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G = C53(C23⋊C8)  order 320 = 26·5

The semidirect product of C5 and C23⋊C8 acting via C23⋊C8/C22⋊C8=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C53(C23⋊C8), C22⋊C81D5, (C22×D5)⋊1C8, C22.3(C8×D5), (C2×C4).107D20, (C2×C20).438D4, (C23×D5).1C4, C23.40(C4×D5), (C22×C4).1D10, C20.55D420C2, C2.5(D101C8), C10.18(C22⋊C8), C10.25(C23⋊C4), C10.9(C4.D4), C22.3(C8⋊D5), (C2×C10).10M4(2), (C22×Dic5).1C4, C2.1(C20.46D4), (C22×C20).322C22, C2.1(C23.1D10), C22.32(D10⋊C4), (C5×C22⋊C8)⋊1C2, (C2×C10).16(C2×C8), (C2×C4).209(C5⋊D4), (C22×C10).94(C2×C4), (C2×D10⋊C4).23C2, (C2×C10).104(C22⋊C4), SmallGroup(320,26)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C53(C23⋊C8)
C1C5C10C2×C10C2×C20C22×C20C2×D10⋊C4 — C53(C23⋊C8)
C5C10C2×C10 — C53(C23⋊C8)
C1C22C22×C4C22⋊C8

Generators and relations for C53(C23⋊C8)
 G = < a,b,c,d,e | a5=b2=c2=d2=e8=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe-1=bcd, ece-1=cd=dc, de=ed >

Subgroups: 518 in 98 conjugacy classes, 31 normal (29 characteristic)
C1, C2 [×3], C2 [×4], C4 [×3], C22 [×3], C22 [×10], C5, C8 [×2], C2×C4 [×2], C2×C4 [×3], C23, C23 [×6], D5 [×2], C10 [×3], C10 [×2], C22⋊C4 [×2], C2×C8 [×2], C22×C4, C22×C4, C24, Dic5, C20 [×2], D10 [×8], C2×C10 [×3], C2×C10 [×2], C22⋊C8, C22⋊C8, C2×C22⋊C4, C52C8, C40, C2×Dic5 [×2], C2×C20 [×2], C2×C20, C22×D5 [×2], C22×D5 [×4], C22×C10, C23⋊C8, C2×C52C8, D10⋊C4 [×2], C2×C40, C22×Dic5, C22×C20, C23×D5, C20.55D4, C5×C22⋊C8, C2×D10⋊C4, C53(C23⋊C8)
Quotients: C1, C2 [×3], C4 [×2], C22, C8 [×2], C2×C4, D4 [×2], D5, C22⋊C4, C2×C8, M4(2), D10, C22⋊C8, C23⋊C4, C4.D4, C4×D5, D20, C5⋊D4, C23⋊C8, C8×D5, C8⋊D5, D10⋊C4, C23.1D10, D101C8, C20.46D4, C53(C23⋊C8)

Smallest permutation representation of C53(C23⋊C8)
On 80 points
Generators in S80
(1 77 26 14 48)(2 78 27 15 41)(3 79 28 16 42)(4 80 29 9 43)(5 73 30 10 44)(6 74 31 11 45)(7 75 32 12 46)(8 76 25 13 47)(17 69 59 51 40)(18 70 60 52 33)(19 71 61 53 34)(20 72 62 54 35)(21 65 63 55 36)(22 66 64 56 37)(23 67 57 49 38)(24 68 58 50 39)
(1 5)(3 24)(4 21)(7 20)(8 17)(9 63)(10 26)(11 31)(12 62)(13 59)(14 30)(15 27)(16 58)(18 22)(25 51)(28 50)(29 55)(32 54)(33 66)(34 71)(35 75)(36 80)(37 70)(38 67)(39 79)(40 76)(41 78)(42 68)(43 65)(44 77)(45 74)(46 72)(47 69)(48 73)(49 57)(52 64)(53 61)(56 60)
(1 22)(2 6)(3 24)(4 8)(5 18)(7 20)(9 13)(10 52)(11 15)(12 54)(14 56)(16 50)(17 21)(19 23)(25 29)(26 64)(27 31)(28 58)(30 60)(32 62)(33 44)(34 38)(35 46)(36 40)(37 48)(39 42)(41 45)(43 47)(49 53)(51 55)(57 61)(59 63)(65 69)(66 77)(67 71)(68 79)(70 73)(72 75)(74 78)(76 80)
(1 18)(2 19)(3 20)(4 21)(5 22)(6 23)(7 24)(8 17)(9 55)(10 56)(11 49)(12 50)(13 51)(14 52)(15 53)(16 54)(25 59)(26 60)(27 61)(28 62)(29 63)(30 64)(31 57)(32 58)(33 48)(34 41)(35 42)(36 43)(37 44)(38 45)(39 46)(40 47)(65 80)(66 73)(67 74)(68 75)(69 76)(70 77)(71 78)(72 79)
(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)| (1,77,26,14,48)(2,78,27,15,41)(3,79,28,16,42)(4,80,29,9,43)(5,73,30,10,44)(6,74,31,11,45)(7,75,32,12,46)(8,76,25,13,47)(17,69,59,51,40)(18,70,60,52,33)(19,71,61,53,34)(20,72,62,54,35)(21,65,63,55,36)(22,66,64,56,37)(23,67,57,49,38)(24,68,58,50,39), (1,5)(3,24)(4,21)(7,20)(8,17)(9,63)(10,26)(11,31)(12,62)(13,59)(14,30)(15,27)(16,58)(18,22)(25,51)(28,50)(29,55)(32,54)(33,66)(34,71)(35,75)(36,80)(37,70)(38,67)(39,79)(40,76)(41,78)(42,68)(43,65)(44,77)(45,74)(46,72)(47,69)(48,73)(49,57)(52,64)(53,61)(56,60), (1,22)(2,6)(3,24)(4,8)(5,18)(7,20)(9,13)(10,52)(11,15)(12,54)(14,56)(16,50)(17,21)(19,23)(25,29)(26,64)(27,31)(28,58)(30,60)(32,62)(33,44)(34,38)(35,46)(36,40)(37,48)(39,42)(41,45)(43,47)(49,53)(51,55)(57,61)(59,63)(65,69)(66,77)(67,71)(68,79)(70,73)(72,75)(74,78)(76,80), (1,18)(2,19)(3,20)(4,21)(5,22)(6,23)(7,24)(8,17)(9,55)(10,56)(11,49)(12,50)(13,51)(14,52)(15,53)(16,54)(25,59)(26,60)(27,61)(28,62)(29,63)(30,64)(31,57)(32,58)(33,48)(34,41)(35,42)(36,43)(37,44)(38,45)(39,46)(40,47)(65,80)(66,73)(67,74)(68,75)(69,76)(70,77)(71,78)(72,79), (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( (1,77,26,14,48)(2,78,27,15,41)(3,79,28,16,42)(4,80,29,9,43)(5,73,30,10,44)(6,74,31,11,45)(7,75,32,12,46)(8,76,25,13,47)(17,69,59,51,40)(18,70,60,52,33)(19,71,61,53,34)(20,72,62,54,35)(21,65,63,55,36)(22,66,64,56,37)(23,67,57,49,38)(24,68,58,50,39), (1,5)(3,24)(4,21)(7,20)(8,17)(9,63)(10,26)(11,31)(12,62)(13,59)(14,30)(15,27)(16,58)(18,22)(25,51)(28,50)(29,55)(32,54)(33,66)(34,71)(35,75)(36,80)(37,70)(38,67)(39,79)(40,76)(41,78)(42,68)(43,65)(44,77)(45,74)(46,72)(47,69)(48,73)(49,57)(52,64)(53,61)(56,60), (1,22)(2,6)(3,24)(4,8)(5,18)(7,20)(9,13)(10,52)(11,15)(12,54)(14,56)(16,50)(17,21)(19,23)(25,29)(26,64)(27,31)(28,58)(30,60)(32,62)(33,44)(34,38)(35,46)(36,40)(37,48)(39,42)(41,45)(43,47)(49,53)(51,55)(57,61)(59,63)(65,69)(66,77)(67,71)(68,79)(70,73)(72,75)(74,78)(76,80), (1,18)(2,19)(3,20)(4,21)(5,22)(6,23)(7,24)(8,17)(9,55)(10,56)(11,49)(12,50)(13,51)(14,52)(15,53)(16,54)(25,59)(26,60)(27,61)(28,62)(29,63)(30,64)(31,57)(32,58)(33,48)(34,41)(35,42)(36,43)(37,44)(38,45)(39,46)(40,47)(65,80)(66,73)(67,74)(68,75)(69,76)(70,77)(71,78)(72,79), (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([(1,77,26,14,48),(2,78,27,15,41),(3,79,28,16,42),(4,80,29,9,43),(5,73,30,10,44),(6,74,31,11,45),(7,75,32,12,46),(8,76,25,13,47),(17,69,59,51,40),(18,70,60,52,33),(19,71,61,53,34),(20,72,62,54,35),(21,65,63,55,36),(22,66,64,56,37),(23,67,57,49,38),(24,68,58,50,39)], [(1,5),(3,24),(4,21),(7,20),(8,17),(9,63),(10,26),(11,31),(12,62),(13,59),(14,30),(15,27),(16,58),(18,22),(25,51),(28,50),(29,55),(32,54),(33,66),(34,71),(35,75),(36,80),(37,70),(38,67),(39,79),(40,76),(41,78),(42,68),(43,65),(44,77),(45,74),(46,72),(47,69),(48,73),(49,57),(52,64),(53,61),(56,60)], [(1,22),(2,6),(3,24),(4,8),(5,18),(7,20),(9,13),(10,52),(11,15),(12,54),(14,56),(16,50),(17,21),(19,23),(25,29),(26,64),(27,31),(28,58),(30,60),(32,62),(33,44),(34,38),(35,46),(36,40),(37,48),(39,42),(41,45),(43,47),(49,53),(51,55),(57,61),(59,63),(65,69),(66,77),(67,71),(68,79),(70,73),(72,75),(74,78),(76,80)], [(1,18),(2,19),(3,20),(4,21),(5,22),(6,23),(7,24),(8,17),(9,55),(10,56),(11,49),(12,50),(13,51),(14,52),(15,53),(16,54),(25,59),(26,60),(27,61),(28,62),(29,63),(30,64),(31,57),(32,58),(33,48),(34,41),(35,42),(36,43),(37,44),(38,45),(39,46),(40,47),(65,80),(66,73),(67,74),(68,75),(69,76),(70,77),(71,78),(72,79)], [(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)])

62 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F5A5B8A8B8C8D8E8F8G8H10A···10F10G10H10I10J20A···20H20I20J20K20L40A···40P
order12222222444444558888888810···101010101020···202020202040···40
size111122202022222020224444202020202···244442···244444···4

62 irreducible representations

dim11111112222222224444
type+++++++++++
imageC1C2C2C2C4C4C8D4D5M4(2)D10D20C5⋊D4C4×D5C8×D5C8⋊D5C23⋊C4C4.D4C23.1D10C20.46D4
kernelC53(C23⋊C8)C20.55D4C5×C22⋊C8C2×D10⋊C4C22×Dic5C23×D5C22×D5C2×C20C22⋊C8C2×C10C22×C4C2×C4C2×C4C23C22C22C10C10C2C2
# reps11112282222444881144

Matrix representation of C53(C23⋊C8) in GL6(𝔽41)

17320000
9170000
000100
0040600
000001
0000406
,
100000
0400000
001000
0064000
000010
0000640
,
100000
010000
001000
000100
0000400
0000040
,
100000
010000
0040000
0004000
0000400
0000040
,
300000
030000
000010
000001
00392800
0013200

G:=sub<GL(6,GF(41))| [17,9,0,0,0,0,32,17,0,0,0,0,0,0,0,40,0,0,0,0,1,6,0,0,0,0,0,0,0,40,0,0,0,0,1,6],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,6,0,0,0,0,0,40,0,0,0,0,0,0,1,6,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[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],[3,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,39,13,0,0,0,0,28,2,0,0,1,0,0,0,0,0,0,1,0,0] >;

C53(C23⋊C8) in GAP, Magma, Sage, TeX

C_5\rtimes_3(C_2^3\rtimes C_8)
% in TeX

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

G:=SmallGroup(320,26);
// by ID

G=gap.SmallGroup(320,26);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,141,36,758,100,570,12550]);
// 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,a*e=e*a,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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