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G = C424D10order 320 = 26·5

4th semidirect product of C42 and D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C424D10, C4○D2011C4, (C2×D20)⋊24C4, (C4×C20)⋊3C22, C4.83(C2×D20), D204C42C2, D20.36(C2×C4), (C2×C20).144D4, (C2×C4).147D20, C20.303(C2×D4), C42⋊C24D5, (C2×Dic10)⋊23C4, (C22×C10).78D4, C20.70(C22⋊C4), (C2×C20).794C23, C20.169(C22×C4), Dic10.38(C2×C4), C55(C42⋊C22), C4○D20.38C22, (C22×C4).113D10, C23.21(C5⋊D4), C4.Dic520C22, C4.10(D10⋊C4), (C22×C20).154C22, C22.25(D10⋊C4), C4.68(C2×C4×D5), (C2×C4).46(C4×D5), (C2×C4○D20).8C2, (C2×C20).267(C2×C4), (C5×C42⋊C2)⋊4C2, (C2×C10).461(C2×D4), (C2×C4).45(C5⋊D4), C10.88(C2×C22⋊C4), (C2×C4.Dic5)⋊10C2, C22.27(C2×C5⋊D4), C2.20(C2×D10⋊C4), (C2×C4).708(C22×D5), (C2×C10).81(C22⋊C4), SmallGroup(320,632)

Series: Derived Chief Lower central Upper central

C1C20 — C424D10
C1C5C10C2×C10C2×C20C4○D20C2×C4○D20 — C424D10
C5C10C20 — C424D10
C1C4C22×C4C42⋊C2

Generators and relations for C424D10
 G = < a,b,c,d | a4=b4=c10=d2=1, ab=ba, cac-1=ab2, dad=ab-1, bc=cb, dbd=b-1, dcd=c-1 >

Subgroups: 590 in 154 conjugacy classes, 59 normal (41 characteristic)
C1, C2, C2 [×5], C4 [×4], C4 [×4], C22 [×3], C22 [×5], C5, C8 [×2], C2×C4 [×6], C2×C4 [×7], D4 [×7], Q8 [×3], C23, C23, D5 [×2], C10, C10 [×3], C42 [×2], C22⋊C4, C4⋊C4, C2×C8, M4(2) [×3], C22×C4, C22×C4, C2×D4 [×2], C2×Q8, C4○D4 [×6], Dic5 [×2], C20 [×4], C20 [×2], D10 [×4], C2×C10 [×3], C2×C10, C4≀C2 [×4], C42⋊C2, C2×M4(2), C2×C4○D4, C52C8 [×2], Dic10 [×2], Dic10, C4×D5 [×4], D20 [×2], D20, C2×Dic5, C5⋊D4 [×4], C2×C20 [×6], C2×C20 [×2], C22×D5, C22×C10, C42⋊C22, C2×C52C8, C4.Dic5 [×2], C4.Dic5, C4×C20 [×2], C5×C22⋊C4, C5×C4⋊C4, C2×Dic10, C2×C4×D5, C2×D20, C4○D20 [×4], C4○D20 [×2], C2×C5⋊D4, C22×C20, D204C4 [×4], C2×C4.Dic5, C5×C42⋊C2, C2×C4○D20, C424D10
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, D5, C22⋊C4 [×4], C22×C4, C2×D4 [×2], D10 [×3], C2×C22⋊C4, C4×D5 [×2], D20 [×2], C5⋊D4 [×2], C22×D5, C42⋊C22, D10⋊C4 [×4], C2×C4×D5, C2×D20, C2×C5⋊D4, C2×D10⋊C4, C424D10

Smallest permutation representation of C424D10
On 80 points
Generators in S80
(1 63 26 46)(2 69 27 42)(3 65 28 48)(4 61 29 44)(5 67 30 50)(6 66 24 49)(7 62 25 45)(8 68 21 41)(9 64 22 47)(10 70 23 43)(11 71 39 57)(12 77 40 53)(13 73 36 59)(14 79 37 55)(15 75 38 51)(16 78 31 54)(17 74 32 60)(18 80 33 56)(19 76 34 52)(20 72 35 58)
(1 16 8 13)(2 17 9 14)(3 18 10 15)(4 19 6 11)(5 20 7 12)(21 36 26 31)(22 37 27 32)(23 38 28 33)(24 39 29 34)(25 40 30 35)(41 59 46 54)(42 60 47 55)(43 51 48 56)(44 52 49 57)(45 53 50 58)(61 76 66 71)(62 77 67 72)(63 78 68 73)(64 79 69 74)(65 80 70 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 5)(2 4)(6 9)(7 8)(11 17)(12 16)(13 20)(14 19)(15 18)(21 30)(22 29)(23 28)(24 27)(25 26)(31 35)(32 34)(36 40)(37 39)(41 53)(42 52)(43 51)(44 60)(45 59)(46 58)(47 57)(48 56)(49 55)(50 54)(61 79)(62 78)(63 77)(64 76)(65 75)(66 74)(67 73)(68 72)(69 71)(70 80)

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

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

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

62 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E4F4G4H4I4J4K5A5B8A8B8C8D10A···10F10G10H10I10J20A···20H20I···20AB
order12222224444444444455888810···101010101020···2020···20
size112222020112224444202022202020202···244442···24···4

62 irreducible representations

dim1111111122222222244
type+++++++++++
imageC1C2C2C2C2C4C4C4D4D4D5D10D10C4×D5D20C5⋊D4C5⋊D4C42⋊C22C424D10
kernelC424D10D204C4C2×C4.Dic5C5×C42⋊C2C2×C4○D20C2×Dic10C2×D20C4○D20C2×C20C22×C10C42⋊C2C42C22×C4C2×C4C2×C4C2×C4C23C5C1
# reps1411122431242884428

Matrix representation of C424D10 in GL4(𝔽41) generated by

00400
00040
17100
402400
,
113200
93000
001132
00930
,
40700
34700
00134
00734
,
74000
73400
001411
002727
G:=sub<GL(4,GF(41))| [0,0,17,40,0,0,1,24,40,0,0,0,0,40,0,0],[11,9,0,0,32,30,0,0,0,0,11,9,0,0,32,30],[40,34,0,0,7,7,0,0,0,0,1,7,0,0,34,34],[7,7,0,0,40,34,0,0,0,0,14,27,0,0,11,27] >;

C424D10 in GAP, Magma, Sage, TeX

C_4^2\rtimes_4D_{10}
% in TeX

G:=Group("C4^2:4D10");
// GroupNames label

G:=SmallGroup(320,632);
// by ID

G=gap.SmallGroup(320,632);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,422,387,58,1123,1684,102,12550]);
// Polycyclic

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

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