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G = C424F5order 320 = 26·5

1st semidirect product of C42 and F5 acting via F5/D5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C424F5, C20.28C42, Dic5.8C42, (C4×F5)⋊6C4, (C4×C20)⋊11C4, C5⋊(C424C4), C4.21(C4×F5), (C4×Dic5)⋊19C4, C10.5(C2×C42), (D5×C42).32C2, D10.21(C4○D4), D10.27(C22×C4), D10.3Q8.9C2, C10.5(C42⋊C2), C22.31(C22×F5), D5.1(C42⋊C2), (C22×F5).12C22, (C22×D5).263C23, C2.3(D10.C23), C2.7(C2×C4×F5), (C2×C4×F5).8C2, (C2×F5).5(C2×C4), (C4×D5).72(C2×C4), (C2×C4).161(C2×F5), (C2×C20).170(C2×C4), (C2×C4×D5).360C22, (C2×C10).24(C22×C4), (C2×Dic5).172(C2×C4), SmallGroup(320,1024)

Series: Derived Chief Lower central Upper central

C1C10 — C424F5
C1C5D5D10C22×D5C22×F5C2×C4×F5 — C424F5
C5C10 — C424F5
C1C2×C4C42

Generators and relations for C424F5
 G = < a,b,c,d | a4=b4=c5=d4=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=a2b, dcd-1=c3 >

Subgroups: 618 in 178 conjugacy classes, 76 normal (16 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×14], C22, C22 [×6], C5, C2×C4, C2×C4 [×2], C2×C4 [×27], C23, D5 [×4], C10, C10 [×2], C42, C42 [×11], C22×C4 [×7], Dic5 [×2], Dic5 [×2], C20 [×2], C20 [×2], F5 [×8], D10 [×6], C2×C10, C2.C42 [×4], C2×C42 [×3], C4×D5 [×4], C4×D5 [×4], C2×Dic5, C2×Dic5 [×2], C2×C20, C2×C20 [×2], C2×F5 [×8], C2×F5 [×8], C22×D5, C424C4, C4×Dic5, C4×Dic5 [×2], C4×C20, C4×F5 [×8], C2×C4×D5, C2×C4×D5 [×2], C22×F5 [×4], D10.3Q8 [×4], D5×C42, C2×C4×F5 [×2], C424F5
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], C23, C42 [×4], C22×C4 [×3], C4○D4 [×4], F5, C2×C42, C42⋊C2 [×6], C2×F5 [×3], C424C4, C4×F5 [×2], C22×F5, C2×C4×F5, D10.C23 [×2], C424F5

Smallest permutation representation of C424F5
On 80 points
Generators in S80
(1 46 6 41)(2 47 7 42)(3 48 8 43)(4 49 9 44)(5 50 10 45)(11 56 16 51)(12 57 17 52)(13 58 18 53)(14 59 19 54)(15 60 20 55)(21 66 26 61)(22 67 27 62)(23 68 28 63)(24 69 29 64)(25 70 30 65)(31 76 36 71)(32 77 37 72)(33 78 38 73)(34 79 39 74)(35 80 40 75)
(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 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 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 3 5 4)(7 8 10 9)(12 13 15 14)(17 18 20 19)(21 26)(22 28 25 29)(23 30 24 27)(31 36)(32 38 35 39)(33 40 34 37)(42 43 45 44)(47 48 50 49)(52 53 55 54)(57 58 60 59)(61 66)(62 68 65 69)(63 70 64 67)(71 76)(72 78 75 79)(73 80 74 77)

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

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

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

56 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J4K4L4M···4AF 5 10A10B10C20A···20L
order122222224444444444444···4510101020···20
size1111555511112222555510···1044444···4

56 irreducible representations

dim111111124444
type++++++
imageC1C2C2C2C4C4C4C4○D4F5C2×F5C4×F5D10.C23
kernelC424F5D10.3Q8D5×C42C2×C4×F5C4×Dic5C4×C20C4×F5D10C42C2×C4C4C2
# reps1412621681348

Matrix representation of C424F5 in GL6(𝔽41)

900000
090000
009000
000900
000090
000009
,
100000
2400000
003402727
00147140
00014714
002727034
,
100000
010000
0040404040
001000
000100
000010
,
9320000
0320000
001000
000001
000100
0040404040

G:=sub<GL(6,GF(41))| [9,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[1,2,0,0,0,0,0,40,0,0,0,0,0,0,34,14,0,27,0,0,0,7,14,27,0,0,27,14,7,0,0,0,27,0,14,34],[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],[9,0,0,0,0,0,32,32,0,0,0,0,0,0,1,0,0,40,0,0,0,0,1,40,0,0,0,0,0,40,0,0,0,1,0,40] >;

C424F5 in GAP, Magma, Sage, TeX

C_4^2\rtimes_4F_5
% in TeX

G:=Group("C4^2:4F5");
// GroupNames label

G:=SmallGroup(320,1024);
// by ID

G=gap.SmallGroup(320,1024);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,232,758,100,6278,1595]);
// 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=a^2*b,d*c*d^-1=c^3>;
// generators/relations

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