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## G = D5×C42⋊2C2order 320 = 26·5

### Direct product of D5 and C42⋊2C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — D5×C42⋊2C2
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C23×D5 — D5×C22⋊C4 — D5×C42⋊2C2
 Lower central C5 — C2×C10 — D5×C42⋊2C2
 Upper central C1 — C22 — C42⋊2C2

Generators and relations for D5×C422C2
G = < a,b,c,d,e | a5=b2=c4=d4=e2=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=cd2, ede=c2d-1 >

Subgroups: 894 in 246 conjugacy classes, 101 normal (16 characteristic)
C1, C2 [×3], C2 [×6], C4 [×12], C22, C22 [×16], C5, C2×C4 [×6], C2×C4 [×18], C23, C23 [×8], D5 [×4], D5, C10 [×3], C10, C42, C42 [×3], C22⋊C4 [×3], C22⋊C4 [×9], C4⋊C4 [×3], C4⋊C4 [×9], C22×C4 [×6], C24, Dic5 [×6], C20 [×6], D10 [×6], D10 [×7], C2×C10, C2×C10 [×3], C2×C42, C2×C22⋊C4 [×3], C2×C4⋊C4 [×3], C422C2, C422C2 [×7], C4×D5 [×12], C2×Dic5 [×6], C2×C20 [×6], C22×D5 [×2], C22×D5 [×6], C22×C10, C2×C422C2, C4×Dic5 [×3], C10.D4 [×6], C4⋊Dic5 [×3], D10⋊C4 [×6], C23.D5 [×3], C4×C20, C5×C22⋊C4 [×3], C5×C4⋊C4 [×3], C2×C4×D5 [×6], C23×D5, D5×C42, C422D5, C23.D10 [×3], D5×C22⋊C4 [×3], D5×C4⋊C4 [×3], C4⋊C4⋊D5 [×3], C5×C422C2, D5×C422C2
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×6], C24, D10 [×7], C422C2 [×4], C2×C4○D4 [×3], C22×D5 [×7], C2×C422C2, C23×D5, D5×C4○D4 [×3], D5×C422C2

Smallest permutation representation of D5×C422C2
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 69 9 64)(2 70 10 65)(3 66 6 61)(4 67 7 62)(5 68 8 63)(11 76 16 71)(12 77 17 72)(13 78 18 73)(14 79 19 74)(15 80 20 75)(21 51 26 56)(22 52 27 57)(23 53 28 58)(24 54 29 59)(25 55 30 60)(31 41 36 46)(32 42 37 47)(33 43 38 48)(34 44 39 49)(35 45 40 50)
(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 76 56 61)(42 77 57 62)(43 78 58 63)(44 79 59 64)(45 80 60 65)(46 71 51 66)(47 72 52 67)(48 73 53 68)(49 74 54 69)(50 75 55 70)
(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(41 46)(42 47)(43 48)(44 49)(45 50)(51 56)(52 57)(53 58)(54 59)(55 60)(61 76)(62 77)(63 78)(64 79)(65 80)(66 71)(67 72)(68 73)(69 74)(70 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), (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), (1,69,9,64)(2,70,10,65)(3,66,6,61)(4,67,7,62)(5,68,8,63)(11,76,16,71)(12,77,17,72)(13,78,18,73)(14,79,19,74)(15,80,20,75)(21,51,26,56)(22,52,27,57)(23,53,28,58)(24,54,29,59)(25,55,30,60)(31,41,36,46)(32,42,37,47)(33,43,38,48)(34,44,39,49)(35,45,40,50), (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)(41,76,56,61)(42,77,57,62)(43,78,58,63)(44,79,59,64)(45,80,60,65)(46,71,51,66)(47,72,52,67)(48,73,53,68)(49,74,54,69)(50,75,55,70), (21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(41,46)(42,47)(43,48)(44,49)(45,50)(51,56)(52,57)(53,58)(54,59)(55,60)(61,76)(62,77)(63,78)(64,79)(65,80)(66,71)(67,72)(68,73)(69,74)(70,75)>;

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

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

56 conjugacy classes

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

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 4 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 D5 C4○D4 D10 D10 D10 D5×C4○D4 kernel D5×C42⋊2C2 D5×C42 C42⋊2D5 C23.D10 D5×C22⋊C4 D5×C4⋊C4 C4⋊C4⋊D5 C5×C42⋊2C2 C42⋊2C2 D10 C42 C22⋊C4 C4⋊C4 C2 # reps 1 1 1 3 3 3 3 1 2 12 2 6 6 12

Matrix representation of D5×C422C2 in GL6(𝔽41)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 40 6 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 0 1 0 0 0 0 1 0 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 9 37 0 0 0 0 0 32 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 9 23 0 0 0 0 9 32
,
 32 0 0 0 0 0 0 32 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 39 0 0 0 0 1 40
,
 1 0 0 0 0 0 25 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 1 40

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,1,6,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[9,0,0,0,0,0,37,32,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,9,9,0,0,0,0,23,32],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,39,40],[1,25,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,40] >;

D5×C422C2 in GAP, Magma, Sage, TeX

D_5\times C_4^2\rtimes_2C_2
% in TeX

G:=Group("D5xC4^2:2C2");
// GroupNames label

G:=SmallGroup(320,1375);
// by ID

G=gap.SmallGroup(320,1375);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,387,100,346,297,136,12550]);
// Polycyclic

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

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