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## G = (D4×C10)⋊18C4order 320 = 26·5

### 2nd 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)⋊18C4
 Chief series C1 — C5 — C10 — C2×C10 — C2×C20 — C4⋊Dic5 — C23.21D10 — (D4×C10)⋊18C4
 Lower central C5 — C10 — C20 — (D4×C10)⋊18C4
 Upper central C1 — C22 — C22×C4 — C22×D4

Generators and relations for (D4×C10)⋊18C4
G = < a,b,c,d | a10=b4=c2=d4=1, ab=ba, ac=ca, dad-1=a-1b2, cbc=dbd-1=b-1, dcd-1=b-1c >

Subgroups: 526 in 190 conjugacy classes, 71 normal (23 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, D4, D4, C23, C23, C10, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C2×D4, C2×D4, C24, Dic5, C20, C20, C2×C10, C2×C10, C2×C10, D4⋊C4, C42⋊C2, C2×M4(2), C22×D4, C52C8, C2×Dic5, C2×C20, C2×C20, C5×D4, C5×D4, C22×C10, C22×C10, C23.37D4, C2×C52C8, C4.Dic5, C4×Dic5, C4⋊Dic5, C23.D5, C22×C20, D4×C10, D4×C10, C23×C10, D4⋊Dic5, C2×C4.Dic5, C23.21D10, D4×C2×C10, (D4×C10)⋊18C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C22⋊C4, C22×C4, C2×D4, Dic5, D10, C2×C22⋊C4, C8⋊C22, C2×Dic5, C5⋊D4, C22×D5, C23.37D4, C23.D5, C22×Dic5, C2×C5⋊D4, D4.D10, C2×C23.D5, (D4×C10)⋊18C4

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

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

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

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

62 conjugacy classes

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

62 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + - + + image C1 C2 C2 C2 C2 C4 D4 D4 D5 D10 Dic5 D10 C5⋊D4 C5⋊D4 C8⋊C22 D4.D10 kernel (D4×C10)⋊18C4 D4⋊Dic5 C2×C4.Dic5 C23.21D10 D4×C2×C10 D4×C10 C2×C20 C22×C10 C22×D4 C22×C4 C2×D4 C2×D4 C2×C4 C23 C10 C2 # reps 1 4 1 1 1 8 3 1 2 2 8 4 12 4 2 8

Matrix representation of (D4×C10)⋊18C4 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 10 0 0 0 0 0 0 10
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 39 0 0 0 0 1 1 0 0 0 0 0 0 1 2 0 0 0 0 40 40
,
 1 0 0 0 0 0 0 40 0 0 0 0 0 0 40 39 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 40 40
,
 0 40 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,10,0,0,0,0,0,0,10],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,1,0,0,0,0,39,1,0,0,0,0,0,0,1,40,0,0,0,0,2,40],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,39,1,0,0,0,0,0,0,1,40,0,0,0,0,0,40],[0,1,0,0,0,0,40,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0] >;

(D4×C10)⋊18C4 in GAP, Magma, Sage, TeX

(D_4\times C_{10})\rtimes_{18}C_4
% in TeX

G:=Group("(D4xC10):18C4");
// GroupNames label

G:=SmallGroup(320,842);
// by ID

G=gap.SmallGroup(320,842);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,232,422,387,1684,438,102,12550]);
// 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=d*b*d^-1=b^-1,d*c*d^-1=b^-1*c>;
// generators/relations

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