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## G = C2×D10.3Q8order 320 = 26·5

### Direct product of C2 and D10.3Q8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C2×D10.3Q8
 Chief series C1 — C5 — D5 — D10 — C22×D5 — C22×F5 — C23×F5 — C2×D10.3Q8
 Lower central C5 — C10 — C2×D10.3Q8
 Upper central C1 — C23 — C22×C4

Generators and relations for C2×D10.3Q8
G = < a,b,c,d,e | a2=b10=c2=d4=1, e2=b4cd2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, ebe-1=b3, cd=dc, ece-1=b2c, ede-1=b5d-1 >

Subgroups: 1242 in 330 conjugacy classes, 124 normal (22 characteristic)
C1, C2 [×3], C2 [×4], C2 [×8], C4 [×12], C22, C22 [×6], C22 [×28], C5, C2×C4 [×2], C2×C4 [×46], C23, C23 [×14], D5 [×8], C10 [×3], C10 [×4], C22×C4, C22×C4 [×29], C24, Dic5 [×2], C20 [×2], F5 [×8], D10 [×2], D10 [×26], C2×C10, C2×C10 [×6], C2.C42 [×4], C23×C4 [×3], C4×D5 [×8], C2×Dic5 [×2], C2×Dic5 [×2], C2×C20 [×2], C2×C20 [×2], C2×F5 [×8], C2×F5 [×24], C22×D5 [×2], C22×D5 [×12], C22×C10, C2×C2.C42, C2×C4×D5 [×4], C2×C4×D5 [×4], C22×Dic5, C22×C20, C22×F5 [×12], C22×F5 [×8], C23×D5, D10.3Q8 [×4], D5×C22×C4, C23×F5 [×2], C2×D10.3Q8
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], D4 [×6], Q8 [×2], C23, C42 [×4], C22⋊C4 [×12], C4⋊C4 [×12], C22×C4 [×3], C2×D4 [×3], C2×Q8, F5, C2.C42 [×8], C2×C42, C2×C22⋊C4 [×3], C2×C4⋊C4 [×3], C2×F5 [×3], C2×C2.C42, C4×F5 [×2], C4⋊F5 [×2], C22⋊F5 [×2], C22×F5, D10.3Q8 [×4], C2×C4×F5, C2×C4⋊F5, C2×C22⋊F5, C2×D10.3Q8

Smallest permutation representation of C2×D10.3Q8
On 80 points
Generators in S80
(1 52)(2 53)(3 54)(4 55)(5 56)(6 57)(7 58)(8 59)(9 60)(10 51)(11 37)(12 38)(13 39)(14 40)(15 31)(16 32)(17 33)(18 34)(19 35)(20 36)(21 69)(22 70)(23 61)(24 62)(25 63)(26 64)(27 65)(28 66)(29 67)(30 68)(41 73)(42 74)(43 75)(44 76)(45 77)(46 78)(47 79)(48 80)(49 71)(50 72)
(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 21)(2 30)(3 29)(4 28)(5 27)(6 26)(7 25)(8 24)(9 23)(10 22)(11 76)(12 75)(13 74)(14 73)(15 72)(16 71)(17 80)(18 79)(19 78)(20 77)(31 50)(32 49)(33 48)(34 47)(35 46)(36 45)(37 44)(38 43)(39 42)(40 41)(51 70)(52 69)(53 68)(54 67)(55 66)(56 65)(57 64)(58 63)(59 62)(60 61)
(1 77 27 16)(2 78 28 17)(3 79 29 18)(4 80 30 19)(5 71 21 20)(6 72 22 11)(7 73 23 12)(8 74 24 13)(9 75 25 14)(10 76 26 15)(31 51 44 64)(32 52 45 65)(33 53 46 66)(34 54 47 67)(35 55 48 68)(36 56 49 69)(37 57 50 70)(38 58 41 61)(39 59 42 62)(40 60 43 63)
(2 8 10 4)(3 5 9 7)(11 77)(12 74 20 80)(13 71 19 73)(14 78 18 76)(15 75 17 79)(16 72)(21 25 23 29)(24 26 30 28)(31 43 33 47)(32 50)(34 44 40 46)(35 41 39 49)(36 48 38 42)(37 45)(51 55 53 59)(54 56 60 58)(61 67 69 63)(62 64 68 66)

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

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

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

56 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2O 4A 4B 4C 4D 4E ··· 4X 5 10A ··· 10G 20A ··· 20H order 1 2 ··· 2 2 ··· 2 4 4 4 4 4 ··· 4 5 10 ··· 10 20 ··· 20 size 1 1 ··· 1 5 ··· 5 2 2 2 2 10 ··· 10 4 4 ··· 4 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 4 4 4 4 4 4 type + + + + + - + + + + image C1 C2 C2 C2 C4 C4 C4 C4 D4 Q8 F5 C2×F5 C2×F5 C4×F5 C4⋊F5 C22⋊F5 kernel C2×D10.3Q8 D10.3Q8 D5×C22×C4 C23×F5 C2×C4×D5 C22×Dic5 C22×C20 C22×F5 C22×D5 C22×D5 C22×C4 C2×C4 C23 C22 C22 C22 # reps 1 4 1 2 4 2 2 16 6 2 1 2 1 4 4 4

Matrix representation of C2×D10.3Q8 in GL8(𝔽41)

 40 0 0 0 0 0 0 0 0 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 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 0 1 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 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 1 1 1 1 0 0 0 0 40 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 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0
,
 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 0 9 2 0 0 0 0 0 0 1 32 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 9
,
 1 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 1 32 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0

G:=sub<GL(8,GF(41))| [40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,1,40,0,0,0,0,0,0,1,0,0,0,0,0,40,0,1,0,0,0,0,0,0,40,1,0],[40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,40,0,0,0,0,0,1,40,0],[0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,9,1,0,0,0,0,0,0,2,32,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,9],[1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,9,1,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0] >;

C2×D10.3Q8 in GAP, Magma, Sage, TeX

C_2\times D_{10}._3Q_8
% in TeX

G:=Group("C2xD10.3Q8");
// GroupNames label

G:=SmallGroup(320,1100);
// by ID

G=gap.SmallGroup(320,1100);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,758,100,6278,1595]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^10=c^2=d^4=1,e^2=b^4*c*d^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,e*b*e^-1=b^3,c*d=d*c,e*c*e^-1=b^2*c,e*d*e^-1=b^5*d^-1>;
// generators/relations

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