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## G = (D4×C10).29C4order 320 = 26·5

### 10th non-split extension by D4×C10 of C4 acting via C4/C2=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — (D4×C10).29C4
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C4.Dic5 — C2×C4.Dic5 — (D4×C10).29C4
 Lower central C5 — C10 — C2×C10 — (D4×C10).29C4
 Upper central C1 — C4 — C22×C4 — C2×C4○D4

Generators and relations for (D4×C10).29C4
G = < a,b,c,d | a10=b4=c2=1, d4=b2, ab=ba, ac=ca, dad-1=a-1b2, cbc=b-1, dbd-1=a5b, dcd-1=b2c >

Subgroups: 350 in 150 conjugacy classes, 67 normal (25 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C10, C10, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C20, C20, C20, C2×C10, C2×C10, C2×C10, C4.D4, C4.10D4, C2×M4(2), C2×C4○D4, C52C8, C2×C20, C2×C20, C2×C20, C5×D4, C5×Q8, C22×C10, C22×C10, M4(2).8C22, C2×C52C8, C4.Dic5, C4.Dic5, C22×C20, C22×C20, D4×C10, D4×C10, Q8×C10, C5×C4○D4, C20.D4, C20.10D4, C2×C4.Dic5, C10×C4○D4, (D4×C10).29C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C22⋊C4, C22×C4, C2×D4, Dic5, D10, C2×C22⋊C4, C2×Dic5, C5⋊D4, C22×D5, M4(2).8C22, C23.D5, C22×Dic5, C2×C5⋊D4, C2×C23.D5, (D4×C10).29C4

Smallest permutation representation of (D4×C10).29C4
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 37 9 34)(2 38 10 35)(3 39 6 31)(4 40 7 32)(5 36 8 33)(11 24 18 30)(12 25 19 26)(13 21 20 27)(14 22 16 28)(15 23 17 29)(41 72 46 77)(42 73 47 78)(43 74 48 79)(44 75 49 80)(45 76 50 71)(51 62 56 67)(52 63 57 68)(53 64 58 69)(54 65 59 70)(55 66 60 61)
(1 30)(2 26)(3 27)(4 28)(5 29)(6 21)(7 22)(8 23)(9 24)(10 25)(11 34)(12 35)(13 31)(14 32)(15 33)(16 40)(17 36)(18 37)(19 38)(20 39)(41 67)(42 68)(43 69)(44 70)(45 61)(46 62)(47 63)(48 64)(49 65)(50 66)(51 77)(52 78)(53 79)(54 80)(55 71)(56 72)(57 73)(58 74)(59 75)(60 76)
(1 45 11 60 9 50 18 55)(2 49 12 54 10 44 19 59)(3 43 13 58 6 48 20 53)(4 47 14 52 7 42 16 57)(5 41 15 56 8 46 17 51)(21 69 39 74 27 64 31 79)(22 63 40 78 28 68 32 73)(23 67 36 72 29 62 33 77)(24 61 37 76 30 66 34 71)(25 65 38 80 26 70 35 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), (1,37,9,34)(2,38,10,35)(3,39,6,31)(4,40,7,32)(5,36,8,33)(11,24,18,30)(12,25,19,26)(13,21,20,27)(14,22,16,28)(15,23,17,29)(41,72,46,77)(42,73,47,78)(43,74,48,79)(44,75,49,80)(45,76,50,71)(51,62,56,67)(52,63,57,68)(53,64,58,69)(54,65,59,70)(55,66,60,61), (1,30)(2,26)(3,27)(4,28)(5,29)(6,21)(7,22)(8,23)(9,24)(10,25)(11,34)(12,35)(13,31)(14,32)(15,33)(16,40)(17,36)(18,37)(19,38)(20,39)(41,67)(42,68)(43,69)(44,70)(45,61)(46,62)(47,63)(48,64)(49,65)(50,66)(51,77)(52,78)(53,79)(54,80)(55,71)(56,72)(57,73)(58,74)(59,75)(60,76), (1,45,11,60,9,50,18,55)(2,49,12,54,10,44,19,59)(3,43,13,58,6,48,20,53)(4,47,14,52,7,42,16,57)(5,41,15,56,8,46,17,51)(21,69,39,74,27,64,31,79)(22,63,40,78,28,68,32,73)(23,67,36,72,29,62,33,77)(24,61,37,76,30,66,34,71)(25,65,38,80,26,70,35,75)>;

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

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

62 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 4E 4F 4G 5A 5B 8A ··· 8H 10A ··· 10F 10G ··· 10R 20A ··· 20H 20I ··· 20T order 1 2 2 2 2 2 2 4 4 4 4 4 4 4 5 5 8 ··· 8 10 ··· 10 10 ··· 10 20 ··· 20 20 ··· 20 size 1 1 2 2 2 4 4 1 1 2 2 2 4 4 2 2 20 ··· 20 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4

62 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + + - + - + + image C1 C2 C2 C2 C2 C4 C4 D4 D5 Dic5 D10 Dic5 D10 D10 C5⋊D4 M4(2).8C22 (D4×C10).29C4 kernel (D4×C10).29C4 C20.D4 C20.10D4 C2×C4.Dic5 C10×C4○D4 C22×C20 D4×C10 C2×C20 C2×C4○D4 C22×C4 C22×C4 C2×D4 C2×D4 C2×Q8 C2×C4 C5 C1 # reps 1 2 2 2 1 4 4 4 2 4 2 4 2 2 16 2 8

Matrix representation of (D4×C10).29C4 in GL6(𝔽41)

 31 0 0 0 0 0 0 4 0 0 0 0 0 0 40 0 5 37 0 0 0 40 4 13 0 0 0 0 1 0 0 0 0 0 0 1
,
 40 0 0 0 0 0 0 1 0 0 0 0 0 0 32 0 0 5 0 0 0 9 5 0 0 0 0 0 32 0 0 0 0 0 0 9
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 0 32 36 40 0 0 9 0 1 36 0 0 0 0 0 9 0 0 0 0 32 0
,
 0 1 0 0 0 0 40 0 0 0 0 0 0 0 37 36 40 4 0 0 13 37 37 31 0 0 0 39 4 13 0 0 2 0 36 4

G:=sub<GL(6,GF(41))| [31,0,0,0,0,0,0,4,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,5,4,1,0,0,0,37,13,0,1],[40,0,0,0,0,0,0,1,0,0,0,0,0,0,32,0,0,0,0,0,0,9,0,0,0,0,0,5,32,0,0,0,5,0,0,9],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,9,0,0,0,0,32,0,0,0,0,0,36,1,0,32,0,0,40,36,9,0],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,37,13,0,2,0,0,36,37,39,0,0,0,40,37,4,36,0,0,4,31,13,4] >;

(D4×C10).29C4 in GAP, Magma, Sage, TeX

(D_4\times C_{10})._{29}C_4
% in TeX

G:=Group("(D4xC10).29C4");
// GroupNames label

G:=SmallGroup(320,864);
// by ID

G=gap.SmallGroup(320,864);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,422,387,297,136,1684,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^10=b^4=c^2=1,d^4=b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1*b^2,c*b*c=b^-1,d*b*d^-1=a^5*b,d*c*d^-1=b^2*c>;
// generators/relations

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