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G = C4xC5:F5  order 400 = 24·52

Direct product of C4 and C5:F5

direct product, metabelian, supersoluble, monomial, A-group

Aliases: C4xC5:F5, C20:3F5, C52:6C42, C5:2(C4xF5), (C5xC20):7C4, C52:6C4:6C4, C10.16(C2xF5), C5:D5.8(C2xC4), (C4xC5:D5).11C2, C2.2(C2xC5:F5), (C5xC10).29(C2xC4), (C2xC5:F5).5C2, (C2xC5:D5).21C22, SmallGroup(400,151)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C52 — C4xC5:F5
Chief series
C1 — C5 — C52 — C5:D5 — C2xC5:D5 — C2xC5:F5 — C4xC5:F5
Lower central
C52 — C4xC5:F5
Upper central
C1 — C4

Generators and relations for C4xC5: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, C2xC4, D5, C10, C42, Dic5, C20, F5, D10, C52, C4xD5, C2xF5, C5:D5, C5xC10, C4xF5, C52:6C4, C5xC20, C5:F5, C2xC5:D5, C4xC5:D5, C2xC5:F5, C4xC5:F5
Quotients: C1, C2, C4, C22, C2xC4, C42, F5, C2xF5, C4xF5, C5:F5, C2xC5:F5, C4xC5:F5

Smallest permutation representation of C4xC5: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 2A2B2C4A4B4C···4L5A···5F10A···10F20A···20L
order1222444···45···510···1020···20
size1125251125···254···44···44···4

40 irreducible representations

dim111111444
type+++++
imageC1C2C2C4C4C4F5C2xF5C4xF5
kernelC4xC5:F5C4xC5:D5C2xC5:F5C52:6C4C5xC20C5:F5C20C10C5
# reps1122286612

Matrix representation of C4xC5:F5 ►in GL8(F41)

320000000
032000000
003200000
000320000
00009000
00000900
00000090
00000009
,
10000000
01000000
00100000
00010000
000040404040
00001000
00000100
00000010
,
404040400000
10000000
01000000
00100000
000040404040
00001000
00000100
00000010
,
90000000
00090000
09000000
323232320000
00000099
00009900
00003232320
00000990

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] >;
 

C4xC5: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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