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## G = C10×C4≀C2order 320 = 26·5

### Direct product of C10 and C4≀C2

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C10×C4≀C2
 Chief series C1 — C2 — C4 — C2×C4 — C2×C20 — C5×M4(2) — C5×C4≀C2 — C10×C4≀C2
 Lower central C1 — C2 — C4 — C10×C4≀C2
 Upper central C1 — C2×C20 — C22×C20 — C10×C4≀C2

Generators and relations for C10×C4≀C2
G = < a,b,c,d | a10=b4=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, dcd-1=b-1c >

Subgroups: 274 in 170 conjugacy classes, 82 normal (46 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×4], C4 [×6], C22 [×3], C22 [×6], C5, C8 [×2], C2×C4 [×6], C2×C4 [×11], D4 [×2], D4 [×5], Q8 [×2], Q8, C23, C23, C10, C10 [×2], C10 [×4], C42 [×2], C42, C2×C8, M4(2) [×2], M4(2), C22×C4, C22×C4 [×2], C2×D4, C2×D4, C2×Q8, C4○D4 [×4], C4○D4 [×2], C20 [×4], C20 [×6], C2×C10 [×3], C2×C10 [×6], C4≀C2 [×4], C2×C42, C2×M4(2), C2×C4○D4, C40 [×2], C2×C20 [×6], C2×C20 [×11], C5×D4 [×2], C5×D4 [×5], C5×Q8 [×2], C5×Q8, C22×C10, C22×C10, C2×C4≀C2, C4×C20 [×2], C4×C20, C2×C40, C5×M4(2) [×2], C5×M4(2), C22×C20, C22×C20 [×2], D4×C10, D4×C10, Q8×C10, C5×C4○D4 [×4], C5×C4○D4 [×2], C5×C4≀C2 [×4], C2×C4×C20, C10×M4(2), C10×C4○D4, C10×C4≀C2
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C5, C2×C4 [×6], D4 [×4], C23, C10 [×7], C22⋊C4 [×4], C22×C4, C2×D4 [×2], C20 [×4], C2×C10 [×7], C4≀C2 [×2], C2×C22⋊C4, C2×C20 [×6], C5×D4 [×4], C22×C10, C2×C4≀C2, C5×C22⋊C4 [×4], C22×C20, D4×C10 [×2], C5×C4≀C2 [×2], C10×C22⋊C4, C10×C4≀C2

Smallest permutation representation of C10×C4≀C2
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 56 69 50)(2 57 70 41)(3 58 61 42)(4 59 62 43)(5 60 63 44)(6 51 64 45)(7 52 65 46)(8 53 66 47)(9 54 67 48)(10 55 68 49)(11 33 29 80)(12 34 30 71)(13 35 21 72)(14 36 22 73)(15 37 23 74)(16 38 24 75)(17 39 25 76)(18 40 26 77)(19 31 27 78)(20 32 28 79)
(1 28)(2 29)(3 30)(4 21)(5 22)(6 23)(7 24)(8 25)(9 26)(10 27)(11 70)(12 61)(13 62)(14 63)(15 64)(16 65)(17 66)(18 67)(19 68)(20 69)(31 55)(32 56)(33 57)(34 58)(35 59)(36 60)(37 51)(38 52)(39 53)(40 54)(41 80)(42 71)(43 72)(44 73)(45 74)(46 75)(47 76)(48 77)(49 78)(50 79)
(1 64)(2 65)(3 66)(4 67)(5 68)(6 69)(7 70)(8 61)(9 62)(10 63)(11 38 29 75)(12 39 30 76)(13 40 21 77)(14 31 22 78)(15 32 23 79)(16 33 24 80)(17 34 25 71)(18 35 26 72)(19 36 27 73)(20 37 28 74)(41 52)(42 53)(43 54)(44 55)(45 56)(46 57)(47 58)(48 59)(49 60)(50 51)

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

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

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

140 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E ··· 4N 4O 4P 5A 5B 5C 5D 8A 8B 8C 8D 10A ··· 10L 10M ··· 10T 10U ··· 10AB 20A ··· 20P 20Q ··· 20BD 20BE ··· 20BL 40A ··· 40P order 1 2 2 2 2 2 2 2 4 4 4 4 4 ··· 4 4 4 5 5 5 5 8 8 8 8 10 ··· 10 10 ··· 10 10 ··· 10 20 ··· 20 20 ··· 20 20 ··· 20 40 ··· 40 size 1 1 1 1 2 2 4 4 1 1 1 1 2 ··· 2 4 4 1 1 1 1 4 4 4 4 1 ··· 1 2 ··· 2 4 ··· 4 1 ··· 1 2 ··· 2 4 ··· 4 4 ··· 4

140 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + + + image C1 C2 C2 C2 C2 C4 C4 C4 C5 C10 C10 C10 C10 C20 C20 C20 D4 D4 C4≀C2 C5×D4 C5×D4 C5×C4≀C2 kernel C10×C4≀C2 C5×C4≀C2 C2×C4×C20 C10×M4(2) C10×C4○D4 D4×C10 Q8×C10 C5×C4○D4 C2×C4≀C2 C4≀C2 C2×C42 C2×M4(2) C2×C4○D4 C2×D4 C2×Q8 C4○D4 C2×C20 C22×C10 C10 C2×C4 C23 C2 # reps 1 4 1 1 1 2 2 4 4 16 4 4 4 8 8 16 3 1 8 12 4 32

Matrix representation of C10×C4≀C2 in GL4(𝔽41) generated by

 4 0 0 0 0 4 0 0 0 0 40 0 0 0 0 40
,
 40 0 0 0 0 40 0 0 0 0 9 0 0 0 0 32
,
 4 39 0 0 28 37 0 0 0 0 0 32 0 0 9 0
,
 40 0 0 0 37 1 0 0 0 0 1 0 0 0 0 9
G:=sub<GL(4,GF(41))| [4,0,0,0,0,4,0,0,0,0,40,0,0,0,0,40],[40,0,0,0,0,40,0,0,0,0,9,0,0,0,0,32],[4,28,0,0,39,37,0,0,0,0,0,9,0,0,32,0],[40,37,0,0,0,1,0,0,0,0,1,0,0,0,0,9] >;

C10×C4≀C2 in GAP, Magma, Sage, TeX

C_{10}\times C_4\wr C_2
% in TeX

G:=Group("C10xC4wrC2");
// GroupNames label

G:=SmallGroup(320,921);
// by ID

G=gap.SmallGroup(320,921);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,560,589,7004,3511,172,124]);
// Polycyclic

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

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