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G = C4⋊C436D10order 320 = 26·5

2nd semidirect product of C4⋊C4 and D10 acting via D10/C10=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C4⋊C436D10, (C2×D20)⋊23C4, (C2×C4).46D20, C4.62(C2×D20), D20.39(C2×C4), C20.142(C2×D4), (C2×C20).472D4, C42⋊C21D5, D206C427C2, (C22×C10).74D4, C2.2(D4⋊D10), C20.94(C22⋊C4), C20.123(C22×C4), (C2×C20).328C23, (C22×D20).13C2, (C22×C4).110D10, C23.54(C5⋊D4), C53(C23.37D4), C4.23(D10⋊C4), C10.105(C8⋊C22), (C2×D20).242C22, (C22×C20).150C22, C22.23(D10⋊C4), C4.51(C2×C4×D5), (C2×C4).44(C4×D5), (C5×C4⋊C4)⋊41C22, (C2×C52C8)⋊4C22, (C2×C4.Dic5)⋊9C2, (C2×C20).264(C2×C4), (C5×C42⋊C2)⋊1C2, (C2×C10).457(C2×D4), C10.85(C2×C22⋊C4), C22.72(C2×C5⋊D4), C2.17(C2×D10⋊C4), (C2×C4).241(C5⋊D4), (C2×C4).428(C22×D5), (C2×C10).78(C22⋊C4), SmallGroup(320,628)

Series: Derived Chief Lower central Upper central

C1C20 — C4⋊C436D10
C1C5C10C2×C10C2×C20C2×D20C22×D20 — C4⋊C436D10
C5C10C20 — C4⋊C436D10
C1C22C22×C4C42⋊C2

Generators and relations for C4⋊C436D10
 G = < a,b,c,d | a4=b4=c10=d2=1, bab-1=dad=a-1, ac=ca, cbc-1=a2b, dbd=ab-1, dcd=c-1 >

Subgroups: 974 in 190 conjugacy classes, 63 normal (23 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×2], C4 [×2], C4 [×2], C22, C22 [×2], C22 [×18], C5, C8 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×2], D4 [×10], C23, C23 [×10], D5 [×4], C10, C10 [×2], C10 [×2], C42, C22⋊C4, C4⋊C4 [×2], C2×C8 [×2], M4(2) [×2], C22×C4, C2×D4 [×9], C24, C20 [×2], C20 [×2], C20 [×2], D10 [×16], C2×C10, C2×C10 [×2], C2×C10 [×2], D4⋊C4 [×4], C42⋊C2, C2×M4(2), C22×D4, C52C8 [×2], D20 [×4], D20 [×6], C2×C20 [×2], C2×C20 [×4], C2×C20 [×2], C22×D5 [×10], C22×C10, C23.37D4, C2×C52C8 [×2], C4.Dic5 [×2], C4×C20, C5×C22⋊C4, C5×C4⋊C4 [×2], C2×D20 [×6], C2×D20 [×3], C22×C20, C23×D5, D206C4 [×4], C2×C4.Dic5, C5×C42⋊C2, C22×D20, C4⋊C436D10
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, C8⋊C22 [×2], C4×D5 [×2], D20 [×2], C5⋊D4 [×2], C22×D5, C23.37D4, D10⋊C4 [×4], C2×C4×D5, C2×D20, C2×C5⋊D4, C2×D10⋊C4, D4⋊D10 [×2], C4⋊C436D10

Smallest permutation representation of C4⋊C436D10
On 80 points
Generators in S80
(1 13 33 50)(2 14 34 41)(3 15 35 42)(4 16 36 43)(5 17 37 44)(6 18 38 45)(7 19 39 46)(8 20 40 47)(9 11 31 48)(10 12 32 49)(21 68 55 73)(22 69 56 74)(23 70 57 75)(24 61 58 76)(25 62 59 77)(26 63 60 78)(27 64 51 79)(28 65 52 80)(29 66 53 71)(30 67 54 72)
(1 73 38 78)(2 69 39 64)(3 75 40 80)(4 61 31 66)(5 77 32 72)(6 63 33 68)(7 79 34 74)(8 65 35 70)(9 71 36 76)(10 67 37 62)(11 53 43 58)(12 30 44 25)(13 55 45 60)(14 22 46 27)(15 57 47 52)(16 24 48 29)(17 59 49 54)(18 26 50 21)(19 51 41 56)(20 28 42 23)
(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 10)(7 9)(11 46)(12 45)(13 44)(14 43)(15 42)(16 41)(17 50)(18 49)(19 48)(20 47)(21 67)(22 66)(23 65)(24 64)(25 63)(26 62)(27 61)(28 70)(29 69)(30 68)(31 39)(32 38)(33 37)(34 36)(51 76)(52 75)(53 74)(54 73)(55 72)(56 71)(57 80)(58 79)(59 78)(60 77)

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

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

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

62 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G4H5A5B8A8B8C8D10A···10F10G10H10I10J20A···20H20I···20AB
order12222222224444444455888810···101010101020···2020···20
size111122202020202222444422202020202···244442···24···4

62 irreducible representations

dim11111122222222244
type+++++++++++++
imageC1C2C2C2C2C4D4D4D5D10D10C4×D5D20C5⋊D4C5⋊D4C8⋊C22D4⋊D10
kernelC4⋊C436D10D206C4C2×C4.Dic5C5×C42⋊C2C22×D20C2×D20C2×C20C22×C10C42⋊C2C4⋊C4C22×C4C2×C4C2×C4C2×C4C23C10C2
# reps14111831242884428

Matrix representation of C4⋊C436D10 in GL6(𝔽41)

100000
010000
000100
0040000
0004040
0037010
,
11320000
9300000
00834370
00348037
002813337
001328733
,
4070000
3470000
0040000
0004000
00372410
00243701
,
100000
7400000
001000
0004000
003114040
003127400

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,37,0,0,1,0,4,0,0,0,0,0,0,1,0,0,0,0,40,0],[11,9,0,0,0,0,32,30,0,0,0,0,0,0,8,34,28,13,0,0,34,8,13,28,0,0,37,0,33,7,0,0,0,37,7,33],[40,34,0,0,0,0,7,7,0,0,0,0,0,0,40,0,37,24,0,0,0,40,24,37,0,0,0,0,1,0,0,0,0,0,0,1],[1,7,0,0,0,0,0,40,0,0,0,0,0,0,1,0,31,31,0,0,0,40,14,27,0,0,0,0,0,40,0,0,0,0,40,0] >;

C4⋊C436D10 in GAP, Magma, Sage, TeX

C_4\rtimes C_4\rtimes_{36}D_{10}
% in TeX

G:=Group("C4:C4:36D10");
// GroupNames label

G:=SmallGroup(320,628);
// by ID

G=gap.SmallGroup(320,628);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,422,387,58,1684,438,102,12550]);
// Polycyclic

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

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