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G = C40.47C23order 320 = 26·5

40th non-split extension by C40 of C23 acting via C23/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C40.47C23, C20.70C24, M4(2)⋊26D10, (C2×C8)⋊22D10, C4○D20.9C4, (C2×C40)⋊24C22, D20.43(C2×C4), (C2×D20).29C4, (C8×D5)⋊11C22, C54(Q8○M4(2)), C23.22(C4×D5), C4.69(C23×D5), C8.44(C22×D5), C8⋊D519C22, (D5×M4(2))⋊10C2, (C2×M4(2))⋊16D5, C10.54(C23×C4), C52C8.32C23, (C4×D5).72C23, D20.3C415C2, D20.2C412C2, (C10×M4(2))⋊10C2, (C2×C20).510C23, C20.152(C22×C4), (C2×Dic10).30C4, Dic10.45(C2×C4), C4○D20.59C22, D10.23(C22×C4), (C22×C4).264D10, C4.Dic540C22, (C5×M4(2))⋊26C22, Dic5.22(C22×C4), (C22×C20).265C22, C4.95(C2×C4×D5), (C2×C4).58(C4×D5), C5⋊D4.8(C2×C4), C2.34(D5×C22×C4), C22.28(C2×C4×D5), (C4×D5).10(C2×C4), (C2×C5⋊D4).24C4, (C2×C20).305(C2×C4), (C2×C52C8)⋊12C22, (C2×C4○D20).22C2, (C2×C4.Dic5)⋊25C2, (C2×C4×D5).162C22, (C2×Dic5).39(C2×C4), (C22×D5).32(C2×C4), (C2×C4).605(C22×D5), (C2×C10).127(C22×C4), (C22×C10).147(C2×C4), SmallGroup(320,1417)

Series: Derived Chief Lower central Upper central

C1C10 — C40.47C23
C1C5C10C20C4×D5C2×C4×D5C2×C4○D20 — C40.47C23
C5C10 — C40.47C23
C1C4C2×M4(2)

Generators and relations for C40.47C23
 G = < a,b,c,d,e | a20=b2=c2=d2=1, e2=a5, bab=a9, ac=ca, ad=da, ae=ea, bc=cb, dbd=a10b, be=eb, cd=dc, ece-1=a10c, de=ed >

Subgroups: 718 in 258 conjugacy classes, 147 normal (41 characteristic)
C1, C2, C2 [×7], C4 [×4], C4 [×4], C22 [×3], C22 [×7], C5, C8 [×4], C8 [×4], C2×C4 [×6], C2×C4 [×10], D4 [×12], Q8 [×4], C23, C23 [×2], D5 [×4], C10, C10 [×3], C2×C8 [×2], C2×C8 [×10], M4(2) [×4], M4(2) [×12], C22×C4, C22×C4 [×2], C2×D4 [×3], C2×Q8, C4○D4 [×8], Dic5 [×4], C20 [×4], D10 [×4], D10 [×2], C2×C10 [×3], C2×C10, C2×M4(2), C2×M4(2) [×5], C8○D4 [×8], C2×C4○D4, C52C8 [×4], C40 [×4], Dic10 [×4], C4×D5 [×8], D20 [×4], C2×Dic5 [×2], C5⋊D4 [×8], C2×C20 [×6], C22×D5 [×2], C22×C10, Q8○M4(2), C8×D5 [×8], C8⋊D5 [×8], C2×C52C8 [×2], C4.Dic5 [×4], C2×C40 [×2], C5×M4(2) [×4], C2×Dic10, C2×C4×D5 [×2], C2×D20, C4○D20 [×8], C2×C5⋊D4 [×2], C22×C20, D20.3C4 [×4], D5×M4(2) [×4], D20.2C4 [×4], C2×C4.Dic5, C10×M4(2), C2×C4○D20, C40.47C23
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, C40.47C23

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

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

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

74 conjugacy classes

class 1 2A2B2C2D2E2F2G2H4A4B4C4D4E4F4G4H4I5A5B8A···8H8I···8P10A···10F10G10H10I10J20A···20H20I20J20K20L40A···40P
order122222222444444444558···88···810···101010101020···202020202040···40
size11222101010101122210101010222···210···102···244442···244444···4

74 irreducible representations

dim1111111111122222244
type+++++++++++
imageC1C2C2C2C2C2C2C4C4C4C4D5D10D10D10C4×D5C4×D5Q8○M4(2)C40.47C23
kernelC40.47C23D20.3C4D5×M4(2)D20.2C4C2×C4.Dic5C10×M4(2)C2×C4○D20C2×Dic10C2×D20C4○D20C2×C5⋊D4C2×M4(2)C2×C8M4(2)C22×C4C2×C4C23C5C1
# reps14441112284248212428

Matrix representation of C40.47C23 in GL4(𝔽41) generated by

193200
9000
001932
0090
,
1000
344000
0010
003440
,
1000
0100
00400
00040
,
174000
12400
001740
00124
,
0010
0001
9000
0900
G:=sub<GL(4,GF(41))| [19,9,0,0,32,0,0,0,0,0,19,9,0,0,32,0],[1,34,0,0,0,40,0,0,0,0,1,34,0,0,0,40],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[17,1,0,0,40,24,0,0,0,0,17,1,0,0,40,24],[0,0,9,0,0,0,0,9,1,0,0,0,0,1,0,0] >;

C40.47C23 in GAP, Magma, Sage, TeX

C_{40}._{47}C_2^3
% in TeX

G:=Group("C40.47C2^3");
// GroupNames label

G:=SmallGroup(320,1417);
// by ID

G=gap.SmallGroup(320,1417);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,570,80,102,12550]);
// Polycyclic

G:=Group<a,b,c,d,e|a^20=b^2=c^2=d^2=1,e^2=a^5,b*a*b=a^9,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d=a^10*b,b*e=e*b,c*d=d*c,e*c*e^-1=a^10*c,d*e=e*d>;
// generators/relations

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