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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 [×2], C2 [×18], C4 [×4], C4 [×8], C22, C22 [×10], C22 [×50], C5, C2×C4 [×6], C2×C4 [×36], D4 [×16], D4 [×56], Q8 [×8], C23, C23 [×12], C23 [×32], D5 [×8], C10, C10 [×2], C10 [×10], C22×C4, C22×C4 [×8], C2×D4 [×12], C2×D4 [×78], C2×Q8 [×2], C4○D4 [×48], C24 [×2], C24 [×4], Dic5 [×8], C20 [×4], D10 [×8], D10 [×24], C2×C10, C2×C10 [×10], C2×C10 [×18], C22×D4, C22×D4 [×8], C2×C4○D4 [×6], 2+ 1+4 [×16], Dic10 [×8], C4×D5 [×16], D20 [×8], C2×Dic5 [×20], C5⋊D4 [×48], C2×C20 [×6], C5×D4 [×16], C22×D5 [×20], C22×D5 [×8], C22×C10, C22×C10 [×12], C22×C10 [×4], C2×2+ 1+4, C2×Dic10 [×2], C2×C4×D5 [×4], C2×D20 [×2], C4○D20 [×16], D4×D5 [×32], D42D5 [×32], C22×Dic5 [×4], C2×C5⋊D4 [×44], C22×C20, D4×C10 [×12], C23×D5 [×4], C23×C10 [×2], C2×C4○D20 [×2], C2×D4×D5 [×4], C2×D42D5 [×4], D46D10 [×16], C22×C5⋊D4 [×4], D4×C2×C10, C2×D46D10
Quotients: C1, C2 [×31], C22 [×155], C23 [×155], D5, C24 [×31], D10 [×15], 2+ 1+4 [×2], C25, C22×D5 [×35], C2×2+ 1+4, C23×D5 [×15], D46D10 [×2], D5×C24, C2×D46D10

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

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

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

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

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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