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## G = C2×D4⋊2Dic5order 320 = 26·5

### Direct product of C2 and D4⋊2Dic5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C2×D4⋊2Dic5
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C4.Dic5 — C2×C4.Dic5 — C2×D4⋊2Dic5
 Lower central C5 — C10 — C20 — C2×D4⋊2Dic5
 Upper central C1 — C2×C4 — C22×C4 — C2×C4○D4

Generators and relations for C2×D42Dic5
G = < a,b,c,d,e | a2=b4=d10=1, c2=b2, e2=d5, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, dcd-1=b2c, ece-1=b-1c, ede-1=d-1 >

Subgroups: 446 in 170 conjugacy classes, 71 normal (43 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C10, C10, C10, C42, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4○D4, Dic5, C20, C20, C2×C10, C2×C10, C4≀C2, C2×C42, C2×M4(2), C2×C4○D4, C52C8, C2×Dic5, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C5×Q8, C22×C10, C22×C10, C2×C4≀C2, C2×C52C8, C4.Dic5, C4.Dic5, C4×Dic5, C4×Dic5, C22×Dic5, C22×C20, C22×C20, D4×C10, D4×C10, Q8×C10, C5×C4○D4, C5×C4○D4, D42Dic5, C2×C4.Dic5, C2×C4×Dic5, C10×C4○D4, C2×D42Dic5
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C22⋊C4, C22×C4, C2×D4, Dic5, D10, C4≀C2, C2×C22⋊C4, C2×Dic5, C5⋊D4, C22×D5, C2×C4≀C2, C23.D5, C22×Dic5, C2×C5⋊D4, D42Dic5, C2×C23.D5, C2×D42Dic5

Smallest permutation representation of C2×D42Dic5
On 80 points
Generators in S80
(1 34)(2 35)(3 31)(4 32)(5 33)(6 28)(7 29)(8 30)(9 26)(10 27)(11 20)(12 16)(13 17)(14 18)(15 19)(21 39)(22 40)(23 36)(24 37)(25 38)(41 69)(42 70)(43 61)(44 62)(45 63)(46 64)(47 65)(48 66)(49 67)(50 68)(51 73)(52 74)(53 75)(54 76)(55 77)(56 78)(57 79)(58 80)(59 71)(60 72)
(1 7 18 37)(2 8 19 38)(3 9 20 39)(4 10 16 40)(5 6 17 36)(11 21 31 26)(12 22 32 27)(13 23 33 28)(14 24 34 29)(15 25 35 30)(41 56 46 51)(42 57 47 52)(43 58 48 53)(44 59 49 54)(45 60 50 55)(61 80 66 75)(62 71 67 76)(63 72 68 77)(64 73 69 78)(65 74 70 79)
(1 67 18 62)(2 63 19 68)(3 69 20 64)(4 65 16 70)(5 61 17 66)(6 75 36 80)(7 71 37 76)(8 77 38 72)(9 73 39 78)(10 79 40 74)(11 46 31 41)(12 42 32 47)(13 48 33 43)(14 44 34 49)(15 50 35 45)(21 56 26 51)(22 52 27 57)(23 58 28 53)(24 54 29 59)(25 60 30 55)
(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 5)(2 4)(6 7)(8 10)(12 15)(13 14)(16 19)(17 18)(22 25)(23 24)(27 30)(28 29)(32 35)(33 34)(36 37)(38 40)(41 51 46 56)(42 60 47 55)(43 59 48 54)(44 58 49 53)(45 57 50 52)(61 71 66 76)(62 80 67 75)(63 79 68 74)(64 78 69 73)(65 77 70 72)

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

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

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

68 conjugacy classes

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

68 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 type + + + + + + + + + - - - + image C1 C2 C2 C2 C2 C4 C4 C4 D4 D4 D5 D10 Dic5 Dic5 Dic5 D10 C4≀C2 C5⋊D4 C5⋊D4 D4⋊2Dic5 kernel C2×D4⋊2Dic5 D4⋊2Dic5 C2×C4.Dic5 C2×C4×Dic5 C10×C4○D4 D4×C10 Q8×C10 C5×C4○D4 C2×C20 C22×C10 C2×C4○D4 C22×C4 C2×D4 C2×Q8 C4○D4 C4○D4 C10 C2×C4 C23 C2 # reps 1 4 1 1 1 2 2 4 3 1 2 2 2 2 4 4 8 12 4 8

Matrix representation of C2×D42Dic5 in GL4(𝔽41) generated by

 40 0 0 0 0 40 0 0 0 0 1 0 0 0 0 1
,
 40 0 0 0 0 40 0 0 0 0 9 0 0 0 0 32
,
 17 35 0 0 7 24 0 0 0 0 0 9 0 0 9 0
,
 40 40 0 0 8 7 0 0 0 0 40 0 0 0 0 1
,
 7 6 0 0 33 34 0 0 0 0 32 0 0 0 0 1
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[40,0,0,0,0,40,0,0,0,0,9,0,0,0,0,32],[17,7,0,0,35,24,0,0,0,0,0,9,0,0,9,0],[40,8,0,0,40,7,0,0,0,0,40,0,0,0,0,1],[7,33,0,0,6,34,0,0,0,0,32,0,0,0,0,1] >;

C2×D42Dic5 in GAP, Magma, Sage, TeX

C_2\times D_4\rtimes_2{\rm Dic}_5
% in TeX

G:=Group("C2xD4:2Dic5");
// GroupNames label

G:=SmallGroup(320,862);
// by ID

G=gap.SmallGroup(320,862);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,422,136,1684,438,102,12550]);
// Polycyclic

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

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