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## G = C40.50D4order 320 = 26·5

### 50th non-split extension by C40 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C40.50D4
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C4○D20 — D4.8D10 — C40.50D4
 Lower central C5 — C10 — C20 — C40.50D4
 Upper central C1 — C4 — C2×C8 — C8○D4

Generators and relations for C40.50D4
G = < a,b,c | a40=c2=1, b4=a20, bab-1=a29, cac=a9, cbc=a20b3 >

Subgroups: 326 in 104 conjugacy classes, 47 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, D5, C10, C10, C42, C2×C8, C2×C8, M4(2), M4(2), D8, SD16, Q16, C4○D4, C4○D4, Dic5, C20, C20, D10, C2×C10, C2×C10, C8⋊C4, C4≀C2, C8.C4, C8○D4, C8○D4, C4○D8, C52C8, C52C8, C40, C40, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C8.26D4, C8×D5, C8⋊D5, C2×C52C8, C4.Dic5, C4×Dic5, D4⋊D5, D4.D5, Q8⋊D5, C5⋊Q16, C2×C40, C2×C40, C5×M4(2), C5×M4(2), C4○D20, C5×C4○D4, C408C4, C20.53D4, D207C4, D42Dic5, D20.3C4, D4.8D10, C5×C8○D4, C40.50D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C22×C4, C2×D4, C4○D4, D10, C4×D4, C4×D5, C5⋊D4, C22×D5, C8.26D4, C2×C4×D5, C4○D20, C2×C5⋊D4, C4×C5⋊D4, C40.50D4

Smallest permutation representation of C40.50D4
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 16 11 26 21 36 31 6)(2 5 12 15 22 25 32 35)(3 34 13 4 23 14 33 24)(7 30 17 40 27 10 37 20)(8 19 18 29 28 39 38 9)(41 74 71 64 61 54 51 44)(42 63 72 53 62 43 52 73)(45 70 75 60 65 50 55 80)(46 59 76 49 66 79 56 69)(47 48 77 78 67 68 57 58)
(1 70)(2 79)(3 48)(4 57)(5 66)(6 75)(7 44)(8 53)(9 62)(10 71)(11 80)(12 49)(13 58)(14 67)(15 76)(16 45)(17 54)(18 63)(19 72)(20 41)(21 50)(22 59)(23 68)(24 77)(25 46)(26 55)(27 64)(28 73)(29 42)(30 51)(31 60)(32 69)(33 78)(34 47)(35 56)(36 65)(37 74)(38 43)(39 52)(40 61)

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

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

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

62 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 4F 4G 5A 5B 8A 8B 8C 8D 8E 8F 8G 8H 8I 8J 10A 10B 10C ··· 10H 20A 20B 20C 20D 20E ··· 20J 40A ··· 40H 40I ··· 40T order 1 2 2 2 2 4 4 4 4 4 4 4 5 5 8 8 8 8 8 8 8 8 8 8 10 10 10 ··· 10 20 20 20 20 20 ··· 20 40 ··· 40 40 ··· 40 size 1 1 2 4 20 1 1 2 4 20 20 20 2 2 2 2 2 2 4 4 20 20 20 20 2 2 4 ··· 4 2 2 2 2 4 ··· 4 2 ··· 2 4 ··· 4

62 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C4 C4 C4 C4 D4 D5 C4○D4 D10 D10 D10 C5⋊D4 C4×D5 C4×D5 C4○D20 C8.26D4 C40.50D4 kernel C40.50D4 C40⋊8C4 C20.53D4 D20⋊7C4 D4⋊2Dic5 D20.3C4 D4.8D10 C5×C8○D4 D4⋊D5 D4.D5 Q8⋊D5 C5⋊Q16 C40 C8○D4 C2×C10 C2×C8 M4(2) C4○D4 C8 D4 Q8 C22 C5 C1 # reps 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 8 4 4 8 2 8

Matrix representation of C40.50D4 in GL4(𝔽41) generated by

 34 34 0 0 7 35 0 0 0 0 7 7 0 0 34 6
,
 13 19 0 0 23 28 0 0 0 0 35 7 0 0 2 6
,
 0 0 35 7 0 0 2 6 13 19 0 0 23 28 0 0
G:=sub<GL(4,GF(41))| [34,7,0,0,34,35,0,0,0,0,7,34,0,0,7,6],[13,23,0,0,19,28,0,0,0,0,35,2,0,0,7,6],[0,0,13,23,0,0,19,28,35,2,0,0,7,6,0,0] >;

C40.50D4 in GAP, Magma, Sage, TeX

C_{40}._{50}D_4
% in TeX

G:=Group("C40.50D4");
// GroupNames label

G:=SmallGroup(320,772);
// by ID

G=gap.SmallGroup(320,772);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,387,58,136,1684,851,102,12550]);
// Polycyclic

G:=Group<a,b,c|a^40=c^2=1,b^4=a^20,b*a*b^-1=a^29,c*a*c=a^9,c*b*c=a^20*b^3>;
// generators/relations

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