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G = C40.C23order 320 = 26·5

6th non-split extension by C40 of C23 acting faithfully

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Q165D10, SD167D10, D20.43D4, D404C22, C40.6C23, C20.25C24, M4(2)⋊13D10, Dic10.43D4, D20.18C23, Dic10.18C23, C5⋊D4.6D4, C8⋊D104C2, D40⋊C24C2, (C2×Q8)⋊12D10, (D5×SD16)⋊4C2, C55(D4○SD16), C4.117(D4×D5), (C8×D5)⋊6C22, C8.C224D5, D48D108C2, Q8.D102C2, (Q8×D5)⋊4C22, C8.6(C22×D5), D4⋊D516C22, Q16⋊D53C2, C4○D4.14D10, D10.57(C2×D4), C20.246(C2×D4), C4○D209C22, C40⋊C27C22, C8⋊D57C22, Q8⋊D515C22, (C5×Q16)⋊3C22, (D4×D5).4C22, C4.25(C23×D5), C22.16(D4×D5), D20.2C43C2, D4.8D105C2, C52C8.27C23, D4.D515C22, Dic5.63(C2×D4), Q82D54C22, (Q8×C10)⋊22C22, (C5×SD16)⋊7C22, (C4×D5).16C23, (C5×D4).18C23, D4.18(C22×D5), C5⋊Q1614C22, Q8.18(C22×D5), (C5×Q8).18C23, (C2×C20).116C23, Q8.10D105C2, C10.126(C22×D4), (C5×M4(2))⋊7C22, (C2×D20).188C22, C2.99(C2×D4×D5), (C2×Q8⋊D5)⋊29C2, (C2×C10).71(C2×D4), (C5×C8.C22)⋊3C2, (C2×C52C8)⋊19C22, (C5×C4○D4).27C22, (C2×C4).100(C22×D5), SmallGroup(320,1450)

Series: Derived Chief Lower central Upper central

C1C20 — C40.C23
C1C5C10C20C4×D5C4○D20Q8.10D10 — C40.C23
C5C10C20 — C40.C23
C1C2C2×C4C8.C22

Generators and relations for C40.C23
 G = < a,b,c,d | a40=b2=1, c2=d2=a20, bab=a29, cac-1=a31, dad-1=a11, bc=cb, dbd-1=a20b, cd=dc >

Subgroups: 1062 in 258 conjugacy classes, 99 normal (45 characteristic)
C1, C2, C2 [×7], C4 [×2], C4 [×6], C22, C22 [×9], C5, C8 [×2], C8 [×2], C2×C4, C2×C4 [×11], D4, D4 [×15], Q8, Q8 [×2], Q8 [×5], C23 [×3], D5 [×5], C10, C10 [×2], C2×C8 [×3], M4(2), M4(2) [×2], D8 [×3], SD16 [×2], SD16 [×8], Q16 [×2], Q16, C2×D4 [×6], C2×Q8, C2×Q8 [×3], C4○D4, C4○D4 [×10], Dic5 [×2], Dic5, C20 [×2], C20 [×3], D10 [×2], D10 [×6], C2×C10, C2×C10, C8○D4, C2×SD16 [×3], C4○D8 [×3], C8⋊C22 [×3], C8.C22, C8.C22 [×2], 2+ 1+4, 2- 1+4, C52C8 [×2], C40 [×2], Dic10 [×2], Dic10 [×2], C4×D5 [×2], C4×D5 [×7], D20 [×2], D20 [×2], D20 [×5], C5⋊D4 [×2], C5⋊D4 [×3], C2×C20, C2×C20 [×2], C5×D4, C5×D4, C5×Q8, C5×Q8 [×2], C5×Q8, C22×D5 [×3], D4○SD16, C8×D5 [×2], C8⋊D5 [×2], C40⋊C2 [×2], D40 [×2], C2×C52C8, D4⋊D5, D4.D5, Q8⋊D5, Q8⋊D5 [×4], C5⋊Q16, C5×M4(2), C5×SD16 [×2], C5×Q16 [×2], C2×D20, C2×D20, C4○D20 [×2], C4○D20 [×3], D4×D5 [×2], D4×D5 [×2], Q8×D5 [×2], Q8×D5, Q82D5 [×4], Q82D5, Q8×C10, C5×C4○D4, D20.2C4, C8⋊D10, D5×SD16 [×2], D40⋊C2 [×2], Q16⋊D5 [×2], Q8.D10 [×2], C2×Q8⋊D5, D4.8D10, C5×C8.C22, Q8.10D10, D48D10, C40.C23
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C22×D4, C22×D5 [×7], D4○SD16, D4×D5 [×2], C23×D5, C2×D4×D5, C40.C23

Smallest permutation representation of C40.C23
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 53)(43 71)(44 60)(45 49)(46 78)(47 67)(48 56)(50 74)(51 63)(54 70)(55 59)(57 77)(58 66)(61 73)(64 80)(65 69)(68 76)(75 79)
(1 52 21 72)(2 43 22 63)(3 74 23 54)(4 65 24 45)(5 56 25 76)(6 47 26 67)(7 78 27 58)(8 69 28 49)(9 60 29 80)(10 51 30 71)(11 42 31 62)(12 73 32 53)(13 64 33 44)(14 55 34 75)(15 46 35 66)(16 77 36 57)(17 68 37 48)(18 59 38 79)(19 50 39 70)(20 41 40 61)
(1 36 21 16)(2 7 22 27)(3 18 23 38)(4 29 24 9)(5 40 25 20)(6 11 26 31)(8 33 28 13)(10 15 30 35)(12 37 32 17)(14 19 34 39)(41 56 61 76)(42 67 62 47)(43 78 63 58)(44 49 64 69)(45 60 65 80)(46 71 66 51)(48 53 68 73)(50 75 70 55)(52 57 72 77)(54 79 74 59)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E2F2G2H4A4B4C4D4E4F4G4H5A5B8A8B8C8D8E10A10B10C10D10E10F20A20B20C20D20E···20J40A40B40C40D
order1222222224444444455888881010101010102020202020···2040404040
size1124101020202022444101020224410102022448844448···88888

44 irreducible representations

dim1111111111112222222224448
type++++++++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2D4D4D4D5D10D10D10D10D10D4○SD16D4×D5D4×D5C40.C23
kernelC40.C23D20.2C4C8⋊D10D5×SD16D40⋊C2Q16⋊D5Q8.D10C2×Q8⋊D5D4.8D10C5×C8.C22Q8.10D10D48D10Dic10D20C5⋊D4C8.C22M4(2)SD16Q16C2×Q8C4○D4C5C4C22C1
# reps1112222111111122244222222

Matrix representation of C40.C23 in GL8(𝔽41)

003470000
003410000
734000000
740000000
0000015015
0000001515
00003002615
000030112615
,
347000000
407000000
003470000
004070000
000010401
00000100
000000400
000000040
,
00100000
00010000
10000000
01000000
00003026015
000030112615
0000001526
0000002626
,
004000000
000400000
400000000
040000000
00001115026
0000002626
00001101526
000030112615

G:=sub<GL(8,GF(41))| [0,0,7,7,0,0,0,0,0,0,34,40,0,0,0,0,34,34,0,0,0,0,0,0,7,1,0,0,0,0,0,0,0,0,0,0,0,0,30,30,0,0,0,0,15,0,0,11,0,0,0,0,0,15,26,26,0,0,0,0,15,15,15,15],[34,40,0,0,0,0,0,0,7,7,0,0,0,0,0,0,0,0,34,40,0,0,0,0,0,0,7,7,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,40,0,0,0,0,0,1,0,0,40],[0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,30,30,0,0,0,0,0,0,26,11,0,0,0,0,0,0,0,26,15,26,0,0,0,0,15,15,26,26],[0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,11,0,11,30,0,0,0,0,15,0,0,11,0,0,0,0,0,26,15,26,0,0,0,0,26,26,26,15] >;

C40.C23 in GAP, Magma, Sage, TeX

C_{40}.C_2^3
% in TeX

G:=Group("C40.C2^3");
// GroupNames label

G:=SmallGroup(320,1450);
// by ID

G=gap.SmallGroup(320,1450);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,184,570,185,438,235,102,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^40=b^2=1,c^2=d^2=a^20,b*a*b=a^29,c*a*c^-1=a^31,d*a*d^-1=a^11,b*c=c*b,d*b*d^-1=a^20*b,c*d=d*c>;
// generators/relations

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