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G = C5⋊(C23⋊C8)  order 320 = 26·5

The semidirect product of C5 and C23⋊C8 acting via C23⋊C8/C22×C4=C4

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C51(C23⋊C8), (C22×D5)⋊2C8, (C22×C4).2F5, (C22×C20).8C4, (C23×D5).4C4, C23.33(C2×F5), C10.7(C22⋊C8), C22.4(D5⋊C8), (C2×C10).6M4(2), C23.2F51C2, C2.8(D10⋊C8), C10.10(C23⋊C4), C2.1(C23.F5), C22.5(C4.F5), (C2×Dic5).103D4, C10.1(C4.D4), C2.2(D10.D4), C22.37(C22⋊F5), (C22×Dic5).173C22, (C2×C10).9(C2×C8), (C2×D10⋊C4).1C2, (C22×C10).44(C2×C4), (C2×C10).28(C22⋊C4), SmallGroup(320,253)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C5⋊(C23⋊C8)
C1C5C10C2×C10C2×Dic5C22×Dic5C23.2F5 — C5⋊(C23⋊C8)
C5C10C2×C10 — C5⋊(C23⋊C8)
C1C22C23C22×C4

Generators and relations for C5⋊(C23⋊C8)
 G = < a,b,c,d,e | a5=b2=c2=d2=e8=1, bab=a-1, ac=ca, ad=da, eae-1=a3, bc=cb, bd=db, ebe-1=bcd, ece-1=cd=dc, de=ed >

Subgroups: 546 in 98 conjugacy classes, 28 normal (22 characteristic)
C1, C2 [×3], C2 [×4], C4 [×3], C22 [×3], C22 [×10], C5, C8 [×2], C2×C4 [×5], C23, C23 [×6], D5 [×2], C10 [×3], C10 [×2], C22⋊C4 [×2], C2×C8 [×2], C22×C4, C22×C4, C24, Dic5 [×2], C20, D10 [×8], C2×C10 [×3], C2×C10 [×2], C22⋊C8 [×2], C2×C22⋊C4, C5⋊C8 [×2], C2×Dic5 [×2], C2×Dic5, C2×C20 [×2], C22×D5 [×2], C22×D5 [×4], C22×C10, C23⋊C8, D10⋊C4 [×2], C2×C5⋊C8 [×2], C22×Dic5, C22×C20, C23×D5, C23.2F5 [×2], C2×D10⋊C4, C5⋊(C23⋊C8)
Quotients: C1, C2 [×3], C4 [×2], C22, C8 [×2], C2×C4, D4 [×2], C22⋊C4, C2×C8, M4(2), F5, C22⋊C8, C23⋊C4, C4.D4, C2×F5, C23⋊C8, D5⋊C8, C4.F5, C22⋊F5, D10.D4, D10⋊C8, C23.F5, C5⋊(C23⋊C8)

Smallest permutation representation of C5⋊(C23⋊C8)
On 80 points
Generators in S80
(1 75 51 44 15)(2 45 76 16 52)(3 9 46 53 77)(4 54 10 78 47)(5 79 55 48 11)(6 41 80 12 56)(7 13 42 49 73)(8 50 14 74 43)(17 59 67 40 27)(18 33 60 28 68)(19 29 34 69 61)(20 70 30 62 35)(21 63 71 36 31)(22 37 64 32 72)(23 25 38 65 57)(24 66 26 58 39)
(1 20)(2 6)(4 19)(5 24)(8 23)(9 77)(10 69)(11 66)(12 76)(13 73)(14 65)(15 70)(16 80)(17 21)(25 43)(26 48)(27 63)(28 60)(29 47)(30 44)(31 59)(32 64)(33 68)(34 78)(35 75)(36 67)(37 72)(38 74)(39 79)(40 71)(41 52)(42 49)(45 56)(46 53)(50 57)(51 62)(54 61)(55 58)
(1 5)(2 17)(3 7)(4 19)(6 21)(8 23)(9 13)(10 34)(11 15)(12 36)(14 38)(16 40)(18 22)(20 24)(25 50)(26 30)(27 52)(28 32)(29 54)(31 56)(33 37)(35 39)(41 63)(42 46)(43 57)(44 48)(45 59)(47 61)(49 53)(51 55)(58 62)(60 64)(65 74)(66 70)(67 76)(68 72)(69 78)(71 80)(73 77)(75 79)
(1 20)(2 21)(3 22)(4 23)(5 24)(6 17)(7 18)(8 19)(9 37)(10 38)(11 39)(12 40)(13 33)(14 34)(15 35)(16 36)(25 54)(26 55)(27 56)(28 49)(29 50)(30 51)(31 52)(32 53)(41 59)(42 60)(43 61)(44 62)(45 63)(46 64)(47 57)(48 58)(65 78)(66 79)(67 80)(68 73)(69 74)(70 75)(71 76)(72 77)
(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)

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

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

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

38 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F 5 8A···8H10A···10G20A···20H
order1222222244444458···810···1020···20
size11112220204410101010420···204···44···4

38 irreducible representations

dim11111122444444444
type++++++++++
imageC1C2C2C4C4C8D4M4(2)F5C23⋊C4C4.D4C2×F5D5⋊C8C4.F5C22⋊F5D10.D4C23.F5
kernelC5⋊(C23⋊C8)C23.2F5C2×D10⋊C4C22×C20C23×D5C22×D5C2×Dic5C2×C10C22×C4C10C10C23C22C22C22C2C2
# reps12122822111122244

Matrix representation of C5⋊(C23⋊C8) in GL6(𝔽41)

100000
010000
000100
0040600
0000406
00003535
,
100000
0400000
0040000
0035100
000010
0000640
,
4000000
0400000
0040000
0004000
000010
000001
,
100000
010000
0040000
0004000
0000400
0000040
,
010000
900000
000010
000001
00183500
00202300

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,1,6,0,0,0,0,0,0,40,35,0,0,0,0,6,35],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,35,0,0,0,0,0,1,0,0,0,0,0,0,1,6,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[0,9,0,0,0,0,1,0,0,0,0,0,0,0,0,0,18,20,0,0,0,0,35,23,0,0,1,0,0,0,0,0,0,1,0,0] >;

C5⋊(C23⋊C8) in GAP, Magma, Sage, TeX

C_5\rtimes (C_2^3\rtimes C_8)
% in TeX

G:=Group("C5:(C2^3:C8)");
// GroupNames label

G:=SmallGroup(320,253);
// by ID

G=gap.SmallGroup(320,253);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,120,387,268,1123,6278,3156]);
// Polycyclic

G:=Group<a,b,c,d,e|a^5=b^2=c^2=d^2=e^8=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,e*a*e^-1=a^3,b*c=c*b,b*d=d*b,e*b*e^-1=b*c*d,e*c*e^-1=c*d=d*c,d*e=e*d>;
// generators/relations

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