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## G = C2×D4⋊6D10order 320 = 26·5

### Direct product of C2 and D4⋊6D10

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C2×D4⋊6D10
 Chief series C1 — C5 — C10 — D10 — C22×D5 — C23×D5 — C2×D4×D5 — C2×D4⋊6D10
 Lower central C5 — C10 — C2×D4⋊6D10
 Upper central C1 — C22 — C22×D4

Generators and relations for C2×D46D10
G = < a,b,c,d,e | a2=b4=c2=d10=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=dbd-1=b-1, be=eb, dcd-1=ece=b2c, ede=d-1 >

Subgroups: 2926 in 898 conjugacy classes, 447 normal (13 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, D4, D4, Q8, C23, C23, C23, D5, C10, C10, C10, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, C24, Dic5, C20, D10, D10, C2×C10, C2×C10, C2×C10, C22×D4, C22×D4, C2×C4○D4, 2+ 1+4, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C22×D5, C22×D5, C22×C10, C22×C10, C22×C10, C2×2+ 1+4, C2×Dic10, C2×C4×D5, C2×D20, C4○D20, D4×D5, D42D5, C22×Dic5, C2×C5⋊D4, C22×C20, D4×C10, C23×D5, C23×C10, C2×C4○D20, C2×D4×D5, C2×D42D5, D46D10, C22×C5⋊D4, D4×C2×C10, C2×D46D10
Quotients: C1, C2, C22, C23, D5, C24, D10, 2+ 1+4, C25, C22×D5, C2×2+ 1+4, C23×D5, D46D10, D5×C24, C2×D46D10

Smallest permutation representation of C2×D46D10
On 80 points
Generators in S80
(1 51)(2 52)(3 53)(4 54)(5 55)(6 56)(7 57)(8 58)(9 59)(10 60)(11 69)(12 70)(13 61)(14 62)(15 63)(16 64)(17 65)(18 66)(19 67)(20 68)(21 45)(22 46)(23 47)(24 48)(25 49)(26 50)(27 41)(28 42)(29 43)(30 44)(31 71)(32 72)(33 73)(34 74)(35 75)(36 76)(37 77)(38 78)(39 79)(40 80)
(1 56 37 72)(2 73 38 57)(3 58 39 74)(4 75 40 59)(5 60 31 76)(6 77 32 51)(7 52 33 78)(8 79 34 53)(9 54 35 80)(10 71 36 55)(11 27 46 64)(12 65 47 28)(13 29 48 66)(14 67 49 30)(15 21 50 68)(16 69 41 22)(17 23 42 70)(18 61 43 24)(19 25 44 62)(20 63 45 26)
(1 68)(2 22)(3 70)(4 24)(5 62)(6 26)(7 64)(8 28)(9 66)(10 30)(11 78)(12 53)(13 80)(14 55)(15 72)(16 57)(17 74)(18 59)(19 76)(20 51)(21 37)(23 39)(25 31)(27 33)(29 35)(32 63)(34 65)(36 67)(38 69)(40 61)(41 73)(42 58)(43 75)(44 60)(45 77)(46 52)(47 79)(48 54)(49 71)(50 56)
(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 55)(2 54)(3 53)(4 52)(5 51)(6 60)(7 59)(8 58)(9 57)(10 56)(11 24)(12 23)(13 22)(14 21)(15 30)(16 29)(17 28)(18 27)(19 26)(20 25)(31 77)(32 76)(33 75)(34 74)(35 73)(36 72)(37 71)(38 80)(39 79)(40 78)(41 66)(42 65)(43 64)(44 63)(45 62)(46 61)(47 70)(48 69)(49 68)(50 67)

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

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

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

74 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2M 2N ··· 2U 4A 4B 4C 4D 4E ··· 4L 5A 5B 10A ··· 10N 10O ··· 10AD 20A ··· 20H order 1 2 2 2 2 ··· 2 2 ··· 2 4 4 4 4 4 ··· 4 5 5 10 ··· 10 10 ··· 10 20 ··· 20 size 1 1 1 1 2 ··· 2 10 ··· 10 2 2 2 2 10 ··· 10 2 2 2 ··· 2 4 ··· 4 4 ··· 4

74 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 4 4 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 D5 D10 D10 D10 2+ 1+4 D4⋊6D10 kernel C2×D4⋊6D10 C2×C4○D20 C2×D4×D5 C2×D4⋊2D5 D4⋊6D10 C22×C5⋊D4 D4×C2×C10 C22×D4 C22×C4 C2×D4 C24 C10 C2 # reps 1 2 4 4 16 4 1 2 2 24 4 2 8

Matrix representation of C2×D46D10 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 9 0 0 0 0 1 0 9 0 0 18 0 40 0 0 0 0 18 0 40
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 40 0 0 0 0 40 0 0 0 0 0 0 23 0 1 0 0 23 0 1 0
,
 6 35 0 0 0 0 6 1 0 0 0 0 0 0 1 0 9 0 0 0 0 1 0 9 0 0 0 0 40 0 0 0 0 0 0 40
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 1

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,18,0,0,0,0,1,0,18,0,0,9,0,40,0,0,0,0,9,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,23,0,0,40,0,23,0,0,0,0,0,0,1,0,0,0,0,1,0],[6,6,0,0,0,0,35,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,9,0,40,0,0,0,0,9,0,40],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,1] >;

C2×D46D10 in GAP, Magma, Sage, TeX

C_2\times D_4\rtimes_6D_{10}
% in TeX

G:=Group("C2xD4:6D10");
// GroupNames label

G:=SmallGroup(320,1614);
// by ID

G=gap.SmallGroup(320,1614);
# by ID

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

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

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