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G = C40.49C23  order 320 = 26·5

42nd non-split extension by C40 of C23 acting via C23/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C40.49C23, C20.72C24, M4(2)⋊28D10, C8○D4⋊8D5, (C2×C8)⋊23D10, (D4×D5).1C4, (Q8×D5).1C4, D4.13(C4×D5), Q8.14(C4×D5), (C2×C40)⋊26C22, C4○D4.43D10, D20.35(C2×C4), D4⋊2D5.1C4, (C8×D5)⋊12C22, C5⋊5(Q8○M4(2)), Q8.Dic5⋊8C2, Q8⋊2D5.1C4, C4.71(C23×D5), C8.56(C22×D5), C8⋊D5⋊22C22, (D5×M4(2))⋊11C2, C20.74(C22×C4), C10.56(C23×C4), C5⋊2C8.33C23, (C4×D5).73C23, D20.3C4⋊17C2, D20.2C4⋊13C2, (C2×C20).514C23, Dic10.37(C2×C4), C4○D20.52C22, D10.25(C22×C4), C4.Dic5⋊27C22, (C5×M4(2))⋊28C22, Dic5.24(C22×C4), C4.39(C2×C4×D5), (C5×C8○D4)⋊9C2, C22.5(C2×C4×D5), (D5×C4○D4).3C2, C5⋊D4.6(C2×C4), (C2×C8⋊D5)⋊28C2, C2.36(D5×C22×C4), (C5×D4).31(C2×C4), (C4×D5).11(C2×C4), (C5×Q8).33(C2×C4), (C2×C5⋊2C8)⋊13C22, (C2×C4×D5).163C22, (C2×C10).12(C22×C4), (C2×Dic5).40(C2×C4), (C5×C4○D4).44C22, (C22×D5).33(C2×C4), (C2×C4).607(C22×D5), SmallGroup(320,1422)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C10 — C40.49C23
Chief series
C1 — C5 — C10 — C20 — C4×D5 — C2×C4×D5 — D5×C4○D4 — C40.49C23
Lower central
C5 — C10 — C40.49C23

Subgroups: 734 in 258 conjugacy classes, 147 normal (24 characteristic)
C1, C2, C2 [×7], C4, C4 [×3], C4 [×4], C22 [×3], C22 [×7], C5, C8, C8 [×3], C8 [×4], C2×C4 [×3], C2×C4 [×13], D4 [×3], D4 [×9], Q8, Q8 [×3], C23 [×3], D5 [×4], C10, C10 [×3], C2×C8 [×3], C2×C8 [×9], M4(2) [×3], M4(2) [×13], C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4, C4○D4 [×7], Dic5, Dic5 [×3], C20, C20 [×3], D10, D10 [×3], D10 [×3], C2×C10 [×3], C2×M4(2) [×6], C8○D4, C8○D4 [×7], C2×C4○D4, C5⋊2C8, C5⋊2C8 [×3], C40, C40 [×3], Dic10 [×3], C4×D5, C4×D5 [×9], D20 [×3], C2×Dic5 [×3], C5⋊D4 [×6], C2×C20 [×3], C5×D4 [×3], C5×Q8, C22×D5 [×3], Q8○M4(2), C8×D5 [×6], C8⋊D5, C8⋊D5 [×9], C2×C5⋊2C8 [×3], C4.Dic5 [×3], C2×C40 [×3], C5×M4(2) [×3], C2×C4×D5 [×3], C4○D20 [×3], D4×D5 [×3], D4⋊2D5 [×3], Q8×D5, Q8⋊2D5, C5×C4○D4, C2×C8⋊D5 [×3], D20.3C4 [×3], D5×M4(2) [×3], D20.2C4 [×3], Q8.Dic5, C5×C8○D4, D5×C4○D4, C40.49C23

Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], D5, C22×C4 [×14], C24, D10 [×7], C23×C4, C4×D5 [×4], C22×D5 [×7], Q8○M4(2), C2×C4×D5 [×6], C23×D5, D5×C22×C4, C40.49C23

Generators and relations
 G = < a,b,c,d | a40=b2=c2=d2=1, bab=a29, cac=a21, ad=da, bc=cb, bd=db, dcd=a20c >

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

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

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

Matrix representation ►G ⊆ GL6(𝔽41)

6350000
610000
0003900
0025000
00400402
002040361
,
3560000
160000
001000
0004000
000010
0010140
,
4000000
0400000
000010
00200201
001000
00211210
,
4000000
0400000
001000
000100
0000400
0010040

G:=sub<GL(6,GF(41))| [6,6,0,0,0,0,35,1,0,0,0,0,0,0,0,25,40,20,0,0,39,0,0,40,0,0,0,0,40,36,0,0,0,0,2,1],[35,1,0,0,0,0,6,6,0,0,0,0,0,0,1,0,0,1,0,0,0,40,0,0,0,0,0,0,1,1,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,20,1,21,0,0,0,0,0,1,0,0,1,20,0,21,0,0,0,1,0,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,1,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40] >;
 

74 conjugacy classes

class 1 2A2B2C2D2E2F2G2H4A4B4C4D4E4F4G4H4I5A5B8A···8H8I···8P10A10B10C···10H20A20B20C20D20E···20J40A···40H40I···40T
order122222222444444444558···88···8101010···102020202020···2040···4040···40
size11222101010101122210101010222···210···10224···422224···42···24···4

74 irreducible representations

dim11111111111122222244
type++++++++++++
imageC1C2C2C2C2C2C2C2C4C4C4C4D5D10D10D10C4×D5C4×D5Q8○M4(2)C40.49C23
kernelC40.49C23C2×C8⋊D5D20.3C4D5×M4(2)D20.2C4Q8.Dic5C5×C8○D4D5×C4○D4D4×D5D4⋊2D5Q8×D5Q8⋊2D5C8○D4C2×C8M4(2)C4○D4D4Q8C5C1
# reps133331116622266212428

In GAP, Magma, Sage, TeX 

C_{40}._{49}C_2^3
 
% in TeX
 
G:=Group("C40.49C2^3");
 
// GroupNames label
 
G:=SmallGroup(320,1422);
 
// by ID
 
G=gap.SmallGroup(320,1422);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,387,1123,80,102,12550]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^40=b^2=c^2=d^2=1,b*a*b=a^29,c*a*c=a^21,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=a^20*c>;
 
// generators/relations
 

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