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## G = C4⋊C4×F5order 320 = 26·5

### Direct product of C4⋊C4 and F5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C4⋊C4×F5
 Chief series C1 — C5 — D5 — D10 — C22×D5 — C22×F5 — C2×C4×F5 — C4⋊C4×F5
 Lower central C5 — C10 — C4⋊C4×F5
 Upper central C1 — C22 — C4⋊C4

Generators and relations for C4⋊C4×F5
G = < a,b,c,d | a4=b4=c5=d4=1, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c3 >

Subgroups: 666 in 194 conjugacy classes, 84 normal (34 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×16], C22, C22 [×6], C5, C2×C4, C2×C4 [×2], C2×C4 [×29], C23, D5 [×4], C10 [×3], C42 [×8], C4⋊C4, C4⋊C4 [×7], C22×C4 [×7], Dic5 [×2], Dic5 [×2], C20 [×2], C20 [×2], F5 [×4], F5 [×6], D10 [×6], C2×C10, C2.C42 [×2], C2×C42 [×3], C2×C4⋊C4 [×2], C4×D5 [×4], C4×D5 [×4], C2×Dic5, C2×Dic5 [×2], C2×C20, C2×C20 [×2], C2×F5 [×12], C2×F5 [×6], C22×D5, C4×C4⋊C4, C10.D4 [×2], C4⋊Dic5, C5×C4⋊C4, C4×F5 [×4], C4×F5 [×4], C4⋊F5 [×4], C2×C4×D5, C2×C4×D5 [×2], C22×F5 [×2], C22×F5 [×2], D10.3Q8 [×2], D5×C4⋊C4, C2×C4×F5, C2×C4×F5 [×2], C2×C4⋊F5, C4⋊C4×F5
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], D4 [×2], Q8 [×2], C23, C42 [×4], C4⋊C4 [×4], C22×C4 [×3], C2×D4, C2×Q8, C4○D4 [×2], F5, C2×C42, C2×C4⋊C4, C42⋊C2, C4×D4 [×2], C4×Q8 [×2], C2×F5 [×3], C4×C4⋊C4, C4×F5 [×2], C22×F5, C2×C4×F5, D4×F5, Q8×F5, C4⋊C4×F5

Smallest permutation representation of C4⋊C4×F5
On 80 points
Generators in S80
(1 56 6 51)(2 57 7 52)(3 58 8 53)(4 59 9 54)(5 60 10 55)(11 46 16 41)(12 47 17 42)(13 48 18 43)(14 49 19 44)(15 50 20 45)(21 76 26 71)(22 77 27 72)(23 78 28 73)(24 79 29 74)(25 80 30 75)(31 66 36 61)(32 67 37 62)(33 68 38 63)(34 69 39 64)(35 70 40 65)
(1 31 11 21)(2 32 12 22)(3 33 13 23)(4 34 14 24)(5 35 15 25)(6 36 16 26)(7 37 17 27)(8 38 18 28)(9 39 19 29)(10 40 20 30)(41 76 51 66)(42 77 52 67)(43 78 53 68)(44 79 54 69)(45 80 55 70)(46 71 56 61)(47 72 57 62)(48 73 58 63)(49 74 59 64)(50 75 60 65)
(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 56 6 51)(2 58 10 54)(3 60 9 52)(4 57 8 55)(5 59 7 53)(11 46 16 41)(12 48 20 44)(13 50 19 42)(14 47 18 45)(15 49 17 43)(21 71 26 76)(22 73 30 79)(23 75 29 77)(24 72 28 80)(25 74 27 78)(31 61 36 66)(32 63 40 69)(33 65 39 67)(34 62 38 70)(35 64 37 68)

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

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

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

50 conjugacy classes

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

50 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 4 4 4 8 8 type + + + + + + - + + + - image C1 C2 C2 C2 C2 C4 C4 C4 C4 C4 D4 Q8 C4○D4 F5 C2×F5 C4×F5 D4×F5 Q8×F5 kernel C4⋊C4×F5 D10.3Q8 D5×C4⋊C4 C2×C4×F5 C2×C4⋊F5 C10.D4 C4⋊Dic5 C5×C4⋊C4 C4×F5 C4⋊F5 C2×F5 C2×F5 D10 C4⋊C4 C2×C4 C4 C2 C2 # reps 1 2 1 3 1 4 2 2 8 8 2 2 4 1 3 4 1 1

Matrix representation of C4⋊C4×F5 in GL8(𝔽41)

 32 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 32 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40
,
 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 32 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 40 0 0 0 0 1 0 0 40 0 0 0 0 0 1 0 40 0 0 0 0 0 0 1 40
,
 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 0 0 40 0 0 0 0 0 40 0 0 0 0 0 0 0 0 0 0 40 0 0 0 0 0 40 0 0

G:=sub<GL(8,GF(41))| [32,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,40,40,40,40],[40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,40,0,0,0,0,40,0,0,0,0,0,0,0,0,0,40,0] >;

C4⋊C4×F5 in GAP, Magma, Sage, TeX

C_4\rtimes C_4\times F_5
% in TeX

G:=Group("C4:C4xF5");
// GroupNames label

G:=SmallGroup(320,1048);
// by ID

G=gap.SmallGroup(320,1048);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,232,387,100,6278,1595]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=c^5=d^4=1,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^3>;
// generators/relations

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