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## G = D5×C4.4D4order 320 = 26·5

### Direct product of D5 and C4.4D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — D5×C4.4D4
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C23×D5 — D5×C22⋊C4 — D5×C4.4D4
 Lower central C5 — C2×C10 — D5×C4.4D4
 Upper central C1 — C22 — C4.4D4

Generators and relations for D5×C4.4D4
G = < a,b,c,d,e | a5=b2=c4=d4=1, e2=c2, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=c-1, ede-1=c2d-1 >

Subgroups: 1358 in 330 conjugacy classes, 111 normal (29 characteristic)
C1, C2, C2 [×2], C2 [×8], C4 [×2], C4 [×10], C22, C22 [×26], C5, C2×C4, C2×C4 [×4], C2×C4 [×17], D4 [×8], Q8 [×8], C23 [×2], C23 [×15], D5 [×4], D5 [×2], C10, C10 [×2], C10 [×2], C42, C42 [×3], C22⋊C4 [×4], C22⋊C4 [×12], C22×C4 [×5], C2×D4, C2×D4 [×7], C2×Q8, C2×Q8 [×7], C24 [×2], Dic5 [×2], Dic5 [×4], C20 [×2], C20 [×4], D10 [×6], D10 [×14], C2×C10, C2×C10 [×6], C2×C42, C2×C22⋊C4 [×4], C4.4D4, C4.4D4 [×7], C22×D4, C22×Q8, Dic10 [×6], C4×D5 [×4], C4×D5 [×8], D20 [×2], C2×Dic5, C2×Dic5 [×4], C5⋊D4 [×4], C2×C20, C2×C20 [×4], C5×D4 [×2], C5×Q8 [×2], C22×D5, C22×D5 [×2], C22×D5 [×12], C22×C10 [×2], C2×C4.4D4, C4×Dic5, C4×Dic5 [×2], D10⋊C4 [×8], C23.D5 [×4], C4×C20, C5×C22⋊C4 [×4], C2×Dic10, C2×Dic10 [×2], C2×C4×D5, C2×C4×D5 [×4], C2×D20, D4×D5 [×4], Q8×D5 [×4], C2×C5⋊D4 [×2], D4×C10, Q8×C10, C23×D5 [×2], D5×C42, C4.D20, D5×C22⋊C4 [×4], Dic5.5D4 [×4], C20.17D4, C20.23D4, C5×C4.4D4, C2×D4×D5, C2×Q8×D5, D5×C4.4D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C4○D4 [×4], C24, D10 [×7], C4.4D4 [×4], C22×D4, C2×C4○D4 [×2], C22×D5 [×7], C2×C4.4D4, D4×D5 [×2], C23×D5, C2×D4×D5, D5×C4○D4 [×2], D5×C4.4D4

Smallest permutation representation of D5×C4.4D4
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 13)(2 12)(3 11)(4 15)(5 14)(6 16)(7 20)(8 19)(9 18)(10 17)(21 31)(22 35)(23 34)(24 33)(25 32)(26 36)(27 40)(28 39)(29 38)(30 37)(41 51)(42 55)(43 54)(44 53)(45 52)(46 56)(47 60)(48 59)(49 58)(50 57)(61 71)(62 75)(63 74)(64 73)(65 72)(66 76)(67 80)(68 79)(69 78)(70 77)
(1 79 19 64)(2 80 20 65)(3 76 16 61)(4 77 17 62)(5 78 18 63)(6 71 11 66)(7 72 12 67)(8 73 13 68)(9 74 14 69)(10 75 15 70)(21 51 36 46)(22 52 37 47)(23 53 38 48)(24 54 39 49)(25 55 40 50)(26 56 31 41)(27 57 32 42)(28 58 33 43)(29 59 34 44)(30 60 35 45)
(1 44 14 54)(2 45 15 55)(3 41 11 51)(4 42 12 52)(5 43 13 53)(6 46 16 56)(7 47 17 57)(8 48 18 58)(9 49 19 59)(10 50 20 60)(21 61 31 71)(22 62 32 72)(23 63 33 73)(24 64 34 74)(25 65 35 75)(26 66 36 76)(27 67 37 77)(28 68 38 78)(29 69 39 79)(30 70 40 80)
(1 39 19 24)(2 40 20 25)(3 36 16 21)(4 37 17 22)(5 38 18 23)(6 31 11 26)(7 32 12 27)(8 33 13 28)(9 34 14 29)(10 35 15 30)(41 71 56 66)(42 72 57 67)(43 73 58 68)(44 74 59 69)(45 75 60 70)(46 76 51 61)(47 77 52 62)(48 78 53 63)(49 79 54 64)(50 80 55 65)

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

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

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

56 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 4A ··· 4F 4G 4H 4I ··· 4N 4O 4P 5A 5B 10A ··· 10F 10G 10H 10I 10J 20A ··· 20L 20M 20N 20O 20P order 1 2 2 2 2 2 2 2 2 2 2 2 4 ··· 4 4 4 4 ··· 4 4 4 5 5 10 ··· 10 10 10 10 10 20 ··· 20 20 20 20 20 size 1 1 1 1 4 4 5 5 5 5 20 20 2 ··· 2 4 4 10 ··· 10 20 20 2 2 2 ··· 2 8 8 8 8 4 ··· 4 8 8 8 8

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 D4 D5 C4○D4 D10 D10 D10 D10 D4×D5 D5×C4○D4 kernel D5×C4.4D4 D5×C42 C4.D20 D5×C22⋊C4 Dic5.5D4 C20.17D4 C20.23D4 C5×C4.4D4 C2×D4×D5 C2×Q8×D5 C4×D5 C4.4D4 D10 C42 C22⋊C4 C2×D4 C2×Q8 C4 C2 # reps 1 1 1 4 4 1 1 1 1 1 4 2 8 2 8 2 2 4 8

Matrix representation of D5×C4.4D4 in GL6(𝔽41)

 35 1 0 0 0 0 5 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 40 40 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 34 36 0 0 0 0 10 7 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 19 37 0 0 0 0 8 22 0 0 0 0 0 0 32 18 0 0 0 0 0 9
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 32 0 0 0 0 0 17 9 0 0 0 0 0 0 32 18 0 0 0 0 32 9

G:=sub<GL(6,GF(41))| [35,5,0,0,0,0,1,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,40,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,34,10,0,0,0,0,36,7,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,19,8,0,0,0,0,37,22,0,0,0,0,0,0,32,0,0,0,0,0,18,9],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,32,17,0,0,0,0,0,9,0,0,0,0,0,0,32,32,0,0,0,0,18,9] >;

D5×C4.4D4 in GAP, Magma, Sage, TeX

D_5\times C_4._4D_4
% in TeX

G:=Group("D5xC4.4D4");
// GroupNames label

G:=SmallGroup(320,1345);
// by ID

G=gap.SmallGroup(320,1345);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,387,100,346,136,12550]);
// Polycyclic

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

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