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## G = C4×C5⋊F5order 400 = 24·52

### Direct product of C4 and C5⋊F5

Aliases: C4×C5⋊F5, C203F5, C526C42, C52(C4×F5), (C5×C20)⋊7C4, C526C46C4, C10.16(C2×F5), C5⋊D5.8(C2×C4), (C4×C5⋊D5).11C2, C2.2(C2×C5⋊F5), (C5×C10).29(C2×C4), (C2×C5⋊F5).5C2, (C2×C5⋊D5).21C22, SmallGroup(400,151)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — C4×C5⋊F5
 Chief series C1 — C5 — C52 — C5⋊D5 — C2×C5⋊D5 — C2×C5⋊F5 — C4×C5⋊F5
 Lower central C52 — C4×C5⋊F5
 Upper central C1 — C4

Generators and relations for C4×C5⋊F5
G = < a,b,c,d | a4=b5=c5=d4=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b3, dcd-1=c3 >

Subgroups: 696 in 120 conjugacy classes, 36 normal (10 characteristic)
C1, C2, C2, C4, C4, C22, C5, C2×C4, D5, C10, C42, Dic5, C20, F5, D10, C52, C4×D5, C2×F5, C5⋊D5, C5×C10, C4×F5, C526C4, C5×C20, C5⋊F5, C2×C5⋊D5, C4×C5⋊D5, C2×C5⋊F5, C4×C5⋊F5
Quotients: C1, C2, C4, C22, C2×C4, C42, F5, C2×F5, C4×F5, C5⋊F5, C2×C5⋊F5, C4×C5⋊F5

Smallest permutation representation of C4×C5⋊F5
On 100 points
Generators in S100
(1 12 56 81)(2 13 57 82)(3 14 58 83)(4 15 59 84)(5 11 60 85)(6 55 80 35)(7 51 76 31)(8 52 77 32)(9 53 78 33)(10 54 79 34)(16 65 90 40)(17 61 86 36)(18 62 87 37)(19 63 88 38)(20 64 89 39)(21 70 95 45)(22 66 91 41)(23 67 92 42)(24 68 93 43)(25 69 94 44)(26 98 48 73)(27 99 49 74)(28 100 50 75)(29 96 46 71)(30 97 47 72)
(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)(81 82 83 84 85)(86 87 88 89 90)(91 92 93 94 95)(96 97 98 99 100)
(1 42 33 39 28)(2 43 34 40 29)(3 44 35 36 30)(4 45 31 37 26)(5 41 32 38 27)(6 17 97 14 25)(7 18 98 15 21)(8 19 99 11 22)(9 20 100 12 23)(10 16 96 13 24)(46 57 68 54 65)(47 58 69 55 61)(48 59 70 51 62)(49 60 66 52 63)(50 56 67 53 64)(71 82 93 79 90)(72 83 94 80 86)(73 84 95 76 87)(74 85 91 77 88)(75 81 92 78 89)
(1 81 56 12)(2 83 60 15)(3 85 59 13)(4 82 58 11)(5 84 57 14)(6 27 87 68)(7 29 86 66)(8 26 90 69)(9 28 89 67)(10 30 88 70)(16 44 77 48)(17 41 76 46)(18 43 80 49)(19 45 79 47)(20 42 78 50)(21 34 72 63)(22 31 71 61)(23 33 75 64)(24 35 74 62)(25 32 73 65)(36 91 51 96)(37 93 55 99)(38 95 54 97)(39 92 53 100)(40 94 52 98)

G:=sub<Sym(100)| (1,12,56,81)(2,13,57,82)(3,14,58,83)(4,15,59,84)(5,11,60,85)(6,55,80,35)(7,51,76,31)(8,52,77,32)(9,53,78,33)(10,54,79,34)(16,65,90,40)(17,61,86,36)(18,62,87,37)(19,63,88,38)(20,64,89,39)(21,70,95,45)(22,66,91,41)(23,67,92,42)(24,68,93,43)(25,69,94,44)(26,98,48,73)(27,99,49,74)(28,100,50,75)(29,96,46,71)(30,97,47,72), (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)(81,82,83,84,85)(86,87,88,89,90)(91,92,93,94,95)(96,97,98,99,100), (1,42,33,39,28)(2,43,34,40,29)(3,44,35,36,30)(4,45,31,37,26)(5,41,32,38,27)(6,17,97,14,25)(7,18,98,15,21)(8,19,99,11,22)(9,20,100,12,23)(10,16,96,13,24)(46,57,68,54,65)(47,58,69,55,61)(48,59,70,51,62)(49,60,66,52,63)(50,56,67,53,64)(71,82,93,79,90)(72,83,94,80,86)(73,84,95,76,87)(74,85,91,77,88)(75,81,92,78,89), (1,81,56,12)(2,83,60,15)(3,85,59,13)(4,82,58,11)(5,84,57,14)(6,27,87,68)(7,29,86,66)(8,26,90,69)(9,28,89,67)(10,30,88,70)(16,44,77,48)(17,41,76,46)(18,43,80,49)(19,45,79,47)(20,42,78,50)(21,34,72,63)(22,31,71,61)(23,33,75,64)(24,35,74,62)(25,32,73,65)(36,91,51,96)(37,93,55,99)(38,95,54,97)(39,92,53,100)(40,94,52,98)>;

G:=Group( (1,12,56,81)(2,13,57,82)(3,14,58,83)(4,15,59,84)(5,11,60,85)(6,55,80,35)(7,51,76,31)(8,52,77,32)(9,53,78,33)(10,54,79,34)(16,65,90,40)(17,61,86,36)(18,62,87,37)(19,63,88,38)(20,64,89,39)(21,70,95,45)(22,66,91,41)(23,67,92,42)(24,68,93,43)(25,69,94,44)(26,98,48,73)(27,99,49,74)(28,100,50,75)(29,96,46,71)(30,97,47,72), (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)(81,82,83,84,85)(86,87,88,89,90)(91,92,93,94,95)(96,97,98,99,100), (1,42,33,39,28)(2,43,34,40,29)(3,44,35,36,30)(4,45,31,37,26)(5,41,32,38,27)(6,17,97,14,25)(7,18,98,15,21)(8,19,99,11,22)(9,20,100,12,23)(10,16,96,13,24)(46,57,68,54,65)(47,58,69,55,61)(48,59,70,51,62)(49,60,66,52,63)(50,56,67,53,64)(71,82,93,79,90)(72,83,94,80,86)(73,84,95,76,87)(74,85,91,77,88)(75,81,92,78,89), (1,81,56,12)(2,83,60,15)(3,85,59,13)(4,82,58,11)(5,84,57,14)(6,27,87,68)(7,29,86,66)(8,26,90,69)(9,28,89,67)(10,30,88,70)(16,44,77,48)(17,41,76,46)(18,43,80,49)(19,45,79,47)(20,42,78,50)(21,34,72,63)(22,31,71,61)(23,33,75,64)(24,35,74,62)(25,32,73,65)(36,91,51,96)(37,93,55,99)(38,95,54,97)(39,92,53,100)(40,94,52,98) );

G=PermutationGroup([[(1,12,56,81),(2,13,57,82),(3,14,58,83),(4,15,59,84),(5,11,60,85),(6,55,80,35),(7,51,76,31),(8,52,77,32),(9,53,78,33),(10,54,79,34),(16,65,90,40),(17,61,86,36),(18,62,87,37),(19,63,88,38),(20,64,89,39),(21,70,95,45),(22,66,91,41),(23,67,92,42),(24,68,93,43),(25,69,94,44),(26,98,48,73),(27,99,49,74),(28,100,50,75),(29,96,46,71),(30,97,47,72)], [(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),(81,82,83,84,85),(86,87,88,89,90),(91,92,93,94,95),(96,97,98,99,100)], [(1,42,33,39,28),(2,43,34,40,29),(3,44,35,36,30),(4,45,31,37,26),(5,41,32,38,27),(6,17,97,14,25),(7,18,98,15,21),(8,19,99,11,22),(9,20,100,12,23),(10,16,96,13,24),(46,57,68,54,65),(47,58,69,55,61),(48,59,70,51,62),(49,60,66,52,63),(50,56,67,53,64),(71,82,93,79,90),(72,83,94,80,86),(73,84,95,76,87),(74,85,91,77,88),(75,81,92,78,89)], [(1,81,56,12),(2,83,60,15),(3,85,59,13),(4,82,58,11),(5,84,57,14),(6,27,87,68),(7,29,86,66),(8,26,90,69),(9,28,89,67),(10,30,88,70),(16,44,77,48),(17,41,76,46),(18,43,80,49),(19,45,79,47),(20,42,78,50),(21,34,72,63),(22,31,71,61),(23,33,75,64),(24,35,74,62),(25,32,73,65),(36,91,51,96),(37,93,55,99),(38,95,54,97),(39,92,53,100),(40,94,52,98)]])

40 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C ··· 4L 5A ··· 5F 10A ··· 10F 20A ··· 20L order 1 2 2 2 4 4 4 ··· 4 5 ··· 5 10 ··· 10 20 ··· 20 size 1 1 25 25 1 1 25 ··· 25 4 ··· 4 4 ··· 4 4 ··· 4

40 irreducible representations

 dim 1 1 1 1 1 1 4 4 4 type + + + + + image C1 C2 C2 C4 C4 C4 F5 C2×F5 C4×F5 kernel C4×C5⋊F5 C4×C5⋊D5 C2×C5⋊F5 C52⋊6C4 C5×C20 C5⋊F5 C20 C10 C5 # reps 1 1 2 2 2 8 6 6 12

Matrix representation of C4×C5⋊F5 in GL8(𝔽41)

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

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

C4×C5⋊F5 in GAP, Magma, Sage, TeX

C_4\times C_5\rtimes F_5
% in TeX

G:=Group("C4xC5:F5");
// GroupNames label

G:=SmallGroup(400,151);
// by ID

G=gap.SmallGroup(400,151);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-5,-5,24,55,964,496,5765,2897]);
// Polycyclic

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

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