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## G = C5×D4.4D4order 320 = 26·5

### Direct product of C5 and D4.4D4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C4 — C5×D4.4D4
 Chief series C1 — C2 — C4 — C2×C4 — C2×C20 — D4×C10 — C10×D8 — C5×D4.4D4
 Lower central C1 — C2 — C2×C4 — C5×D4.4D4
 Upper central C1 — C10 — C2×C20 — C5×D4.4D4

Generators and relations for C5×D4.4D4
G = < a,b,c,d,e | a5=b4=c2=e2=1, d4=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe=b-1, bd=db, cd=dc, ece=bc, ede=b2d3 >

Subgroups: 226 in 108 conjugacy classes, 50 normal (34 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, D4, Q8, C23, C10, C10, C2×C8, C2×C8, M4(2), M4(2), M4(2), D8, SD16, C2×D4, C4○D4, C20, C20, C2×C10, C2×C10, C4.D4, C8.C4, C8○D4, C2×D8, C8⋊C22, C40, C40, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C22×C10, D4.4D4, C2×C40, C2×C40, C5×M4(2), C5×M4(2), C5×M4(2), C5×D8, C5×SD16, D4×C10, C5×C4○D4, C5×C4.D4, C5×C8.C4, C5×C8○D4, C10×D8, C5×C8⋊C22, C5×D4.4D4
Quotients: C1, C2, C22, C5, D4, C23, C10, C2×D4, C4○D4, C2×C10, C4⋊D4, C5×D4, C22×C10, D4.4D4, D4×C10, C5×C4○D4, C5×C4⋊D4, C5×D4.4D4

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

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

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

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

80 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 5A 5B 5C 5D 8A 8B 8C 8D 8E 8F 8G 10A 10B 10C 10D 10E 10F 10G 10H 10I 10J 10K 10L 10M ··· 10T 20A ··· 20H 20I 20J 20K 20L 40A ··· 40H 40I ··· 40T 40U ··· 40AB order 1 2 2 2 2 2 4 4 4 5 5 5 5 8 8 8 8 8 8 8 10 10 10 10 10 10 10 10 10 10 10 10 10 ··· 10 20 ··· 20 20 20 20 20 40 ··· 40 40 ··· 40 40 ··· 40 size 1 1 2 4 8 8 2 2 4 1 1 1 1 2 2 4 4 4 8 8 1 1 1 1 2 2 2 2 4 4 4 4 8 ··· 8 2 ··· 2 4 4 4 4 2 ··· 2 4 ··· 4 8 ··· 8

80 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C5 C10 C10 C10 C10 C10 D4 D4 D4 C4○D4 C5×D4 C5×D4 C5×D4 C5×C4○D4 D4.4D4 C5×D4.4D4 kernel C5×D4.4D4 C5×C4.D4 C5×C8.C4 C5×C8○D4 C10×D8 C5×C8⋊C22 D4.4D4 C4.D4 C8.C4 C8○D4 C2×D8 C8⋊C22 C40 C5×D4 C5×Q8 C2×C10 C8 D4 Q8 C22 C5 C1 # reps 1 2 1 1 1 2 4 8 4 4 4 8 2 1 1 2 8 4 4 8 2 8

Matrix representation of C5×D4.4D4 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 40 0 0 0 40 40 1 2 0 1 40 40
,
 40 40 1 2 0 0 1 0 0 1 0 0 0 40 1 1
,
 29 29 0 0 12 29 0 0 29 29 0 24 12 0 29 17
,
 29 29 0 0 29 12 0 0 29 29 24 24 0 12 29 17
G:=sub<GL(4,GF(41))| [16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[0,40,40,0,1,0,40,1,0,0,1,40,0,0,2,40],[40,0,0,0,40,0,1,40,1,1,0,1,2,0,0,1],[29,12,29,12,29,29,29,0,0,0,0,29,0,0,24,17],[29,29,29,0,29,12,29,12,0,0,24,29,0,0,24,17] >;

C5×D4.4D4 in GAP, Magma, Sage, TeX

C_5\times D_4._4D_4
% in TeX

G:=Group("C5xD4.4D4");
// GroupNames label

G:=SmallGroup(320,973);
// by ID

G=gap.SmallGroup(320,973);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,589,1408,1766,7004,172,10085,2539,124]);
// Polycyclic

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

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