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## G = C5×D42order 320 = 26·5

### Direct product of C5, D4 and D4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C5×D42
 Chief series C1 — C2 — C22 — C2×C10 — C22×C10 — D4×C10 — C5×C22≀C2 — C5×D42
 Lower central C1 — C22 — C5×D42
 Upper central C1 — C2×C10 — C5×D42

Generators and relations for C5×D42
G = < a,b,c,d,e | a5=b4=c2=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 778 in 428 conjugacy classes, 182 normal (14 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, D4, C23, C23, C10, C10, C10, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, C24, C20, C20, C2×C10, C2×C10, C2×C10, C4×D4, C22≀C2, C4⋊D4, C41D4, C22×D4, C2×C20, C2×C20, C2×C20, C5×D4, C5×D4, C22×C10, C22×C10, D42, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C22×C20, D4×C10, D4×C10, C23×C10, D4×C20, C5×C22≀C2, C5×C4⋊D4, C5×C41D4, D4×C2×C10, C5×D42
Quotients: C1, C2, C22, C5, D4, C23, C10, C2×D4, C24, C2×C10, C22×D4, 2+ 1+4, C5×D4, C22×C10, D42, D4×C10, C23×C10, D4×C2×C10, C5×2+ 1+4, C5×D42

Smallest permutation representation of C5×D42
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)
(1 16 21 7)(2 17 22 8)(3 18 23 9)(4 19 24 10)(5 20 25 6)(11 28 77 31)(12 29 78 32)(13 30 79 33)(14 26 80 34)(15 27 76 35)(36 67 41 75)(37 68 42 71)(38 69 43 72)(39 70 44 73)(40 66 45 74)(46 60 54 65)(47 56 55 61)(48 57 51 62)(49 58 52 63)(50 59 53 64)
(1 7)(2 8)(3 9)(4 10)(5 6)(11 28)(12 29)(13 30)(14 26)(15 27)(16 21)(17 22)(18 23)(19 24)(20 25)(31 77)(32 78)(33 79)(34 80)(35 76)(36 75)(37 71)(38 72)(39 73)(40 74)(41 67)(42 68)(43 69)(44 70)(45 66)(46 65)(47 61)(48 62)(49 63)(50 64)(51 57)(52 58)(53 59)(54 60)(55 56)
(1 41 35 47)(2 42 31 48)(3 43 32 49)(4 44 33 50)(5 45 34 46)(6 66 80 65)(7 67 76 61)(8 68 77 62)(9 69 78 63)(10 70 79 64)(11 57 17 71)(12 58 18 72)(13 59 19 73)(14 60 20 74)(15 56 16 75)(21 36 27 55)(22 37 28 51)(23 38 29 52)(24 39 30 53)(25 40 26 54)
(1 21)(2 22)(3 23)(4 24)(5 25)(6 20)(7 16)(8 17)(9 18)(10 19)(11 77)(12 78)(13 79)(14 80)(15 76)(26 34)(27 35)(28 31)(29 32)(30 33)(36 47)(37 48)(38 49)(39 50)(40 46)(41 55)(42 51)(43 52)(44 53)(45 54)(56 67)(57 68)(58 69)(59 70)(60 66)(61 75)(62 71)(63 72)(64 73)(65 74)

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

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

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

125 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2K 2L 2M 2N 2O 4A 4B 4C 4D 4E ··· 4I 5A 5B 5C 5D 10A ··· 10L 10M ··· 10AR 10AS ··· 10BH 20A ··· 20P 20Q ··· 20AJ order 1 2 2 2 2 ··· 2 2 2 2 2 4 4 4 4 4 ··· 4 5 5 5 5 10 ··· 10 10 ··· 10 10 ··· 10 20 ··· 20 20 ··· 20 size 1 1 1 1 2 ··· 2 4 4 4 4 2 2 2 2 4 ··· 4 1 1 1 1 1 ··· 1 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4

125 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 4 4 type + + + + + + + + image C1 C2 C2 C2 C2 C2 C5 C10 C10 C10 C10 C10 D4 C5×D4 2+ 1+4 C5×2+ 1+4 kernel C5×D42 D4×C20 C5×C22≀C2 C5×C4⋊D4 C5×C4⋊1D4 D4×C2×C10 D42 C4×D4 C22≀C2 C4⋊D4 C4⋊1D4 C22×D4 C5×D4 D4 C10 C2 # reps 1 2 4 4 1 4 4 8 16 16 4 16 8 32 1 4

Matrix representation of C5×D42 in GL5(𝔽41)

 37 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 31 39 0 0 0 30 10
,
 40 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 31 39 0 0 0 29 10
,
 40 0 0 0 0 0 0 40 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 40
,
 40 0 0 0 0 0 1 0 0 0 0 0 40 0 0 0 0 0 40 0 0 0 0 0 40

G:=sub<GL(5,GF(41))| [37,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,31,30,0,0,0,39,10],[40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,31,29,0,0,0,39,10],[40,0,0,0,0,0,0,1,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,40],[40,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40] >;

C5×D42 in GAP, Magma, Sage, TeX

C_5\times D_4^2
% in TeX

G:=Group("C5xD4^2");
// GroupNames label

G:=SmallGroup(320,1547);
// by ID

G=gap.SmallGroup(320,1547);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,1149,3446,1242]);
// Polycyclic

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

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