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## G = C2×D4⋊D10order 320 = 26·5

### Direct product of C2 and D4⋊D10

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C2×D4⋊D10
 Chief series C1 — C5 — C10 — C20 — D20 — C2×D20 — C22×D20 — C2×D4⋊D10
 Lower central C5 — C10 — C20 — C2×D4⋊D10
 Upper central C1 — C22 — C22×C4 — C2×C4○D4

Generators and relations for C2×D4⋊D10
G = < a,b,c,d,e | a2=b4=c2=d10=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe=b-1, bd=db, dcd-1=b2c, ece=b-1c, ede=d-1 >

Subgroups: 1214 in 298 conjugacy classes, 111 normal (27 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, D5, C10, C10, C10, C2×C8, M4(2), D8, SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4○D4, C24, C20, C20, C20, D10, C2×C10, C2×C10, C2×C10, C2×M4(2), C2×D8, C2×SD16, C8⋊C22, C22×D4, C2×C4○D4, C52C8, D20, D20, C2×C20, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C5×Q8, C22×D5, C22×C10, C22×C10, C2×C8⋊C22, C2×C52C8, C4.Dic5, D4⋊D5, Q8⋊D5, C2×D20, C2×D20, C22×C20, C22×C20, D4×C10, D4×C10, Q8×C10, C5×C4○D4, C5×C4○D4, C23×D5, C2×C4.Dic5, C2×D4⋊D5, C2×Q8⋊D5, D4⋊D10, C22×D20, C10×C4○D4, C2×D4⋊D10
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C24, D10, C8⋊C22, C22×D4, C5⋊D4, C22×D5, C2×C8⋊C22, C2×C5⋊D4, C23×D5, D4⋊D10, C22×C5⋊D4, C2×D4⋊D10

Smallest permutation representation of C2×D4⋊D10
On 80 points
Generators in S80
(1 14)(2 15)(3 11)(4 12)(5 13)(6 34)(7 35)(8 31)(9 32)(10 33)(16 30)(17 26)(18 27)(19 28)(20 29)(21 39)(22 40)(23 36)(24 37)(25 38)(41 78)(42 79)(43 80)(44 71)(45 72)(46 73)(47 74)(48 75)(49 76)(50 77)(51 69)(52 70)(53 61)(54 62)(55 63)(56 64)(57 65)(58 66)(59 67)(60 68)
(1 17 7 24)(2 18 8 25)(3 19 9 21)(4 20 10 22)(5 16 6 23)(11 28 32 39)(12 29 33 40)(13 30 34 36)(14 26 35 37)(15 27 31 38)(41 69 46 64)(42 70 47 65)(43 61 48 66)(44 62 49 67)(45 63 50 68)(51 73 56 78)(52 74 57 79)(53 75 58 80)(54 76 59 71)(55 77 60 72)
(1 42)(2 48)(3 44)(4 50)(5 46)(6 41)(7 47)(8 43)(9 49)(10 45)(11 71)(12 77)(13 73)(14 79)(15 75)(16 69)(17 65)(18 61)(19 67)(20 63)(21 62)(22 68)(23 64)(24 70)(25 66)(26 57)(27 53)(28 59)(29 55)(30 51)(31 80)(32 76)(33 72)(34 78)(35 74)(36 56)(37 52)(38 58)(39 54)(40 60)
(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 13)(2 12)(3 11)(4 15)(5 14)(6 35)(7 34)(8 33)(9 32)(10 31)(16 37)(17 36)(18 40)(19 39)(20 38)(21 28)(22 27)(23 26)(24 30)(25 29)(41 57)(42 56)(43 55)(44 54)(45 53)(46 52)(47 51)(48 60)(49 59)(50 58)(61 72)(62 71)(63 80)(64 79)(65 78)(66 77)(67 76)(68 75)(69 74)(70 73)

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

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

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

62 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 4A 4B 4C 4D 4E 4F 5A 5B 8A 8B 8C 8D 10A ··· 10F 10G ··· 10R 20A ··· 20H 20I ··· 20T order 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 5 5 8 8 8 8 10 ··· 10 10 ··· 10 20 ··· 20 20 ··· 20 size 1 1 1 1 2 2 4 4 20 20 20 20 2 2 2 2 4 4 2 2 20 20 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 2 4 4 type + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 D4 D4 D5 D10 D10 D10 D10 C5⋊D4 C5⋊D4 C8⋊C22 D4⋊D10 kernel C2×D4⋊D10 C2×C4.Dic5 C2×D4⋊D5 C2×Q8⋊D5 D4⋊D10 C22×D20 C10×C4○D4 C2×C20 C22×C10 C2×C4○D4 C22×C4 C2×D4 C2×Q8 C4○D4 C2×C4 C23 C10 C2 # reps 1 1 2 2 8 1 1 3 1 2 2 2 2 8 12 4 2 8

Matrix representation of C2×D4⋊D10 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 30 28 0 0 0 0 22 11 0 0 0 0 19 24 11 13 0 0 13 32 19 30
,
 1 16 0 0 0 0 0 40 0 0 0 0 0 0 22 17 19 15 0 0 28 9 3 22 0 0 10 27 19 24 0 0 30 40 13 32
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 0 6 0 0 0 0 34 34 0 0 0 0 37 9 0 35 0 0 10 6 7 7
,
 40 0 0 0 0 0 36 1 0 0 0 0 0 0 1 0 0 0 0 0 33 40 0 0 0 0 0 0 11 13 0 0 0 0 16 30

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,30,22,19,13,0,0,28,11,24,32,0,0,0,0,11,19,0,0,0,0,13,30],[1,0,0,0,0,0,16,40,0,0,0,0,0,0,22,28,10,30,0,0,17,9,27,40,0,0,19,3,19,13,0,0,15,22,24,32],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,34,37,10,0,0,6,34,9,6,0,0,0,0,0,7,0,0,0,0,35,7],[40,36,0,0,0,0,0,1,0,0,0,0,0,0,1,33,0,0,0,0,0,40,0,0,0,0,0,0,11,16,0,0,0,0,13,30] >;

C2×D4⋊D10 in GAP, Magma, Sage, TeX

C_2\times D_4\rtimes D_{10}
% in TeX

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

G:=SmallGroup(320,1492);
// by ID

G=gap.SmallGroup(320,1492);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,675,297,1684,235,102,12550]);
// 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=e*b*e=b^-1,b*d=d*b,d*c*d^-1=b^2*c,e*c*e=b^-1*c,e*d*e=d^-1>;
// generators/relations

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