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## G = C10×C8⋊C22order 320 = 26·5

### Direct product of C10 and C8⋊C22

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C10×C8⋊C22
 Chief series C1 — C2 — C4 — C20 — C5×D4 — C5×D8 — C5×C8⋊C22 — C10×C8⋊C22
 Lower central C1 — C2 — C4 — C10×C8⋊C22
 Upper central C1 — C2×C10 — C22×C20 — C10×C8⋊C22

Generators and relations for C10×C8⋊C22
G = < a,b,c,d | a10=b8=c2=d2=1, ab=ba, ac=ca, ad=da, cbc=b3, dbd=b5, cd=dc >

Subgroups: 530 in 298 conjugacy classes, 162 normal (30 characteristic)
C1, C2, C2 [×2], C2 [×8], C4 [×2], C4 [×2], C4 [×2], C22, C22 [×2], C22 [×22], C5, C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×5], D4 [×6], D4 [×11], Q8 [×2], Q8, C23, C23 [×11], C10, C10 [×2], C10 [×8], C2×C8 [×2], M4(2) [×4], D8 [×8], SD16 [×8], C22×C4, C22×C4, C2×D4, C2×D4 [×6], C2×D4 [×4], C2×Q8, C4○D4 [×4], C4○D4 [×2], C24, C20 [×2], C20 [×2], C20 [×2], C2×C10, C2×C10 [×2], C2×C10 [×22], C2×M4(2), C2×D8 [×2], C2×SD16 [×2], C8⋊C22 [×8], C22×D4, C2×C4○D4, C40 [×4], C2×C20 [×2], C2×C20 [×4], C2×C20 [×5], C5×D4 [×6], C5×D4 [×11], C5×Q8 [×2], C5×Q8, C22×C10, C22×C10 [×11], C2×C8⋊C22, C2×C40 [×2], C5×M4(2) [×4], C5×D8 [×8], C5×SD16 [×8], C22×C20, C22×C20, D4×C10, D4×C10 [×6], D4×C10 [×4], Q8×C10, C5×C4○D4 [×4], C5×C4○D4 [×2], C23×C10, C10×M4(2), C10×D8 [×2], C10×SD16 [×2], C5×C8⋊C22 [×8], D4×C2×C10, C10×C4○D4, C10×C8⋊C22
Quotients: C1, C2 [×15], C22 [×35], C5, D4 [×4], C23 [×15], C10 [×15], C2×D4 [×6], C24, C2×C10 [×35], C8⋊C22 [×2], C22×D4, C5×D4 [×4], C22×C10 [×15], C2×C8⋊C22, D4×C10 [×6], C23×C10, C5×C8⋊C22 [×2], D4×C2×C10, C10×C8⋊C22

Smallest permutation representation of C10×C8⋊C22
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 28 56 79 70 20 50 32)(2 29 57 80 61 11 41 33)(3 30 58 71 62 12 42 34)(4 21 59 72 63 13 43 35)(5 22 60 73 64 14 44 36)(6 23 51 74 65 15 45 37)(7 24 52 75 66 16 46 38)(8 25 53 76 67 17 47 39)(9 26 54 77 68 18 48 40)(10 27 55 78 69 19 49 31)
(11 33)(12 34)(13 35)(14 36)(15 37)(16 38)(17 39)(18 40)(19 31)(20 32)(21 72)(22 73)(23 74)(24 75)(25 76)(26 77)(27 78)(28 79)(29 80)(30 71)(41 57)(42 58)(43 59)(44 60)(45 51)(46 52)(47 53)(48 54)(49 55)(50 56)
(1 70)(2 61)(3 62)(4 63)(5 64)(6 65)(7 66)(8 67)(9 68)(10 69)(41 57)(42 58)(43 59)(44 60)(45 51)(46 52)(47 53)(48 54)(49 55)(50 56)

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

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

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

110 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F ··· 2K 4A 4B 4C 4D 4E 4F 5A 5B 5C 5D 8A 8B 8C 8D 10A ··· 10L 10M ··· 10T 10U ··· 10AR 20A ··· 20P 20Q ··· 20X 40A ··· 40P order 1 2 2 2 2 2 2 ··· 2 4 4 4 4 4 4 5 5 5 5 8 8 8 8 10 ··· 10 10 ··· 10 10 ··· 10 20 ··· 20 20 ··· 20 40 ··· 40 size 1 1 1 1 2 2 4 ··· 4 2 2 2 2 4 4 1 1 1 1 4 4 4 4 1 ··· 1 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4 4 ··· 4

110 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 4 4 type + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C5 C10 C10 C10 C10 C10 C10 D4 D4 C5×D4 C5×D4 C8⋊C22 C5×C8⋊C22 kernel C10×C8⋊C22 C10×M4(2) C10×D8 C10×SD16 C5×C8⋊C22 D4×C2×C10 C10×C4○D4 C2×C8⋊C22 C2×M4(2) C2×D8 C2×SD16 C8⋊C22 C22×D4 C2×C4○D4 C2×C20 C22×C10 C2×C4 C23 C10 C2 # reps 1 1 2 2 8 1 1 4 4 8 8 32 4 4 3 1 12 4 2 8

Matrix representation of C10×C8⋊C22 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16
,
 40 2 0 0 0 0 40 1 0 0 0 0 0 0 0 0 1 0 0 0 6 0 0 40 0 0 0 1 0 0 0 0 1 0 6 0
,
 1 0 0 0 0 0 1 40 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 6 0 0 40 0 0 6 0 40 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 29 0 0 1

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[40,40,0,0,0,0,2,1,0,0,0,0,0,0,0,6,0,1,0,0,0,0,1,0,0,0,1,0,0,6,0,0,0,40,0,0],[1,1,0,0,0,0,0,40,0,0,0,0,0,0,1,0,6,6,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,40,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,29,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C10×C8⋊C22 in GAP, Magma, Sage, TeX

C_{10}\times C_8\rtimes C_2^2
% in TeX

G:=Group("C10xC8:C2^2");
// GroupNames label

G:=SmallGroup(320,1575);
// by ID

G=gap.SmallGroup(320,1575);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,1149,3446,10085,5052,124]);
// Polycyclic

G:=Group<a,b,c,d|a^10=b^8=c^2=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^3,d*b*d=b^5,c*d=d*c>;
// generators/relations

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