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G = C4○D20⋊C4order 320 = 26·5

3rd semidirect product of C4○D20 and C4 acting faithfully

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C4○D43F5, C4○D203C4, D4.9(C2×F5), Q8.9(C2×F5), D42D59C4, D20⋊C45C2, Q82D59C4, Q8⋊F55C2, D20.9(C2×C4), (C4×D5).56D4, D5.5(C4○D8), C5⋊(C23.24D4), C4⋊F5.10C22, C4.23(C22×F5), D10.101(C2×D4), C20.23(C22×C4), D5⋊C8.16C22, Dic10.9(C2×C4), Dic5.11(C2×D4), (C22×D5).73D4, (D4×D5).15C22, (C4×D5).45C23, C4.47(C22⋊F5), (Q8×D5).13C22, C20.47(C22⋊C4), (C2×Dic5).125D4, C22.6(C22⋊F5), D10.18(C22⋊C4), D10.C236C2, Dic5.50(C22⋊C4), (C2×D5⋊C8)⋊4C2, (C5×C4○D4)⋊3C4, (D5×C4○D4).7C2, (C5×D4).9(C2×C4), (C2×C4).90(C2×F5), (C5×Q8).9(C2×C4), (C2×C20).68(C2×C4), (C4×D5).29(C2×C4), C2.36(C2×C22⋊F5), C10.35(C2×C22⋊C4), (C2×C4×D5).212C22, (C2×C10).6(C22⋊C4), SmallGroup(320,1132)

Series: Derived Chief Lower central Upper central

C1C20 — C4○D20⋊C4
C1C5C10Dic5C4×D5D5⋊C8C2×D5⋊C8 — C4○D20⋊C4
C5C10C20 — C4○D20⋊C4
C1C4C2×C4C4○D4

Generators and relations for C4○D20⋊C4
 G = < a,b,c,d | a4=c2=d4=1, b10=a2, ab=ba, ac=ca, ad=da, cbc=a2b9, dbd-1=b7, dcd-1=a2bc >

Subgroups: 682 in 158 conjugacy classes, 50 normal (40 characteristic)
C1, C2, C2 [×6], C4 [×2], C4 [×6], C22, C22 [×8], C5, C8 [×2], C2×C4, C2×C4 [×12], D4, D4 [×6], Q8, Q8 [×2], C23 [×2], D5 [×2], D5 [×2], C10, C10 [×2], C42, C22⋊C4, C4⋊C4 [×2], C2×C8 [×4], C22×C4 [×2], C2×D4 [×2], C2×Q8, C4○D4, C4○D4 [×5], Dic5 [×2], Dic5, C20 [×2], C20, F5 [×2], D10 [×2], D10 [×5], C2×C10, C2×C10, D4⋊C4 [×2], Q8⋊C4 [×2], C42⋊C2, C22×C8, C2×C4○D4, C5⋊C8 [×2], Dic10, Dic10, C4×D5 [×4], C4×D5 [×3], D20, D20, C2×Dic5, C2×Dic5, C5⋊D4 [×3], C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C2×F5 [×2], C22×D5, C22×D5, C23.24D4, D5⋊C8 [×2], D5⋊C8, C4×F5, C4⋊F5 [×2], C2×C5⋊C8, C22⋊F5, C2×C4×D5, C2×C4×D5, C4○D20, C4○D20, D4×D5, D4×D5, D42D5, D42D5, Q8×D5, Q82D5, C5×C4○D4, D20⋊C4 [×2], Q8⋊F5 [×2], C2×D5⋊C8, D10.C23, D5×C4○D4, C4○D20⋊C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], F5, C2×C22⋊C4, C4○D8 [×2], C2×F5 [×3], C23.24D4, C22⋊F5 [×2], C22×F5, C2×C22⋊F5, C4○D20⋊C4

Smallest permutation representation of C4○D20⋊C4
On 80 points
Generators in S80
(1 47 11 57)(2 48 12 58)(3 49 13 59)(4 50 14 60)(5 51 15 41)(6 52 16 42)(7 53 17 43)(8 54 18 44)(9 55 19 45)(10 56 20 46)(21 66 31 76)(22 67 32 77)(23 68 33 78)(24 69 34 79)(25 70 35 80)(26 71 36 61)(27 72 37 62)(28 73 38 63)(29 74 39 64)(30 75 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 22)(2 21)(3 40)(4 39)(5 38)(6 37)(7 36)(8 35)(9 34)(10 33)(11 32)(12 31)(13 30)(14 29)(15 28)(16 27)(17 26)(18 25)(19 24)(20 23)(41 73)(42 72)(43 71)(44 70)(45 69)(46 68)(47 67)(48 66)(49 65)(50 64)(51 63)(52 62)(53 61)(54 80)(55 79)(56 78)(57 77)(58 76)(59 75)(60 74)
(1 6)(2 9 10 13)(3 12 19 20)(4 15 8 7)(5 18 17 14)(11 16)(21 27 25 39)(22 30 34 26)(23 33)(24 36 32 40)(29 31 37 35)(41 54 53 50)(42 57)(43 60 51 44)(45 46 49 58)(47 52)(48 55 56 59)(61 77 65 69)(62 80 74 76)(64 66 72 70)(67 75 79 71)(68 78)

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

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

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

38 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F4G4H···4L 5 8A···8H10A10B10C10D20A20B20C20D20E
order1222222244444444···458···8101010102020202020
size11245510201124551020···20410···10488844888

38 irreducible representations

dim111111111122224444448
type+++++++++++++++
imageC1C2C2C2C2C2C4C4C4C4D4D4D4C4○D8F5C2×F5C2×F5C2×F5C22⋊F5C22⋊F5C4○D20⋊C4
kernelC4○D20⋊C4D20⋊C4Q8⋊F5C2×D5⋊C8D10.C23D5×C4○D4C4○D20D42D5Q82D5C5×C4○D4C4×D5C2×Dic5C22×D5D5C4○D4C2×C4D4Q8C4C22C1
# reps122111222221181111222

Matrix representation of C4○D20⋊C4 in GL6(𝔽41)

3200000
0320000
001000
000100
000010
000001
,
1160000
5400000
000100
000010
000001
0040404040
,
0280000
2200000
0004000
0040000
001111
0000040
,
32200000
090000
001000
000010
0040404040
000100

G:=sub<GL(6,GF(41))| [32,0,0,0,0,0,0,32,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,5,0,0,0,0,16,40,0,0,0,0,0,0,0,0,0,40,0,0,1,0,0,40,0,0,0,1,0,40,0,0,0,0,1,40],[0,22,0,0,0,0,28,0,0,0,0,0,0,0,0,40,1,0,0,0,40,0,1,0,0,0,0,0,1,0,0,0,0,0,1,40],[32,0,0,0,0,0,20,9,0,0,0,0,0,0,1,0,40,0,0,0,0,0,40,1,0,0,0,1,40,0,0,0,0,0,40,0] >;

C4○D20⋊C4 in GAP, Magma, Sage, TeX

C_4\circ D_{20}\rtimes C_4
% in TeX

G:=Group("C4oD20:C4");
// GroupNames label

G:=SmallGroup(320,1132);
// by ID

G=gap.SmallGroup(320,1132);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,422,387,297,136,438,102,6278,1595]);
// Polycyclic

G:=Group<a,b,c,d|a^4=c^2=d^4=1,b^10=a^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=a^2*b^9,d*b*d^-1=b^7,d*c*d^-1=a^2*b*c>;
// generators/relations

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