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## G = C4⋊C4⋊Dic5order 320 = 26·5

### 1st semidirect product of C4⋊C4 and Dic5 acting via Dic5/C5=C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — C4⋊C4⋊Dic5
 Chief series C1 — C5 — C10 — C2×C10 — C22×C10 — C22×C20 — C10.10C42 — C4⋊C4⋊Dic5
 Lower central C5 — C2×C10 — C2×C20 — C4⋊C4⋊Dic5
 Upper central C1 — C22 — C22×C4 — C4⋊D4

Generators and relations for C4⋊C4⋊Dic5
G = < a,b,c,d | a4=b4=c10=1, d2=c5, bab-1=a-1, ac=ca, dad-1=a-1b2, cbc-1=b-1, dbd-1=a-1b, dcd-1=c-1 >

Subgroups: 334 in 90 conjugacy classes, 31 normal (all characteristic)
C1, C2 [×3], C2 [×3], C4 [×5], C22 [×3], C22 [×5], C5, C8, C2×C4 [×2], C2×C4 [×6], D4 [×3], C23, C23, C10 [×3], C10 [×3], C22⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×D4, C2×D4, Dic5 [×2], C20 [×3], C2×C10 [×3], C2×C10 [×5], C2.C42, C22⋊C8, C4⋊D4, C52C8, C2×Dic5 [×4], C2×C20 [×2], C2×C20 [×2], C5×D4 [×3], C22×C10, C22×C10, C22.SD16, C2×C52C8, C5×C22⋊C4, C5×C4⋊C4, C22×Dic5, C22×C20, D4×C10, D4×C10, C20.55D4, C10.10C42, C5×C4⋊D4, C4⋊C4⋊Dic5
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], D5, C22⋊C4, D8, SD16, Dic5 [×2], D10, C23⋊C4, D4⋊C4, C4≀C2, C2×Dic5, C5⋊D4 [×2], C22.SD16, D4⋊D5, D4.D5, C23.D5, D4⋊Dic5, C23⋊Dic5, D42Dic5, C4⋊C4⋊Dic5

Smallest permutation representation of C4⋊C4⋊Dic5
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 54 65)(42 72 55 66)(43 73 56 67)(44 74 57 68)(45 75 58 69)(46 76 59 70)(47 77 60 61)(48 78 51 62)(49 79 52 63)(50 80 53 64)
(1 46 14 41)(2 42 15 47)(3 48 11 43)(4 44 12 49)(5 50 13 45)(6 51 16 56)(7 57 17 52)(8 53 18 58)(9 59 19 54)(10 55 20 60)(21 73 31 78)(22 79 32 74)(23 75 33 80)(24 71 34 76)(25 77 35 72)(26 67 36 62)(27 63 37 68)(28 69 38 64)(29 65 39 70)(30 61 40 66)
(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)
(2 5)(3 4)(6 7)(8 10)(11 12)(13 15)(16 17)(18 20)(21 37)(22 36)(23 40)(24 39)(25 38)(26 32)(27 31)(28 35)(29 34)(30 33)(41 76 46 71)(42 75 47 80)(43 74 48 79)(44 73 49 78)(45 72 50 77)(51 63 56 68)(52 62 57 67)(53 61 58 66)(54 70 59 65)(55 69 60 64)

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

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

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

47 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 4E 4F 4G 4H 5A 5B 8A 8B 8C 8D 10A ··· 10F 10G 10H 10I 10J 10K 10L 10M 10N 20A ··· 20H 20I 20J 20K 20L order 1 2 2 2 2 2 2 4 4 4 4 4 4 4 4 5 5 8 8 8 8 10 ··· 10 10 10 10 10 10 10 10 10 20 ··· 20 20 20 20 20 size 1 1 1 1 2 2 8 2 2 4 8 20 20 20 20 2 2 20 20 20 20 2 ··· 2 4 4 4 4 8 8 8 8 4 ··· 4 8 8 8 8

47 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + - + - + + - image C1 C2 C2 C2 C4 C4 D4 D4 D5 D8 SD16 Dic5 D10 Dic5 C4≀C2 C5⋊D4 C5⋊D4 C23⋊C4 D4⋊D5 D4.D5 C23⋊Dic5 D4⋊2Dic5 kernel C4⋊C4⋊Dic5 C20.55D4 C10.10C42 C5×C4⋊D4 C5×C4⋊C4 D4×C10 C2×C20 C22×C10 C4⋊D4 C2×C10 C2×C10 C4⋊C4 C22×C4 C2×D4 C10 C2×C4 C23 C10 C22 C22 C2 C2 # reps 1 1 1 1 2 2 1 1 2 2 2 2 2 2 4 4 4 1 2 2 4 4

Matrix representation of C4⋊C4⋊Dic5 in GL6(𝔽41)

 32 0 0 0 0 0 26 9 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 9 37 0 0 0 0 0 32
,
 3 21 0 0 0 0 21 38 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 3 7 0 0 0 0 34 38
,
 1 0 0 0 0 0 29 40 0 0 0 0 0 0 0 1 0 0 0 0 40 34 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 22 32 0 0 0 0 0 0 1 0 0 0 0 0 34 40 0 0 0 0 0 0 1 0 0 0 0 0 25 40

G:=sub<GL(6,GF(41))| [32,26,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,0,0,0,0,0,37,32],[3,21,0,0,0,0,21,38,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,3,34,0,0,0,0,7,38],[1,29,0,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,1,34,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,22,0,0,0,0,0,32,0,0,0,0,0,0,1,34,0,0,0,0,0,40,0,0,0,0,0,0,1,25,0,0,0,0,0,40] >;

C4⋊C4⋊Dic5 in GAP, Magma, Sage, TeX

C_4\rtimes C_4\rtimes {\rm Dic}_5
% in TeX

G:=Group("C4:C4:Dic5");
// GroupNames label

G:=SmallGroup(320,95);
// by ID

G=gap.SmallGroup(320,95);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,219,1571,570,136,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=c^10=1,d^2=c^5,b*a*b^-1=a^-1,a*c=c*a,d*a*d^-1=a^-1*b^2,c*b*c^-1=b^-1,d*b*d^-1=a^-1*b,d*c*d^-1=c^-1>;
// generators/relations

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