··· 10P 10Q ··· 10AB 20A ··· 20T 20U ··· 20AF order 1 2 2 2 2 2 2 2 4 ··· 4 4 4 4 5 5 5 5 10 10 10 10 10 ··· 10 10 ··· 10 20 ··· 20 20 ··· 20 size 1 1 2 2 2 4 4 4 4 ··· 4 8 8 8 1 1 1 1 1 1 1 1 2 ··· 2 4 ··· 4 4 ··· 4 8 ··· 8

80 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 4 4 type + + + + + + image C1 C2 C2 C2 C5 C10 C10 C10 D4 D4 C5×D4 C5×D4 C23.7D4 C5×C23.7D4 kernel C5×C23.7D4 C5×C23⋊C4 C5×C22.D4 C5×2+ 1+4 C23.7D4 C23⋊C4 C22.D4 2+ 1+4 C2×C20 C22×C10 C2×C4 C23 C5 C1 # reps 1 3 3 1 4 12 12 4 3 3 12 12 2 8

Matrix representation of C5×C23.7D4 in GL4(𝔽41) generated by

 18 0 0 0 0 18 0 0 0 0 18 0 0 0 0 18
,
 1 0 0 0 0 40 0 0 0 0 40 0 0 0 0 1
,
 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0
,
 40 0 0 0 0 40 0 0 0 0 40 0 0 0 0 40
,
 25 25 25 16 16 16 25 16 25 16 25 25 25 16 16 16
,
 25 25 25 16 25 25 16 25 25 16 25 25 16 25 25 25
G:=sub<GL(4,GF(41))| [18,0,0,0,0,18,0,0,0,0,18,0,0,0,0,18],[1,0,0,0,0,40,0,0,0,0,40,0,0,0,0,1],[0,0,0,1,0,0,1,0,0,1,0,0,1,0,0,0],[40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[25,16,25,25,25,16,16,16,25,25,25,16,16,16,25,16],[25,25,25,16,25,25,16,25,25,16,25,25,16,25,25,25] >;

C5×C23.7D4 in GAP, Magma, Sage, TeX

C_5\times C_2^3._7D_4
% in TeX

G:=Group("C5xC2^3.7D4");
// GroupNames label

G:=SmallGroup(320,959);
// by ID

G=gap.SmallGroup(320,959);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,1120,589,1766,1768,5052]);
// Polycyclic

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

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