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## G = C4⋊C4⋊D5order 160 = 25·5

### 6th semidirect product of C4⋊C4 and D5 acting via D5/C5=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C4⋊C4⋊D5
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — D10⋊C4 — C4⋊C4⋊D5
 Lower central C5 — C2×C10 — C4⋊C4⋊D5
 Upper central C1 — C22 — C4⋊C4

Generators and relations for C4⋊C4⋊D5
G = < a,b,c,d | a4=b4=c5=d2=1, bab-1=a-1, ac=ca, dad=ab2, bc=cb, dbd=a2b, dcd=c-1 >

Subgroups: 192 in 60 conjugacy classes, 29 normal (all characteristic)
C1, C2 [×3], C2, C4 [×6], C22, C22 [×3], C5, C2×C4 [×3], C2×C4 [×3], C23, D5, C10 [×3], C42, C22⋊C4 [×3], C4⋊C4, C4⋊C4 [×2], Dic5 [×3], C20 [×3], D10 [×3], C2×C10, C422C2, C2×Dic5 [×3], C2×C20 [×3], C22×D5, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4 [×3], C5×C4⋊C4, C4⋊C4⋊D5
Quotients: C1, C2 [×7], C22 [×7], C23, D5, C4○D4 [×3], D10 [×3], C422C2, C22×D5, C4○D20, D42D5, Q82D5, C4⋊C4⋊D5

Smallest permutation representation of C4⋊C4⋊D5
On 80 points
Generators in S80
(1 39 9 34)(2 40 10 35)(3 36 6 31)(4 37 7 32)(5 38 8 33)(11 26 16 21)(12 27 17 22)(13 28 18 23)(14 29 19 24)(15 30 20 25)(41 71 46 76)(42 72 47 77)(43 73 48 78)(44 74 49 79)(45 75 50 80)(51 61 56 66)(52 62 57 67)(53 63 58 68)(54 64 59 69)(55 65 60 70)
(1 54 14 44)(2 55 15 45)(3 51 11 41)(4 52 12 42)(5 53 13 43)(6 56 16 46)(7 57 17 47)(8 58 18 48)(9 59 19 49)(10 60 20 50)(21 71 31 61)(22 72 32 62)(23 73 33 63)(24 74 34 64)(25 75 35 65)(26 76 36 66)(27 77 37 67)(28 78 38 68)(29 79 39 69)(30 80 40 70)
(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 5)(2 4)(7 10)(8 9)(12 15)(13 14)(17 20)(18 19)(21 31)(22 35)(23 34)(24 33)(25 32)(26 36)(27 40)(28 39)(29 38)(30 37)(41 46)(42 50)(43 49)(44 48)(45 47)(51 56)(52 60)(53 59)(54 58)(55 57)(61 76)(62 80)(63 79)(64 78)(65 77)(66 71)(67 75)(68 74)(69 73)(70 72)

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

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

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

34 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 4F 4G 4H 4I 5A 5B 10A ··· 10F 20A ··· 20L order 1 2 2 2 2 4 4 4 4 4 4 4 4 4 5 5 10 ··· 10 20 ··· 20 size 1 1 1 1 20 2 2 4 4 10 10 10 10 20 2 2 2 ··· 2 4 ··· 4

34 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 4 4 type + + + + + + + + - + image C1 C2 C2 C2 C2 C2 D5 C4○D4 D10 C4○D20 D4⋊2D5 Q8⋊2D5 kernel C4⋊C4⋊D5 C4×Dic5 C10.D4 C4⋊Dic5 D10⋊C4 C5×C4⋊C4 C4⋊C4 C10 C2×C4 C2 C2 C2 # reps 1 1 1 1 3 1 2 6 6 8 2 2

Matrix representation of C4⋊C4⋊D5 in GL4(𝔽41) generated by

 9 0 0 0 16 32 0 0 0 0 23 6 0 0 35 18
,
 32 5 0 0 25 9 0 0 0 0 32 0 0 0 0 32
,
 1 0 0 0 0 1 0 0 0 0 0 1 0 0 40 6
,
 1 0 0 0 20 40 0 0 0 0 0 1 0 0 1 0
G:=sub<GL(4,GF(41))| [9,16,0,0,0,32,0,0,0,0,23,35,0,0,6,18],[32,25,0,0,5,9,0,0,0,0,32,0,0,0,0,32],[1,0,0,0,0,1,0,0,0,0,0,40,0,0,1,6],[1,20,0,0,0,40,0,0,0,0,0,1,0,0,1,0] >;

C4⋊C4⋊D5 in GAP, Magma, Sage, TeX

C_4\rtimes C_4\rtimes D_5
% in TeX

G:=Group("C4:C4:D5");
// GroupNames label

G:=SmallGroup(160,119);
// by ID

G=gap.SmallGroup(160,119);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,217,55,218,188,86,4613]);
// Polycyclic

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

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