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## G = C10.(C4×D4)  order 320 = 26·5

### 7th non-split extension by C10 of C4×D4 acting via C4×D4/D4=C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C10.(C4×D4)
 Chief series C1 — C5 — D5 — D10 — C22×D5 — C22×F5 — C2×C4×F5 — C10.(C4×D4)
 Lower central C5 — C2×C10 — C10.(C4×D4)
 Upper central C1 — C22 — C22⋊C4

Generators and relations for C10.(C4×D4)
G = < a,b,c,d | a10=b4=c4=d2=1, bab-1=a3, ac=ca, ad=da, bc=cb, dbd=a5b, dcd=a5c-1 >

Subgroups: 874 in 190 conjugacy classes, 54 normal (50 characteristic)
C1, C2 [×3], C2 [×6], C4 [×10], C22, C22 [×16], C5, C2×C4 [×2], C2×C4 [×20], C23, C23 [×8], D5 [×4], D5, C10 [×3], C10, C42 [×2], C22⋊C4, C22⋊C4 [×7], C4⋊C4 [×2], C22×C4 [×6], C24, Dic5 [×2], C20 [×2], F5 [×6], D10 [×6], D10 [×7], C2×C10, C2×C10 [×3], C2.C42 [×2], C2×C42, C2×C22⋊C4 [×3], C2×C4⋊C4, C4×D5 [×4], C2×Dic5 [×2], C2×C20 [×2], C2×F5 [×4], C2×F5 [×10], C22×D5 [×2], C22×D5 [×6], C22×C10, C24.C22, D10⋊C4 [×2], C23.D5, C5×C22⋊C4, C4×F5 [×2], C4⋊F5 [×2], C22⋊F5 [×4], C2×C4×D5 [×2], C22×F5 [×4], C23×D5, D10.3Q8 [×2], D5×C22⋊C4, C2×C4×F5, C2×C4⋊F5, C2×C22⋊F5 [×2], C10.(C4×D4)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22×C4, C2×D4 [×2], C4○D4 [×4], F5, C42⋊C2, C4×D4 [×2], C4⋊D4, C22.D4, C4.4D4, C422C2, C2×F5 [×3], C24.C22, C22×F5, D10.C23, D4×F5 [×2], C10.(C4×D4)

Smallest permutation representation of C10.(C4×D4)
On 80 points
Generators in S80
(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 73)(2 80 10 76)(3 77 9 79)(4 74 8 72)(5 71 7 75)(6 78)(11 30)(12 27 20 23)(13 24 19 26)(14 21 18 29)(15 28 17 22)(16 25)(31 59 35 57)(32 56 34 60)(33 53)(36 54 40 52)(37 51 39 55)(38 58)(41 67 49 63)(42 64 48 66)(43 61 47 69)(44 68 46 62)(45 65)(50 70)
(1 53 30 65)(2 54 21 66)(3 55 22 67)(4 56 23 68)(5 57 24 69)(6 58 25 70)(7 59 26 61)(8 60 27 62)(9 51 28 63)(10 52 29 64)(11 45 73 33)(12 46 74 34)(13 47 75 35)(14 48 76 36)(15 49 77 37)(16 50 78 38)(17 41 79 39)(18 42 80 40)(19 43 71 31)(20 44 72 32)
(1 45)(2 46)(3 47)(4 48)(5 49)(6 50)(7 41)(8 42)(9 43)(10 44)(11 58)(12 59)(13 60)(14 51)(15 52)(16 53)(17 54)(18 55)(19 56)(20 57)(21 34)(22 35)(23 36)(24 37)(25 38)(26 39)(27 40)(28 31)(29 32)(30 33)(61 74)(62 75)(63 76)(64 77)(65 78)(66 79)(67 80)(68 71)(69 72)(70 73)

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

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

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

38 conjugacy classes

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

38 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 4 4 4 4 8 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 C4 D4 C4○D4 F5 C2×F5 C2×F5 D10.C23 D4×F5 kernel C10.(C4×D4) D10.3Q8 D5×C22⋊C4 C2×C4×F5 C2×C4⋊F5 C2×C22⋊F5 D10⋊C4 C23.D5 C5×C22⋊C4 C2×F5 D10 C22⋊C4 C2×C4 C23 C2 C2 # reps 1 2 1 1 1 2 4 2 2 4 8 1 2 1 4 2

Matrix representation of C10.(C4×D4) in GL8(𝔽41)

 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 0 0 40 1 0 0 0 0 0 0 40 0 0 0 0 0 1 0 40 0 0 0 0 0 0 1 40 0
,
 0 9 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 40 39 0 0 0 0 0 0 1 1 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
,
 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 32 0 0 0 0 0 0 0 0 32 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 0 40 0 0 0 0 0 0 0 0 1 2 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 0 40

G:=sub<GL(8,GF(41))| [40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,40,40,40,40,0,0,0,0,1,0,0,0],[0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,40,1,0,0,0,0,0,0,39,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,2,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40] >;

C10.(C4×D4) in GAP, Magma, Sage, TeX

C_{10}.(C_4\times D_4)
% in TeX

G:=Group("C10.(C4xD4)");
// GroupNames label

G:=SmallGroup(320,1038);
// by ID

G=gap.SmallGroup(320,1038);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,120,422,387,6278,1595]);
// Polycyclic

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

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