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G = C4223D10order 320 = 26·5

23rd semidirect product of C42 and D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C4223D10, C10.1372+ 1+4, C4⋊C433D10, (C4×D20)⋊13C2, (C4×C20)⋊7C22, C4⋊D2036C2, C4.D208C2, C422C22D5, D208C439C2, C22⋊D2027C2, D10⋊D444C2, D10⋊Q840C2, (C2×D20)⋊29C22, C4⋊Dic561C22, C22⋊C4.40D10, D10.18(C4○D4), D10⋊C47C22, (C2×C20).193C23, (C2×C10).248C24, C59(C22.32C24), (C4×Dic5)⋊38C22, D10.13D438C2, D10.12D448C2, C2.62(D48D10), C23.54(C22×D5), Dic5.5D444C2, (C2×Dic10)⋊11C22, C10.D427C22, (C22×C10).62C23, (C23×D5).68C22, C22.269(C23×D5), C23.D5.64C22, (C2×Dic5).274C23, (C22×D5).111C23, C2.95(D5×C4○D4), (C2×C4×D5)⋊27C22, C4⋊C4⋊D541C2, (C5×C4⋊C4)⋊32C22, (D5×C22⋊C4)⋊20C2, (C5×C422C2)⋊3C2, C10.206(C2×C4○D4), (C2×C4).85(C22×D5), (C2×C5⋊D4).68C22, (C5×C22⋊C4).73C22, SmallGroup(320,1376)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C4223D10
C1C5C10C2×C10C22×D5C23×D5D5×C22⋊C4 — C4223D10
C5C2×C10 — C4223D10
C1C22C422C2

Generators and relations for C4223D10
 G = < a,b,c,d | a4=b4=c10=d2=1, ab=ba, cac-1=dad=a-1b2, cbc-1=a2b, dbd=b-1, dcd=c-1 >

Subgroups: 1134 in 250 conjugacy classes, 93 normal (91 characteristic)
C1, C2 [×3], C2 [×6], C4 [×10], C22, C22 [×20], C5, C2×C4 [×6], C2×C4 [×8], D4 [×9], Q8, C23, C23 [×8], D5 [×5], C10 [×3], C10, C42, C42, C22⋊C4 [×3], C22⋊C4 [×11], C4⋊C4 [×3], C4⋊C4 [×3], C22×C4 [×4], C2×D4 [×7], C2×Q8, C24, Dic5 [×4], C20 [×6], D10 [×2], D10 [×15], C2×C10, C2×C10 [×3], C2×C22⋊C4, C4×D4 [×2], C22≀C2 [×2], C4⋊D4 [×3], C22⋊Q8, C22.D4 [×2], C4.4D4 [×2], C422C2, C422C2, Dic10, C4×D5 [×4], D20 [×7], C2×Dic5 [×4], C5⋊D4 [×2], C2×C20 [×6], C22×D5 [×4], C22×D5 [×4], C22×C10, C22.32C24, C4×Dic5, C10.D4 [×2], C4⋊Dic5, D10⋊C4 [×10], C23.D5, C4×C20, C5×C22⋊C4 [×3], C5×C4⋊C4 [×3], C2×Dic10, C2×C4×D5 [×4], C2×D20 [×5], C2×C5⋊D4 [×2], C23×D5, C4×D20, C4.D20, D5×C22⋊C4, C22⋊D20 [×2], D10.12D4, D10⋊D4, Dic5.5D4, D208C4, D10.13D4, C4⋊D20 [×2], D10⋊Q8, C4⋊C4⋊D5, C5×C422C2, C4223D10
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×2], C24, D10 [×7], C2×C4○D4, 2+ 1+4 [×2], C22×D5 [×7], C22.32C24, C23×D5, D5×C4○D4, D48D10 [×2], C4223D10

Smallest permutation representation of C4223D10
On 80 points
Generators in S80
(1 65 8 44)(2 61 9 50)(3 67 10 46)(4 63 6 42)(5 69 7 48)(11 68 16 47)(12 64 17 43)(13 70 18 49)(14 66 19 45)(15 62 20 41)(21 78 26 57)(22 53 27 74)(23 80 28 59)(24 55 29 76)(25 72 30 51)(31 60 36 71)(32 77 37 56)(33 52 38 73)(34 79 39 58)(35 54 40 75)
(1 38 18 21)(2 34 19 27)(3 40 20 23)(4 36 16 29)(5 32 17 25)(6 31 11 24)(7 37 12 30)(8 33 13 26)(9 39 14 22)(10 35 15 28)(41 80 67 75)(42 60 68 55)(43 72 69 77)(44 52 70 57)(45 74 61 79)(46 54 62 59)(47 76 63 71)(48 56 64 51)(49 78 65 73)(50 58 66 53)
(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 7)(2 6)(3 10)(4 9)(5 8)(11 19)(12 18)(13 17)(14 16)(15 20)(21 37)(22 36)(23 35)(24 34)(25 33)(26 32)(27 31)(28 40)(29 39)(30 38)(41 67)(42 66)(43 65)(44 64)(45 63)(46 62)(47 61)(48 70)(49 69)(50 68)(51 57)(52 56)(53 55)(58 60)(71 79)(72 78)(73 77)(74 76)

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

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

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

50 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A4B4C···4G4H4I4J4K4L5A5B10A···10F10G10H20A···20L20M···20R
order1222222222444···4444445510···10101020···2020···20
size111141010202020224···41010202020222···2884···48···8

50 irreducible representations

dim1111111111111122222444
type++++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2C2C2D5C4○D4D10D10D102+ 1+4D5×C4○D4D48D10
kernelC4223D10C4×D20C4.D20D5×C22⋊C4C22⋊D20D10.12D4D10⋊D4Dic5.5D4D208C4D10.13D4C4⋊D20D10⋊Q8C4⋊C4⋊D5C5×C422C2C422C2D10C42C22⋊C4C4⋊C4C10C2C2
# reps1111211111211124266248

Matrix representation of C4223D10 in GL6(𝔽41)

3200000
0320000
0000119
00003230
0011900
00323000
,
1370000
21400000
000010
000001
001000
000100
,
100000
21400000
007700
00344000
00003434
000071
,
100000
21400000
00343400
001700
00003434
000017

G:=sub<GL(6,GF(41))| [32,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,11,32,0,0,0,0,9,30,0,0,11,32,0,0,0,0,9,30,0,0],[1,21,0,0,0,0,37,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[1,21,0,0,0,0,0,40,0,0,0,0,0,0,7,34,0,0,0,0,7,40,0,0,0,0,0,0,34,7,0,0,0,0,34,1],[1,21,0,0,0,0,0,40,0,0,0,0,0,0,34,1,0,0,0,0,34,7,0,0,0,0,0,0,34,1,0,0,0,0,34,7] >;

C4223D10 in GAP, Magma, Sage, TeX

C_4^2\rtimes_{23}D_{10}
% in TeX

G:=Group("C4^2:23D10");
// GroupNames label

G:=SmallGroup(320,1376);
// by ID

G=gap.SmallGroup(320,1376);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,219,184,675,570,192,12550]);
// Polycyclic

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

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