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## G = C4×C4⋊F5order 320 = 26·5

### Direct product of C4 and C4⋊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 — C2×C4 — C42

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

Subgroups: 666 in 194 conjugacy classes, 86 normal (32 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×14], C22, C22 [×6], C5, C2×C4 [×3], C2×C4 [×29], C23, D5 [×4], C10 [×3], C42, C42 [×7], C4⋊C4 [×8], C22×C4 [×7], Dic5 [×4], Dic5, C20 [×4], C20, F5 [×8], D10 [×2], D10 [×4], C2×C10, C2.C42 [×2], C2×C42 [×3], C2×C4⋊C4 [×2], C4×D5 [×8], C4×D5 [×2], C2×Dic5 [×3], C2×C20 [×3], C2×F5 [×8], C2×F5 [×8], C22×D5, C4×C4⋊C4, C4×Dic5 [×3], C4×C20, C4×F5 [×4], C4⋊F5 [×8], C2×C4×D5 [×3], C22×F5 [×4], D10.3Q8 [×2], D5×C42, C2×C4×F5 [×2], C2×C4⋊F5 [×2], 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], C4⋊F5 [×2], C22×F5, C2×C4×F5, C2×C4⋊F5, D10.C23, C4×C4⋊F5

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

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

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

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

56 conjugacy classes

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

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 4 4 4 4 4 type + + + + + + - + + image C1 C2 C2 C2 C2 C4 C4 C4 D4 Q8 C4○D4 F5 C2×F5 C4×F5 C4⋊F5 D10.C23 kernel C4×C4⋊F5 D10.3Q8 D5×C42 C2×C4×F5 C2×C4⋊F5 C4×Dic5 C4×C20 C4⋊F5 C4×D5 C4×D5 D10 C42 C2×C4 C4 C4 C2 # reps 1 2 1 2 2 6 2 16 2 2 4 1 3 4 4 4

Matrix representation of C4×C4⋊F5 in GL6(𝔽41)

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

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[0,1,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,40,40,40,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[9,0,0,0,0,0,0,32,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,32,0,0,32,0,0,0,0,0,0,0,32,0] >;

C4×C4⋊F5 in GAP, Magma, Sage, TeX

C_4\times C_4\rtimes F_5
% in TeX

G:=Group("C4xC4:F5");
// GroupNames label

G:=SmallGroup(320,1025);
// by ID

G=gap.SmallGroup(320,1025);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,477,232,100,6278,1595]);
// Polycyclic

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

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