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## G = (D4×C10)⋊22C4order 320 = 26·5

### 6th semidirect product of D4×C10 and C4 acting via C4/C2=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — (D4×C10)⋊22C4
 Chief series C1 — C5 — C10 — C2×C10 — C22×C10 — C23.D5 — C23.21D10 — (D4×C10)⋊22C4
 Lower central C5 — C10 — C2×C10 — (D4×C10)⋊22C4
 Upper central C1 — C4 — C22×C4 — C2×C4○D4

Generators and relations for (D4×C10)⋊22C4
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=a5b2c >

Subgroups: 446 in 158 conjugacy classes, 67 normal (25 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C10, C10, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, Dic5, C20, C20, C20, C2×C10, C2×C10, C2×C10, C23⋊C4, C42⋊C2, C2×C4○D4, C2×Dic5, C2×C20, C2×C20, C2×C20, C5×D4, C5×Q8, C22×C10, C22×C10, C23.C23, C4×Dic5, C4⋊Dic5, C23.D5, C22×C20, C22×C20, D4×C10, D4×C10, Q8×C10, C5×C4○D4, C23⋊Dic5, C23.21D10, C10×C4○D4, (D4×C10)⋊22C4
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, C23.C23, C23.D5, C22×Dic5, C2×C5⋊D4, C2×C23.D5, (D4×C10)⋊22C4

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

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

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

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

62 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 4E 4F 4G 4H ··· 4O 5A 5B 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 5 5 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 2 2 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4

62 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + - + - + - image C1 C2 C2 C2 C4 C4 C4 D4 D5 Dic5 D10 Dic5 D10 Dic5 C5⋊D4 C23.C23 (D4×C10)⋊22C4 kernel (D4×C10)⋊22C4 C23⋊Dic5 C23.21D10 C10×C4○D4 C22×C20 D4×C10 Q8×C10 C2×C20 C2×C4○D4 C22×C4 C22×C4 C2×D4 C2×D4 C2×Q8 C2×C4 C5 C1 # reps 1 4 2 1 4 2 2 4 2 4 2 2 4 2 16 2 8

Matrix representation of (D4×C10)⋊22C4 in GL4(𝔽41) generated by

 23 21 0 0 20 20 0 0 0 0 40 4 0 0 20 3
,
 32 0 36 20 0 32 15 21 0 0 9 0 0 0 0 9
,
 32 0 36 20 0 32 15 21 33 33 9 0 35 39 0 9
,
 32 28 4 12 0 9 16 7 0 0 14 16 0 0 16 27
G:=sub<GL(4,GF(41))| [23,20,0,0,21,20,0,0,0,0,40,20,0,0,4,3],[32,0,0,0,0,32,0,0,36,15,9,0,20,21,0,9],[32,0,33,35,0,32,33,39,36,15,9,0,20,21,0,9],[32,0,0,0,28,9,0,0,4,16,14,16,12,7,16,27] >;

(D4×C10)⋊22C4 in GAP, Magma, Sage, TeX

(D_4\times C_{10})\rtimes_{22}C_4
% in TeX

G:=Group("(D4xC10):22C4");
// GroupNames label

G:=SmallGroup(320,867);
// by ID

G=gap.SmallGroup(320,867);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,232,422,297,1684,12550]);
// 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=a^5*b^2*c>;
// generators/relations

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