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## G = C5×C42⋊6C4order 320 = 26·5

### Direct product of C5 and C42⋊6C4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C5×C42⋊6C4
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C22×C20 — C5×C42⋊C2 — C5×C42⋊6C4
 Lower central C1 — C2 — C4 — C5×C42⋊6C4
 Upper central C1 — C2×C20 — C22×C20 — C5×C42⋊6C4

Generators and relations for C5×C426C4
G = < a,b,c,d | a5=b4=c4=d4=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, dcd-1=c-1 >

Subgroups: 170 in 110 conjugacy classes, 58 normal (42 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×4], C4 [×6], C22 [×3], C22 [×2], C5, C8 [×2], C2×C4 [×6], C2×C4 [×8], C23, C10, C10 [×2], C10 [×2], C42 [×2], C42 [×2], C22⋊C4, C4⋊C4 [×2], C2×C8, M4(2) [×2], M4(2), C22×C4, C22×C4, C20 [×4], C20 [×6], C2×C10 [×3], C2×C10 [×2], C2×C42, C42⋊C2, C2×M4(2), C40 [×2], C2×C20 [×6], C2×C20 [×8], C22×C10, C426C4, C4×C20 [×2], C4×C20 [×2], C5×C22⋊C4, C5×C4⋊C4 [×2], C2×C40, C5×M4(2) [×2], C5×M4(2), C22×C20, C22×C20, C2×C4×C20, C5×C42⋊C2, C10×M4(2), C5×C426C4
Quotients: C1, C2 [×3], C4 [×6], C22, C5, C2×C4 [×3], D4 [×3], Q8, C10 [×3], C42, C22⋊C4 [×3], C4⋊C4 [×3], C20 [×6], C2×C10, C2.C42, C4≀C2 [×2], C2×C20 [×3], C5×D4 [×3], C5×Q8, C426C4, C4×C20, C5×C22⋊C4 [×3], C5×C4⋊C4 [×3], C5×C2.C42, C5×C4≀C2 [×2], C5×C426C4

Smallest permutation representation of C5×C426C4
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)
(6 19 64 60)(7 20 65 56)(8 16 61 57)(9 17 62 58)(10 18 63 59)(21 34 79 75)(22 35 80 71)(23 31 76 72)(24 32 77 73)(25 33 78 74)
(1 54 12 42)(2 55 13 43)(3 51 14 44)(4 52 15 45)(5 53 11 41)(6 60 64 19)(7 56 65 20)(8 57 61 16)(9 58 62 17)(10 59 63 18)(21 75 79 34)(22 71 80 35)(23 72 76 31)(24 73 77 32)(25 74 78 33)(26 69 50 40)(27 70 46 36)(28 66 47 37)(29 67 48 38)(30 68 49 39)
(1 24 66 7)(2 25 67 8)(3 21 68 9)(4 22 69 10)(5 23 70 6)(11 76 36 64)(12 77 37 65)(13 78 38 61)(14 79 39 62)(15 80 40 63)(16 55 33 48)(17 51 34 49)(18 52 35 50)(19 53 31 46)(20 54 32 47)(26 59 45 71)(27 60 41 72)(28 56 42 73)(29 57 43 74)(30 58 44 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), (6,19,64,60)(7,20,65,56)(8,16,61,57)(9,17,62,58)(10,18,63,59)(21,34,79,75)(22,35,80,71)(23,31,76,72)(24,32,77,73)(25,33,78,74), (1,54,12,42)(2,55,13,43)(3,51,14,44)(4,52,15,45)(5,53,11,41)(6,60,64,19)(7,56,65,20)(8,57,61,16)(9,58,62,17)(10,59,63,18)(21,75,79,34)(22,71,80,35)(23,72,76,31)(24,73,77,32)(25,74,78,33)(26,69,50,40)(27,70,46,36)(28,66,47,37)(29,67,48,38)(30,68,49,39), (1,24,66,7)(2,25,67,8)(3,21,68,9)(4,22,69,10)(5,23,70,6)(11,76,36,64)(12,77,37,65)(13,78,38,61)(14,79,39,62)(15,80,40,63)(16,55,33,48)(17,51,34,49)(18,52,35,50)(19,53,31,46)(20,54,32,47)(26,59,45,71)(27,60,41,72)(28,56,42,73)(29,57,43,74)(30,58,44,75)>;

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

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

140 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E ··· 4N 4O 4P 4Q 4R 5A 5B 5C 5D 8A 8B 8C 8D 10A ··· 10L 10M ··· 10T 20A ··· 20P 20Q ··· 20BD 20BE ··· 20BT 40A ··· 40P order 1 2 2 2 2 2 4 4 4 4 4 ··· 4 4 4 4 4 5 5 5 5 8 8 8 8 10 ··· 10 10 ··· 10 20 ··· 20 20 ··· 20 20 ··· 20 40 ··· 40 size 1 1 1 1 2 2 1 1 1 1 2 ··· 2 4 4 4 4 1 1 1 1 4 4 4 4 1 ··· 1 2 ··· 2 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 2 2 2 2 2 2 2 2 type + + + + + - + image C1 C2 C2 C2 C4 C4 C4 C5 C10 C10 C10 C20 C20 C20 D4 Q8 D4 C4≀C2 C5×D4 C5×Q8 C5×D4 C5×C4≀C2 kernel C5×C42⋊6C4 C2×C4×C20 C5×C42⋊C2 C10×M4(2) C4×C20 C5×C4⋊C4 C5×M4(2) C42⋊6C4 C2×C42 C42⋊C2 C2×M4(2) C42 C4⋊C4 M4(2) C2×C20 C2×C20 C22×C10 C10 C2×C4 C2×C4 C23 C2 # reps 1 1 1 1 4 4 4 4 4 4 4 16 16 16 2 1 1 8 8 4 4 32

Matrix representation of C5×C426C4 in GL3(𝔽41) generated by

 1 0 0 0 18 0 0 0 18
,
 40 0 0 0 1 5 0 0 9
,
 1 0 0 0 9 40 0 0 32
,
 32 0 0 0 32 0 0 2 9
G:=sub<GL(3,GF(41))| [1,0,0,0,18,0,0,0,18],[40,0,0,0,1,0,0,5,9],[1,0,0,0,9,0,0,40,32],[32,0,0,0,32,2,0,0,9] >;

C5×C426C4 in GAP, Magma, Sage, TeX

C_5\times C_4^2\rtimes_6C_4
% in TeX

G:=Group("C5xC4^2:6C4");
// GroupNames label

G:=SmallGroup(320,144);
// by ID

G=gap.SmallGroup(320,144);
# by ID

G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-2,280,309,568,5043,248,10085]);
// Polycyclic

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

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