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G = C4⋊C421D10order 320 = 26·5

4th semidirect product of C4⋊C4 and D10 acting via D10/D5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C4⋊C421D10, (C4×D5)⋊12D4, (C2×D4)⋊22D10, C4⋊D426D5, C4.182(D4×D5), D103(C4○D4), C23⋊D108C2, C202D417C2, C22⋊C426D10, (D4×Dic5)⋊20C2, D10.74(C2×D4), C20.226(C2×D4), D102Q820C2, Dic54D48C2, (D4×C10)⋊11C22, C4⋊Dic530C22, C10.65(C22×D4), C20.48D433C2, C222(D42D5), (C2×C20).594C23, (C2×C10).150C24, Dic5.118(C2×D4), C54(C22.19C24), (C4×Dic5)⋊20C22, (C22×C4).368D10, D10.12D418C2, C23.D522C22, C23.15(C22×D5), (C2×Dic10)⋊24C22, C10.D428C22, (C22×C10).19C23, (C2×Dic5).71C23, C22.171(C23×D5), D10⋊C4.14C22, (C22×C20).240C22, (C22×Dic5)⋊19C22, (C22×D5).195C23, (C23×D5).119C22, C2.38(C2×D4×D5), (D5×C22×C4)⋊4C2, (C5×C4⋊C4)⋊9C22, C2.38(D5×C4○D4), C4⋊C47D519C2, (C2×C10)⋊5(C4○D4), (C5×C4⋊D4)⋊12C2, (C2×D42D5)⋊12C2, C10.151(C2×C4○D4), C2.36(C2×D42D5), (C2×C4×D5).317C22, (C5×C22⋊C4)⋊11C22, (C2×C4).294(C22×D5), (C2×C5⋊D4).27C22, SmallGroup(320,1278)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C4⋊C421D10
C1C5C10C2×C10C22×D5C23×D5D5×C22×C4 — C4⋊C421D10
C5C2×C10 — C4⋊C421D10
C1C22C4⋊D4

Generators and relations for C4⋊C421D10
 G = < a,b,c,d | a4=b4=c10=d2=1, bab-1=a-1, ac=ca, ad=da, cbc-1=b-1, dbd=a2b, dcd=c-1 >

Subgroups: 1198 in 330 conjugacy classes, 109 normal (43 characteristic)
C1, C2 [×3], C2 [×8], C4 [×2], C4 [×10], C22, C22 [×2], C22 [×24], C5, C2×C4 [×2], C2×C4 [×2], C2×C4 [×24], D4 [×14], Q8 [×2], C23, C23 [×2], C23 [×8], D5 [×4], C10 [×3], C10 [×4], C42 [×2], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4, C4⋊C4 [×5], C22×C4, C22×C4 [×11], C2×D4, C2×D4 [×2], C2×D4 [×4], C2×Q8, C4○D4 [×4], C24, Dic5 [×2], Dic5 [×5], C20 [×2], C20 [×3], D10 [×4], D10 [×12], C2×C10, C2×C10 [×2], C2×C10 [×8], C42⋊C2, C4×D4 [×4], C22≀C2 [×2], C4⋊D4, C4⋊D4, C22⋊Q8 [×2], C22.D4 [×2], C23×C4, C2×C4○D4, Dic10 [×2], C4×D5 [×4], C4×D5 [×6], C2×Dic5 [×2], C2×Dic5 [×4], C2×Dic5 [×6], C5⋊D4 [×8], C2×C20 [×2], C2×C20 [×2], C2×C20 [×2], C5×D4 [×6], C22×D5 [×2], C22×D5 [×6], C22×C10, C22×C10 [×2], C22.19C24, C4×Dic5 [×2], C10.D4 [×2], C4⋊Dic5, C4⋊Dic5 [×2], D10⋊C4 [×4], C23.D5 [×4], C5×C22⋊C4 [×2], C5×C4⋊C4, C2×Dic10, C2×C4×D5 [×4], C2×C4×D5 [×4], D42D5 [×4], C22×Dic5, C22×Dic5 [×2], C2×C5⋊D4 [×4], C22×C20, D4×C10, D4×C10 [×2], C23×D5, Dic54D4 [×2], D10.12D4 [×2], C4⋊C47D5, D102Q8, C20.48D4, D4×Dic5 [×2], C23⋊D10 [×2], C202D4, C5×C4⋊D4, D5×C22×C4, C2×D42D5, C4⋊C421D10
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C4○D4 [×4], C24, D10 [×7], C22×D4, C2×C4○D4 [×2], C22×D5 [×7], C22.19C24, D4×D5 [×2], D42D5 [×2], C23×D5, C2×D4×D5, C2×D42D5, D5×C4○D4, C4⋊C421D10

Smallest permutation representation of C4⋊C421D10
On 80 points
Generators in S80
(1 48 13 58)(2 49 14 59)(3 50 15 60)(4 41 16 51)(5 42 17 52)(6 43 18 53)(7 44 19 54)(8 45 20 55)(9 46 11 56)(10 47 12 57)(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 78 18 67)(2 68 19 79)(3 80 20 69)(4 70 11 71)(5 72 12 61)(6 62 13 73)(7 74 14 63)(8 64 15 75)(9 76 16 65)(10 66 17 77)(21 52 32 47)(22 48 33 53)(23 54 34 49)(24 50 35 55)(25 56 36 41)(26 42 37 57)(27 58 38 43)(28 44 39 59)(29 60 40 45)(30 46 31 51)
(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 17)(2 16)(3 15)(4 14)(5 13)(6 12)(7 11)(8 20)(9 19)(10 18)(21 27)(22 26)(23 25)(28 30)(31 39)(32 38)(33 37)(34 36)(41 59)(42 58)(43 57)(44 56)(45 55)(46 54)(47 53)(48 52)(49 51)(50 60)(61 67)(62 66)(63 65)(68 70)(71 79)(72 78)(73 77)(74 76)

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

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

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

56 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J2K4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P5A5B10A···10F10G10H10I10J10K10L10M10N20A···20H20I20J20K20L
order12222222222244444444444444445510···10101010101010101020···2020202020
size11112244101010102222445555101020202020222···2444488884···48888

56 irreducible representations

dim11111111111122222222444
type+++++++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2D4D5C4○D4C4○D4D10D10D10D10D4×D5D42D5D5×C4○D4
kernelC4⋊C421D10Dic54D4D10.12D4C4⋊C47D5D102Q8C20.48D4D4×Dic5C23⋊D10C202D4C5×C4⋊D4D5×C22×C4C2×D42D5C4×D5C4⋊D4D10C2×C10C22⋊C4C4⋊C4C22×C4C2×D4C4C22C2
# reps12211122111142444226444

Matrix representation of C4⋊C421D10 in GL6(𝔽41)

4000000
0400000
001000
000100
0000320
000009
,
0400000
100000
0040000
0004000
000009
0000320
,
4000000
010000
00353500
0064000
0000400
0000040
,
100000
010000
00353500
0040600
0000400
000001

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,32,0,0,0,0,0,0,9],[0,1,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,32,0,0,0,0,9,0],[40,0,0,0,0,0,0,1,0,0,0,0,0,0,35,6,0,0,0,0,35,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,35,40,0,0,0,0,35,6,0,0,0,0,0,0,40,0,0,0,0,0,0,1] >;

C4⋊C421D10 in GAP, Magma, Sage, TeX

C_4\rtimes C_4\rtimes_{21}D_{10}
% in TeX

G:=Group("C4:C4:21D10");
// GroupNames label

G:=SmallGroup(320,1278);
// by ID

G=gap.SmallGroup(320,1278);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,100,1123,794,297,12550]);
// Polycyclic

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

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