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G = C42⋊26D10order 320 = 26·5

26th semidirect product of C42 and D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C42⋊26D10
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C23×D5 — C2×D4×D5 — C42⋊26D10
 Lower central C5 — C2×C10 — C42⋊26D10
 Upper central C1 — C22 — C4⋊1D4

Generators and relations for C4226D10
G = < a,b,c,d | a4=b4=c10=d2=1, ab=ba, cac-1=dad=a-1, cbc-1=b-1, dbd=a2b-1, dcd=c-1 >

Subgroups: 1470 in 334 conjugacy classes, 103 normal (29 characteristic)
C1, C2, C2 [×2], C2 [×8], C4 [×2], C4 [×8], C22, C22 [×30], C5, C2×C4, C2×C4 [×2], C2×C4 [×13], D4 [×22], Q8 [×2], C23 [×4], C23 [×11], D5 [×4], C10, C10 [×2], C10 [×4], C42, C42, C22⋊C4 [×10], C4⋊C4 [×2], C22×C4 [×3], C2×D4 [×2], C2×D4 [×4], C2×D4 [×13], C2×Q8, C4○D4 [×4], C24 [×2], Dic5 [×2], Dic5 [×4], C20 [×2], C20 [×2], D10 [×2], D10 [×16], C2×C10, C2×C10 [×12], C42⋊C2, C22≀C2 [×4], C4⋊D4 [×4], C4.4D4 [×2], C41D4, C41D4, C22×D4, C2×C4○D4, Dic10 [×2], C4×D5 [×4], D20 [×2], C2×Dic5, C2×Dic5 [×4], C2×Dic5 [×4], C5⋊D4 [×12], C2×C20, C2×C20 [×2], C5×D4 [×8], C22×D5, C22×D5 [×2], C22×D5 [×8], C22×C10 [×4], C22.29C24, C4×Dic5, C10.D4 [×2], D10⋊C4 [×6], C23.D5 [×4], C4×C20, C2×Dic10, C2×C4×D5, C2×D20, D4×D5 [×4], D42D5 [×4], C22×Dic5 [×2], C2×C5⋊D4 [×8], D4×C10 [×2], D4×C10 [×4], C23×D5 [×2], C42⋊D5, C4.D20, C20.17D4, C23⋊D10 [×4], Dic5⋊D4 [×4], C20⋊D4, C5×C41D4, C2×D4×D5, C2×D42D5, C4226D10
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C22×D4, 2+ 1+4 [×2], C22×D5 [×7], C22.29C24, D4×D5 [×2], C23×D5, C2×D4×D5, D46D10 [×2], C4226D10

Smallest permutation representation of C4226D10
On 80 points
Generators in S80
(1 73 16 78)(2 79 17 74)(3 75 18 80)(4 71 19 76)(5 77 20 72)(6 50 14 45)(7 46 15 41)(8 42 11 47)(9 48 12 43)(10 44 13 49)(21 69 38 51)(22 52 39 70)(23 61 40 53)(24 54 31 62)(25 63 32 55)(26 56 33 64)(27 65 34 57)(28 58 35 66)(29 67 36 59)(30 60 37 68)
(1 24 13 29)(2 30 14 25)(3 26 15 21)(4 22 11 27)(5 28 12 23)(6 32 17 37)(7 38 18 33)(8 34 19 39)(9 40 20 35)(10 36 16 31)(41 69 75 56)(42 57 76 70)(43 61 77 58)(44 59 78 62)(45 63 79 60)(46 51 80 64)(47 65 71 52)(48 53 72 66)(49 67 73 54)(50 55 74 68)
(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 8)(2 7)(3 6)(4 10)(5 9)(11 16)(12 20)(13 19)(14 18)(15 17)(21 25)(22 24)(26 30)(27 29)(31 39)(32 38)(33 37)(34 36)(41 79)(42 78)(43 77)(44 76)(45 75)(46 74)(47 73)(48 72)(49 71)(50 80)(51 63)(52 62)(53 61)(54 70)(55 69)(56 68)(57 67)(58 66)(59 65)(60 64)

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

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

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

50 conjugacy classes

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

50 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 D4 D5 D10 D10 2+ 1+4 D4×D5 D4⋊6D10 kernel C42⋊26D10 C42⋊D5 C4.D20 C20.17D4 C23⋊D10 Dic5⋊D4 C20⋊D4 C5×C4⋊1D4 C2×D4×D5 C2×D4⋊2D5 C4×D5 C4⋊1D4 C42 C2×D4 C10 C4 C2 # reps 1 1 1 1 4 4 1 1 1 1 4 2 2 12 2 4 8

Matrix representation of C4226D10 in GL8(𝔽41)

 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 2 0 0 0 0 0 2 1 39 39 0 0 0 0 40 0 1 0 0 0 0 0 1 1 0 40
,
 0 0 1 40 0 0 0 0 35 1 2 35 0 0 0 0 38 21 40 0 0 0 0 0 39 21 40 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 40 40 0 0 0 0 0 0 0 1
,
 0 6 0 0 0 0 0 0 34 7 0 0 0 0 0 0 11 6 35 35 0 0 0 0 16 12 6 40 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 40 0 0 0 0 0 1 1 0 40
,
 7 35 0 0 0 0 0 0 8 34 0 0 0 0 0 0 30 35 6 6 0 0 0 0 25 29 1 35 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 40 0 0 0 0 0 40 40 2 1

G:=sub<GL(8,GF(41))| [40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,2,40,1,0,0,0,0,0,1,0,1,0,0,0,0,2,39,1,0,0,0,0,0,0,39,0,40],[0,35,38,39,0,0,0,0,0,1,21,21,0,0,0,0,1,2,40,40,0,0,0,0,40,35,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,1],[0,34,11,16,0,0,0,0,6,7,6,12,0,0,0,0,0,0,35,6,0,0,0,0,0,0,35,40,0,0,0,0,0,0,0,0,1,0,1,1,0,0,0,0,0,1,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[7,8,30,25,0,0,0,0,35,34,35,29,0,0,0,0,0,0,6,1,0,0,0,0,0,0,6,35,0,0,0,0,0,0,0,0,1,0,1,40,0,0,0,0,0,40,0,40,0,0,0,0,0,0,40,2,0,0,0,0,0,0,0,1] >;

C4226D10 in GAP, Magma, Sage, TeX

C_4^2\rtimes_{26}D_{10}
% in TeX

G:=Group("C4^2:26D10");
// GroupNames label

G:=SmallGroup(320,1387);
// by ID

G=gap.SmallGroup(320,1387);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,219,675,570,297,136,12550]);
// Polycyclic

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

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