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## G = C5×C23.C8order 320 = 26·5

### Direct product of C5 and C23.C8

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C5×C23.C8
 Chief series C1 — C2 — C4 — C8 — C2×C8 — C2×C40 — C5×M5(2) — C5×C23.C8
 Lower central C1 — C2 — C22 — C5×C23.C8
 Upper central C1 — C20 — C2×C40 — C5×C23.C8

Generators and relations for C5×C23.C8
G = < a,b,c,d,e | a5=b2=c2=d2=1, e8=d, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe-1=bcd, ece-1=cd=dc, de=ed >

Subgroups: 90 in 58 conjugacy classes, 34 normal (26 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, C23, C10, C10, C16, C2×C8, M4(2), C22×C4, C20, C20, C2×C10, C2×C10, M5(2), C2×M4(2), C40, C40, C2×C20, C2×C20, C22×C10, C23.C8, C80, C2×C40, C5×M4(2), C22×C20, C5×M5(2), C10×M4(2), C5×C23.C8
Quotients: C1, C2, C4, C22, C5, C8, C2×C4, D4, C10, C22⋊C4, C2×C8, M4(2), C20, C2×C10, C22⋊C8, C40, C2×C20, C5×D4, C23.C8, C5×C22⋊C4, C2×C40, C5×M4(2), C5×C22⋊C8, C5×C23.C8

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

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

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

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

110 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 5A 5B 5C 5D 8A 8B 8C 8D 8E 8F 10A 10B 10C 10D 10E 10F 10G 10H 10I 10J 10K 10L 16A ··· 16H 20A ··· 20H 20I 20J 20K 20L 20M 20N 20O 20P 40A ··· 40P 40Q ··· 40X 80A ··· 80AF order 1 2 2 2 4 4 4 4 5 5 5 5 8 8 8 8 8 8 10 10 10 10 10 10 10 10 10 10 10 10 16 ··· 16 20 ··· 20 20 20 20 20 20 20 20 20 40 ··· 40 40 ··· 40 80 ··· 80 size 1 1 2 4 1 1 2 4 1 1 1 1 2 2 2 2 4 4 1 1 1 1 2 2 2 2 4 4 4 4 4 ··· 4 1 ··· 1 2 2 2 2 4 4 4 4 2 ··· 2 4 ··· 4 4 ··· 4

110 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 4 4 type + + + + image C1 C2 C2 C4 C4 C5 C8 C8 C10 C10 C20 C20 C40 C40 D4 M4(2) C5×D4 C5×M4(2) C23.C8 C5×C23.C8 kernel C5×C23.C8 C5×M5(2) C10×M4(2) C2×C40 C22×C20 C23.C8 C2×C20 C22×C10 M5(2) C2×M4(2) C2×C8 C22×C4 C2×C4 C23 C40 C20 C8 C4 C5 C1 # reps 1 2 1 2 2 4 4 4 8 4 8 8 16 16 2 2 8 8 2 8

Matrix representation of C5×C23.C8 in GL6(𝔽241)

 98 0 0 0 0 0 0 98 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 1
,
 0 240 0 0 0 0 240 0 0 0 0 0 0 0 1 0 0 0 0 0 0 240 0 0 0 0 0 0 1 0 0 0 0 0 0 240
,
 240 0 0 0 0 0 0 240 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 240 0 0 0 0 0 0 240
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 240 0 0 0 0 0 0 240 0 0 0 0 0 0 240 0 0 0 0 0 0 240
,
 0 240 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 177 0 0 0

G:=sub<GL(6,GF(241))| [98,0,0,0,0,0,0,98,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,240,0,0,0,0,240,0,0,0,0,0,0,0,1,0,0,0,0,0,0,240,0,0,0,0,0,0,1,0,0,0,0,0,0,240],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[0,1,0,0,0,0,240,0,0,0,0,0,0,0,0,0,0,177,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0] >;

C5×C23.C8 in GAP, Magma, Sage, TeX

C_5\times C_2^3.C_8
% in TeX

G:=Group("C5xC2^3.C8");
// GroupNames label

G:=SmallGroup(320,154);
// by ID

G=gap.SmallGroup(320,154);
# by ID

G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-2,280,309,5043,3511,102,124]);
// Polycyclic

G:=Group<a,b,c,d,e|a^5=b^2=c^2=d^2=1,e^8=d,a*b=b*a,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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