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G = C429F5order 320 = 26·5

6th semidirect product of C42 and F5 acting via F5/D5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C429F5, (C4×C20)⋊6C4, C5⋊(C428C4), (C4×D5).81D4, C4.11(C4⋊F5), C20.18(C4⋊C4), D10.9(C2×Q8), (C4×D5).23Q8, (C4×Dic5)⋊21C4, D10.26(C2×D4), (D5×C42).22C2, D10.23(C4○D4), Dic5.27(C4⋊C4), D5.1(C4.4D4), D10.3Q8.4C2, D5.1(C42.C2), C10.7(C42⋊C2), (C22×F5).2C22, C22.67(C22×F5), (C22×D5).266C23, C2.10(D10.C23), C2.9(C2×C4⋊F5), C10.6(C2×C4⋊C4), (C2×C4⋊F5).10C2, (C2×C4).101(C2×F5), (C2×C20).125(C2×C4), (C2×C4×D5).361C22, (C2×C10).27(C22×C4), (C2×Dic5).175(C2×C4), SmallGroup(320,1027)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C429F5
C1C5D5D10C22×D5C22×F5D10.3Q8 — C429F5
C5C2×C10 — C429F5
C1C22C42

Generators and relations for C429F5
 G = < a,b,c,d | a4=b4=c5=d4=1, ab=ba, ac=ca, dad-1=ab2, bc=cb, dbd-1=a2b, dcd-1=c3 >

Subgroups: 618 in 154 conjugacy classes, 60 normal (18 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×10], C22, C22 [×6], C5, C2×C4, C2×C4 [×2], C2×C4 [×23], C23, D5 [×4], C10, C10 [×2], C42, C42 [×3], C4⋊C4 [×4], C22×C4 [×7], Dic5 [×2], Dic5 [×2], C20 [×2], C20 [×2], F5 [×4], D10 [×2], D10 [×4], C2×C10, C2.C42 [×4], C2×C42, C2×C4⋊C4 [×2], C4×D5 [×4], C4×D5 [×4], C2×Dic5, C2×Dic5 [×2], C2×C20, C2×C20 [×2], C2×F5 [×12], C22×D5, C428C4, C4×Dic5, C4×Dic5 [×2], C4×C20, C4⋊F5 [×4], C2×C4×D5, C2×C4×D5 [×2], C22×F5 [×4], D10.3Q8 [×4], D5×C42, C2×C4⋊F5 [×2], C429F5
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, C4○D4 [×4], F5, C2×C4⋊C4, C42⋊C2 [×2], C4.4D4 [×2], C42.C2 [×2], C2×F5 [×3], C428C4, C4⋊F5 [×2], C22×F5, C2×C4⋊F5, D10.C23 [×2], C429F5

Smallest permutation representation of C429F5
On 80 points
Generators in S80
(1 51 11 41)(2 52 12 42)(3 53 13 43)(4 54 14 44)(5 55 15 45)(6 56 16 46)(7 57 17 47)(8 58 18 48)(9 59 19 49)(10 60 20 50)(21 71 31 61)(22 72 32 62)(23 73 33 63)(24 74 34 64)(25 75 35 65)(26 76 36 66)(27 77 37 67)(28 78 38 68)(29 79 39 69)(30 80 40 70)
(1 26 6 21)(2 27 7 22)(3 28 8 23)(4 29 9 24)(5 30 10 25)(11 36 16 31)(12 37 17 32)(13 38 18 33)(14 39 19 34)(15 40 20 35)(41 66 46 61)(42 67 47 62)(43 68 48 63)(44 69 49 64)(45 70 50 65)(51 76 56 71)(52 77 57 72)(53 78 58 73)(54 79 59 74)(55 80 60 75)
(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 41 16 56)(2 43 20 59)(3 45 19 57)(4 42 18 60)(5 44 17 58)(6 46 11 51)(7 48 15 54)(8 50 14 52)(9 47 13 55)(10 49 12 53)(21 71 36 66)(22 73 40 69)(23 75 39 67)(24 72 38 70)(25 74 37 68)(26 76 31 61)(27 78 35 64)(28 80 34 62)(29 77 33 65)(30 79 32 63)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E2F2G4A···4F4G···4L4M···4T 5 10A10B10C20A···20L
order122222224···44···44···4510101020···20
size111155552···210···1020···2044444···4

44 irreducible representations

dim1111112224444
type+++++-++
imageC1C2C2C2C4C4D4Q8C4○D4F5C2×F5C4⋊F5D10.C23
kernelC429F5D10.3Q8D5×C42C2×C4⋊F5C4×Dic5C4×C20C4×D5C4×D5D10C42C2×C4C4C2
# reps1412622281348

Matrix representation of C429F5 in GL6(𝔽41)

1120000
21300000
009000
000900
000090
000009
,
1120000
21300000
00223803
00019383
00338190
00303822
,
100000
010000
0000040
0010040
0001040
0000140
,
17180000
34240000
0000320
0032000
0000032
0003200

G:=sub<GL(6,GF(41))| [11,21,0,0,0,0,2,30,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[11,21,0,0,0,0,2,30,0,0,0,0,0,0,22,0,3,3,0,0,38,19,38,0,0,0,0,38,19,38,0,0,3,3,0,22],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,40,40,40,40],[17,34,0,0,0,0,18,24,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,32,0,0,32,0,0,0,0,0,0,0,32,0] >;

C429F5 in GAP, Magma, Sage, TeX

C_4^2\rtimes_9F_5
% in TeX

G:=Group("C4^2:9F5");
// GroupNames label

G:=SmallGroup(320,1027);
// by ID

G=gap.SmallGroup(320,1027);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,477,120,422,184,6278,1595]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=c^5=d^4=1,a*b=b*a,a*c=c*a,d*a*d^-1=a*b^2,b*c=c*b,d*b*d^-1=a^2*b,d*c*d^-1=c^3>;
// generators/relations

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