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G = (C2×C40)⋊C4order 320 = 26·5

11st semidirect product of C2×C40 and C4 acting faithfully

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C2×C40)⋊11C4, (C2×C8)⋊1Dic5, (C2×C4).14D20, C20.42(C4⋊C4), (C2×C20).11Q8, C23.9(C4×D5), (C2×C20).109D4, C4.Dic512C4, (C2×C4).5Dic10, C23.D5.8C4, (C2×C10).42C42, C4.13(C4⋊Dic5), (C22×C4).62D10, (C2×M4(2)).7D5, C20.58(C22⋊C4), C55(M4(2)⋊4C4), C4.28(C23.D5), C4.35(D10⋊C4), C22.10(C4×Dic5), C4.11(C10.D4), (C10×M4(2)).11C2, (C22×C20).125C22, C23.21D10.9C2, C22.6(C10.D4), C22.19(D10⋊C4), C10.34(C2.C42), C2.15(C10.10C42), (C2×C4).21(C4×D5), (C2×C10).36(C4⋊C4), (C2×C20).477(C2×C4), (C2×C4).76(C2×Dic5), (C2×C4).180(C5⋊D4), (C2×C4.Dic5).11C2, (C2×C10).74(C22⋊C4), (C22×C10).100(C2×C4), SmallGroup(320,114)

Series: Derived Chief Lower central Upper central

C1C2×C10 — (C2×C40)⋊C4
C1C5C10C2×C10C2×C20C22×C20C23.21D10 — (C2×C40)⋊C4
C5C10C2×C10 — (C2×C40)⋊C4
C1C4C22×C4C2×M4(2)

Generators and relations for (C2×C40)⋊C4
 G = < a,b,c | a2=b40=c4=1, ab=ba, cac-1=ab20, cbc-1=ab29 >

Subgroups: 262 in 90 conjugacy classes, 47 normal (39 characteristic)
C1, C2, C2 [×3], C4 [×4], C4 [×2], C22 [×3], C22, C5, C8 [×4], C2×C4 [×6], C2×C4 [×2], C23, C10, C10 [×3], C42, C22⋊C4 [×2], C4⋊C4, C2×C8 [×2], C2×C8, M4(2) [×5], C22×C4, Dic5 [×2], C20 [×4], C2×C10 [×3], C2×C10, C42⋊C2, C2×M4(2), C2×M4(2), C52C8 [×2], C40 [×2], C2×Dic5 [×2], C2×C20 [×6], C22×C10, M4(2)⋊4C4, C2×C52C8, C4.Dic5 [×2], C4.Dic5, C4×Dic5, C4⋊Dic5, C23.D5 [×2], C2×C40 [×2], C5×M4(2) [×2], C22×C20, C2×C4.Dic5, C23.21D10, C10×M4(2), (C2×C40)⋊C4
Quotients: C1, C2 [×3], C4 [×6], C22, C2×C4 [×3], D4 [×3], Q8, D5, C42, C22⋊C4 [×3], C4⋊C4 [×3], Dic5 [×2], D10, C2.C42, Dic10, C4×D5 [×2], D20, C2×Dic5, C5⋊D4 [×2], M4(2)⋊4C4, C4×Dic5, C10.D4 [×2], C4⋊Dic5, D10⋊C4 [×2], C23.D5, C10.10C42, (C2×C40)⋊C4

Smallest permutation representation of (C2×C40)⋊C4
On 80 points
Generators in S80
(1 73)(2 74)(3 75)(4 76)(5 77)(6 78)(7 79)(8 80)(9 41)(10 42)(11 43)(12 44)(13 45)(14 46)(15 47)(16 48)(17 49)(18 50)(19 51)(20 52)(21 53)(22 54)(23 55)(24 56)(25 57)(26 58)(27 59)(28 60)(29 61)(30 62)(31 63)(32 64)(33 65)(34 66)(35 67)(36 68)(37 69)(38 70)(39 71)(40 72)
(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)
(2 42 22 62)(3 19)(4 60 24 80)(5 37)(6 78 26 58)(7 15)(8 56 28 76)(9 33)(10 74 30 54)(12 52 32 72)(13 29)(14 70 34 50)(16 48 36 68)(17 25)(18 66 38 46)(20 44 40 64)(23 39)(27 35)(41 45)(43 63)(47 59)(49 77)(51 55)(53 73)(57 69)(61 65)(67 79)(71 75)

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

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

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

62 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E4F4G4H4I5A5B8A8B8C8D8E8F8G8H10A···10F10G10H10I10J20A···20H20I20J20K20L40A···40P
order12222444444444558888888810···101010101020···202020202040···40
size112221122220202020224444202020202···244442···244444···4

62 irreducible representations

dim1111111222222222244
type+++++-+-+-+
imageC1C2C2C2C4C4C4D4Q8D5Dic5D10Dic10C4×D5D20C5⋊D4C4×D5M4(2)⋊4C4(C2×C40)⋊C4
kernel(C2×C40)⋊C4C2×C4.Dic5C23.21D10C10×M4(2)C4.Dic5C23.D5C2×C40C2×C20C2×C20C2×M4(2)C2×C8C22×C4C2×C4C2×C4C2×C4C2×C4C23C5C1
# reps1111444312424448428

Matrix representation of (C2×C40)⋊C4 in GL6(𝔽41)

4000000
0400000
0040000
003711229
000001
000010
,
18210000
20210000
00150635
000001
00592615
000900
,
3200000
2890000
00120350
0004000
0000040
000010

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,37,0,0,0,0,0,1,0,0,0,0,0,12,0,1,0,0,0,29,1,0],[18,20,0,0,0,0,21,21,0,0,0,0,0,0,15,0,5,0,0,0,0,0,9,9,0,0,6,0,26,0,0,0,35,1,15,0],[32,28,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,20,40,0,0,0,0,35,0,0,1,0,0,0,0,40,0] >;

(C2×C40)⋊C4 in GAP, Magma, Sage, TeX

(C_2\times C_{40})\rtimes C_4
% in TeX

G:=Group("(C2xC40):C4");
// GroupNames label

G:=SmallGroup(320,114);
// by ID

G=gap.SmallGroup(320,114);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,253,64,387,136,1684,12550]);
// Polycyclic

G:=Group<a,b,c|a^2=b^40=c^4=1,a*b=b*a,c*a*c^-1=a*b^20,c*b*c^-1=a*b^29>;
// generators/relations

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