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G = C427D10order 320 = 26·5

7th semidirect product of C42 and D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C427D10, C10.932+ 1+4, C4⋊C454D10, (C4×D20)⋊5C2, (C2×D20)⋊30C4, D2036(C2×C4), (C4×C20)⋊4C22, C42⋊C26D5, D208C411C2, (C2×C10).65C24, C10.38(C23×C4), C4⋊Dic582C22, C2.1(D48D10), (C2×C20).583C23, C20.179(C22×C4), C22⋊C4.125D10, C53(C22.11C24), (C4×Dic5)⋊10C22, D10.13(C22×C4), (C22×D20).18C2, (C22×C4).188D10, D10⋊C460C22, C22.27(C23×D5), (C2×D20).263C22, (C23×D5).35C22, C23.153(C22×D5), C23.D5.94C22, C23.21D1024C2, (C22×C10).135C23, (C22×C20).225C22, (C2×Dic5).205C23, (C22×D5).172C23, (C2×C4)⋊6(C4×D5), C4.58(C2×C4×D5), (C2×C20)⋊24(C2×C4), (C2×C4×D5)⋊43C22, (C5×C4⋊C4)⋊51C22, C2.19(D5×C22×C4), C22.27(C2×C4×D5), (D5×C22⋊C4)⋊25C2, (C22×D5)⋊9(C2×C4), (C5×C42⋊C2)⋊7C2, (C2×C4).271(C22×D5), (C2×C10).122(C22×C4), (C5×C22⋊C4).135C22, SmallGroup(320,1193)

Series: Derived Chief Lower central Upper central

C1C10 — C427D10
C1C5C10C2×C10C22×D5C23×D5C22×D20 — C427D10
C5C10 — C427D10
C1C22C42⋊C2

Generators and relations for C427D10
 G = < a,b,c,d | a4=b4=c10=d2=1, ab=ba, ac=ca, dad=a-1, cbc-1=dbd=a2b, dcd=c-1 >

Subgroups: 1358 in 338 conjugacy classes, 151 normal (19 characteristic)
C1, C2, C2 [×2], C2 [×10], C4 [×4], C4 [×8], C22, C22 [×2], C22 [×26], C5, C2×C4 [×2], C2×C4 [×8], C2×C4 [×12], D4 [×16], C23, C23 [×16], D5 [×8], C10, C10 [×2], C10 [×2], C42 [×2], C42 [×2], C22⋊C4 [×2], C22⋊C4 [×10], C4⋊C4 [×2], C4⋊C4 [×2], C22×C4, C22×C4 [×8], C2×D4 [×12], C24 [×2], Dic5 [×4], C20 [×4], C20 [×4], D10 [×8], D10 [×16], C2×C10, C2×C10 [×2], C2×C10 [×2], C2×C22⋊C4 [×4], C42⋊C2, C42⋊C2, C4×D4 [×8], C22×D4, C4×D5 [×8], D20 [×16], C2×Dic5 [×4], C2×C20 [×2], C2×C20 [×8], C22×D5 [×12], C22×D5 [×4], C22×C10, C22.11C24, C4×Dic5 [×2], C4⋊Dic5 [×2], D10⋊C4 [×8], C23.D5 [×2], C4×C20 [×2], C5×C22⋊C4 [×2], C5×C4⋊C4 [×2], C2×C4×D5 [×8], C2×D20 [×12], C22×C20, C23×D5 [×2], C4×D20 [×4], D5×C22⋊C4 [×4], D208C4 [×4], C23.21D10, C5×C42⋊C2, C22×D20, C427D10
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], D5, C22×C4 [×14], C24, D10 [×7], C23×C4, 2+ 1+4 [×2], C4×D5 [×4], C22×D5 [×7], C22.11C24, C2×C4×D5 [×6], C23×D5, D5×C22×C4, D48D10 [×2], C427D10

Smallest permutation representation of C427D10
On 80 points
Generators in S80
(1 65 25 60)(2 66 26 51)(3 67 27 52)(4 68 28 53)(5 69 29 54)(6 70 30 55)(7 61 21 56)(8 62 22 57)(9 63 23 58)(10 64 24 59)(11 45 73 33)(12 46 74 34)(13 47 75 35)(14 48 76 36)(15 49 77 37)(16 50 78 38)(17 41 79 39)(18 42 80 40)(19 43 71 31)(20 44 72 32)
(1 33 6 50)(2 46 7 39)(3 35 8 42)(4 48 9 31)(5 37 10 44)(11 70 78 65)(12 56 79 51)(13 62 80 67)(14 58 71 53)(15 64 72 69)(16 60 73 55)(17 66 74 61)(18 52 75 57)(19 68 76 63)(20 54 77 59)(21 41 26 34)(22 40 27 47)(23 43 28 36)(24 32 29 49)(25 45 30 38)
(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 59)(2 58)(3 57)(4 56)(5 55)(6 54)(7 53)(8 52)(9 51)(10 60)(11 44)(12 43)(13 42)(14 41)(15 50)(16 49)(17 48)(18 47)(19 46)(20 45)(21 68)(22 67)(23 66)(24 65)(25 64)(26 63)(27 62)(28 61)(29 70)(30 69)(31 74)(32 73)(33 72)(34 71)(35 80)(36 79)(37 78)(38 77)(39 76)(40 75)

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

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

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

74 conjugacy classes

class 1 2A2B2C2D2E2F···2M4A···4L4M···4T5A5B10A···10F10G10H10I10J20A···20H20I···20AB
order1222222···24···44···45510···101010101020···2020···20
size11112210···102···210···10222···244442···24···4

74 irreducible representations

dim1111111122222244
type++++++++++++++
imageC1C2C2C2C2C2C2C4D5D10D10D10D10C4×D52+ 1+4D48D10
kernelC427D10C4×D20D5×C22⋊C4D208C4C23.21D10C5×C42⋊C2C22×D20C2×D20C42⋊C2C42C22⋊C4C4⋊C4C22×C4C2×C4C10C2
# reps144411116244421628

Matrix representation of C427D10 in GL6(𝔽41)

4000000
0400000
00303200
0091100
005363032
00136911
,
900000
090000
00320390
003340039
000090
0040181
,
070000
3560000
00343400
007100
0093577
0030273440
,
3570000
3660000
00142700
00112700
0012292714
0027293014

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,30,9,5,1,0,0,32,11,36,36,0,0,0,0,30,9,0,0,0,0,32,11],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,32,33,0,40,0,0,0,40,0,1,0,0,39,0,9,8,0,0,0,39,0,1],[0,35,0,0,0,0,7,6,0,0,0,0,0,0,34,7,9,30,0,0,34,1,35,27,0,0,0,0,7,34,0,0,0,0,7,40],[35,36,0,0,0,0,7,6,0,0,0,0,0,0,14,11,12,27,0,0,27,27,29,29,0,0,0,0,27,30,0,0,0,0,14,14] >;

C427D10 in GAP, Magma, Sage, TeX

C_4^2\rtimes_7D_{10}
% in TeX

G:=Group("C4^2:7D10");
// GroupNames label

G:=SmallGroup(320,1193);
// by ID

G=gap.SmallGroup(320,1193);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,184,570,80,12550]);
// Polycyclic

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

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