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## G = C2×D4.D10order 320 = 26·5

### Direct product of C2 and D4.D10

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C2×D4.D10
 Chief series C1 — C5 — C10 — C20 — D20 — C2×D20 — C2×C4○D20 — C2×D4.D10
 Lower central C5 — C10 — C20 — C2×D4.D10
 Upper central C1 — C22 — C22×C4 — C22×D4

Generators and relations for C2×D4.D10
G = < a,b,c,d,e | a2=b4=c2=1, d10=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b-1, bd=db, dcd-1=b2c, ece-1=b-1c, ede-1=d9 >

Subgroups: 958 in 298 conjugacy classes, 111 normal (25 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, C23, C23, D5, C10, C10, C10, C2×C8, M4(2), D8, SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, Dic5, C20, C20, D10, C2×C10, C2×C10, C2×C10, C2×M4(2), C2×D8, C2×SD16, C8⋊C22, C22×D4, C2×C4○D4, C52C8, Dic10, Dic10, C4×D5, D20, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C5×D4, C22×D5, C22×C10, C22×C10, C2×C8⋊C22, C2×C52C8, C4.Dic5, D4⋊D5, D4.D5, C2×Dic10, C2×C4×D5, C2×D20, C4○D20, C4○D20, C2×C5⋊D4, C22×C20, D4×C10, D4×C10, C23×C10, C2×C4.Dic5, C2×D4⋊D5, D4.D10, C2×D4.D5, C2×C4○D20, D4×C2×C10, C2×D4.D10
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C24, D10, C8⋊C22, C22×D4, C5⋊D4, C22×D5, C2×C8⋊C22, C2×C5⋊D4, C23×D5, D4.D10, C22×C5⋊D4, C2×D4.D10

Smallest permutation representation of C2×D4.D10
On 80 points
Generators in S80
(1 27)(2 28)(3 29)(4 30)(5 31)(6 32)(7 33)(8 34)(9 35)(10 36)(11 37)(12 38)(13 39)(14 40)(15 21)(16 22)(17 23)(18 24)(19 25)(20 26)(41 72)(42 73)(43 74)(44 75)(45 76)(46 77)(47 78)(48 79)(49 80)(50 61)(51 62)(52 63)(53 64)(54 65)(55 66)(56 67)(57 68)(58 69)(59 70)(60 71)
(1 32 11 22)(2 33 12 23)(3 34 13 24)(4 35 14 25)(5 36 15 26)(6 37 16 27)(7 38 17 28)(8 39 18 29)(9 40 19 30)(10 21 20 31)(41 67 51 77)(42 68 52 78)(43 69 53 79)(44 70 54 80)(45 71 55 61)(46 72 56 62)(47 73 57 63)(48 74 58 64)(49 75 59 65)(50 76 60 66)
(1 22)(2 33)(3 24)(4 35)(5 26)(6 37)(7 28)(8 39)(9 30)(10 21)(11 32)(12 23)(13 34)(14 25)(15 36)(16 27)(17 38)(18 29)(19 40)(20 31)(42 52)(44 54)(46 56)(48 58)(50 60)(61 71)(63 73)(65 75)(67 77)(69 79)
(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 64 11 74)(2 73 12 63)(3 62 13 72)(4 71 14 61)(5 80 15 70)(6 69 16 79)(7 78 17 68)(8 67 18 77)(9 76 19 66)(10 65 20 75)(21 59 31 49)(22 48 32 58)(23 57 33 47)(24 46 34 56)(25 55 35 45)(26 44 36 54)(27 53 37 43)(28 42 38 52)(29 51 39 41)(30 60 40 50)

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

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

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

62 conjugacy classes

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

62 irreducible representations

 dim 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 D4 D4 D5 D10 D10 C5⋊D4 C5⋊D4 C8⋊C22 D4.D10 kernel C2×D4.D10 C2×C4.Dic5 C2×D4⋊D5 D4.D10 C2×D4.D5 C2×C4○D20 D4×C2×C10 C2×C20 C22×C10 C22×D4 C22×C4 C2×D4 C2×C4 C23 C10 C2 # reps 1 1 2 8 2 1 1 3 1 2 2 12 12 4 2 8

Matrix representation of C2×D4.D10 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 0 1 0 0 0 0 40 0 0 0 0 0 6 6 1 2 0 0 0 35 40 40
,
 40 0 0 0 0 0 11 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 1 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 23 0 0 0 0 18 0 0 0 0 0 14 14 16 32 0 0 21 6 25 25
,
 1 30 0 0 0 0 0 40 0 0 0 0 0 0 14 14 16 32 0 0 0 0 16 0 0 0 0 23 0 0 0 0 31 8 27 27

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,6,0,0,0,1,0,6,35,0,0,0,0,1,40,0,0,0,0,2,40],[40,11,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,40,1,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,18,14,21,0,0,23,0,14,6,0,0,0,0,16,25,0,0,0,0,32,25],[1,0,0,0,0,0,30,40,0,0,0,0,0,0,14,0,0,31,0,0,14,0,23,8,0,0,16,16,0,27,0,0,32,0,0,27] >;

C2×D4.D10 in GAP, Magma, Sage, TeX

C_2\times D_4.D_{10}
% in TeX

G:=Group("C2xD4.D10");
// GroupNames label

G:=SmallGroup(320,1465);
// by ID

G=gap.SmallGroup(320,1465);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,184,675,297,1684,235,102,12550]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^4=c^2=1,d^10=e^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=e*b*e^-1=b^-1,b*d=d*b,d*c*d^-1=b^2*c,e*c*e^-1=b^-1*c,e*d*e^-1=d^9>;
// generators/relations

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