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## G = (D4×C10)⋊21C4order 320 = 26·5

### 5th semidirect product of D4×C10 and C4 acting via C4/C2=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — (D4×C10)⋊21C4
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C4.Dic5 — C2×C4.Dic5 — (D4×C10)⋊21C4
 Lower central C5 — C10 — C20 — (D4×C10)⋊21C4
 Upper central C1 — C4 — C22×C4 — C2×C4○D4

Generators and relations for (D4×C10)⋊21C4
G = < a,b,c,d | a10=b4=c2=d4=1, ab=ba, ac=ca, dad-1=a-1b2, cbc=b-1, bd=db, dcd-1=bc >

Subgroups: 398 in 154 conjugacy classes, 67 normal (43 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C10, C10, C42, C22⋊C4, C4⋊C4, 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, C42⋊C2, 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, C42⋊C22, C2×C52C8, C4.Dic5, C4.Dic5, C4×Dic5, C4⋊Dic5, C23.D5, C22×C20, C22×C20, D4×C10, D4×C10, Q8×C10, C5×C4○D4, C5×C4○D4, D42Dic5, C2×C4.Dic5, C23.21D10, C10×C4○D4, (D4×C10)⋊21C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C22⋊C4, C22×C4, C2×D4, Dic5, D10, C2×C22⋊C4, C2×Dic5, C5⋊D4, C22×D5, C42⋊C22, C23.D5, C22×Dic5, C2×C5⋊D4, C2×C23.D5, (D4×C10)⋊21C4

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

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

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

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

62 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 5A 5B 8A 8B 8C 8D 10A ··· 10F 10G ··· 10R 20A ··· 20H 20I ··· 20T order 1 2 2 2 2 2 2 4 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 2 2 2 4 4 1 1 2 2 2 4 4 20 20 20 20 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 1 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + - - - + image C1 C2 C2 C2 C2 C4 C4 C4 D4 D4 D5 D10 Dic5 Dic5 Dic5 D10 C5⋊D4 C5⋊D4 C42⋊C22 (D4×C10)⋊21C4 kernel (D4×C10)⋊21C4 D4⋊2Dic5 C2×C4.Dic5 C23.21D10 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 C2×C4 C23 C5 C1 # reps 1 4 1 1 1 2 2 4 3 1 2 2 2 2 4 4 12 4 2 8

Matrix representation of (D4×C10)⋊21C4 in GL4(𝔽41) generated by

 35 15 0 0 3 20 0 0 0 0 35 15 0 0 3 20
,
 32 0 0 0 0 32 0 0 0 0 9 0 0 0 0 9
,
 0 0 39 37 0 0 32 2 2 4 0 0 9 39 0 0
,
 6 6 0 0 1 35 0 0 0 0 28 28 0 0 32 13
G:=sub<GL(4,GF(41))| [35,3,0,0,15,20,0,0,0,0,35,3,0,0,15,20],[32,0,0,0,0,32,0,0,0,0,9,0,0,0,0,9],[0,0,2,9,0,0,4,39,39,32,0,0,37,2,0,0],[6,1,0,0,6,35,0,0,0,0,28,32,0,0,28,13] >;

(D4×C10)⋊21C4 in GAP, Magma, Sage, TeX

(D_4\times C_{10})\rtimes_{21}C_4
% in TeX

G:=Group("(D4xC10):21C4");
// GroupNames label

G:=SmallGroup(320,863);
// by ID

G=gap.SmallGroup(320,863);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,422,387,136,1684,438,102,12550]);
// Polycyclic

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

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