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G = C5×C8.C22order 160 = 25·5

Direct product of C5 and C8.C22

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C5×C8.C22
 Chief series C1 — C2 — C4 — C20 — C5×D4 — C5×SD16 — C5×C8.C22
 Lower central C1 — C2 — C4 — C5×C8.C22
 Upper central C1 — C10 — C2×C20 — C5×C8.C22

Generators and relations for C5×C8.C22
G = < a,b,c,d | a5=b8=c2=d2=1, ab=ba, ac=ca, ad=da, cbc=b3, dbd=b5, dcd=b4c >

Subgroups: 84 in 60 conjugacy classes, 40 normal (24 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, Q8, C10, C10, M4(2), SD16, Q16, C2×Q8, C4○D4, C20, C20, C2×C10, C2×C10, C8.C22, C40, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C5×Q8, C5×Q8, C5×M4(2), C5×SD16, C5×Q16, Q8×C10, C5×C4○D4, C5×C8.C22
Quotients: C1, C2, C22, C5, D4, C23, C10, C2×D4, C2×C10, C8.C22, C5×D4, C22×C10, D4×C10, C5×C8.C22

Smallest permutation representation of C5×C8.C22
On 80 points
Generators in S80
(1 22 69 49 26)(2 23 70 50 27)(3 24 71 51 28)(4 17 72 52 29)(5 18 65 53 30)(6 19 66 54 31)(7 20 67 55 32)(8 21 68 56 25)(9 78 58 35 48)(10 79 59 36 41)(11 80 60 37 42)(12 73 61 38 43)(13 74 62 39 44)(14 75 63 40 45)(15 76 64 33 46)(16 77 57 34 47)
(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 4)(3 7)(6 8)(9 15)(11 13)(12 16)(17 23)(19 21)(20 24)(25 31)(27 29)(28 32)(33 35)(34 38)(37 39)(42 44)(43 47)(46 48)(50 52)(51 55)(54 56)(57 61)(58 64)(60 62)(66 68)(67 71)(70 72)(73 77)(74 80)(76 78)
(1 38)(2 35)(3 40)(4 37)(5 34)(6 39)(7 36)(8 33)(9 70)(10 67)(11 72)(12 69)(13 66)(14 71)(15 68)(16 65)(17 42)(18 47)(19 44)(20 41)(21 46)(22 43)(23 48)(24 45)(25 64)(26 61)(27 58)(28 63)(29 60)(30 57)(31 62)(32 59)(49 73)(50 78)(51 75)(52 80)(53 77)(54 74)(55 79)(56 76)

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

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

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

C5×C8.C22 is a maximal subgroup of   D20.39D4  M4(2).15D10  M4(2).16D10  D20.40D4  D40⋊C22  C40.C23  D20.44D4

55 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 5A 5B 5C 5D 8A 8B 10A 10B 10C 10D 10E 10F 10G 10H 10I 10J 10K 10L 20A ··· 20H 20I ··· 20T 40A ··· 40H order 1 2 2 2 4 4 4 4 4 5 5 5 5 8 8 10 10 10 10 10 10 10 10 10 10 10 10 20 ··· 20 20 ··· 20 40 ··· 40 size 1 1 2 4 2 2 4 4 4 1 1 1 1 4 4 1 1 1 1 2 2 2 2 4 4 4 4 2 ··· 2 4 ··· 4 4 ··· 4

55 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 4 4 type + + + + + + + + - image C1 C2 C2 C2 C2 C2 C5 C10 C10 C10 C10 C10 D4 D4 C5×D4 C5×D4 C8.C22 C5×C8.C22 kernel C5×C8.C22 C5×M4(2) C5×SD16 C5×Q16 Q8×C10 C5×C4○D4 C8.C22 M4(2) SD16 Q16 C2×Q8 C4○D4 C20 C2×C10 C4 C22 C5 C1 # reps 1 1 2 2 1 1 4 4 8 8 4 4 1 1 4 4 1 4

Matrix representation of C5×C8.C22 in GL4(𝔽41) generated by

 10 0 0 0 0 10 0 0 0 0 10 0 0 0 0 10
,
 16 25 16 16 16 16 25 16 25 25 25 16 16 25 25 25
,
 1 0 0 0 0 40 0 0 0 0 40 0 0 0 0 1
,
 0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0
G:=sub<GL(4,GF(41))| [10,0,0,0,0,10,0,0,0,0,10,0,0,0,0,10],[16,16,25,16,25,16,25,25,16,25,25,25,16,16,16,25],[1,0,0,0,0,40,0,0,0,0,40,0,0,0,0,1],[0,0,1,0,0,0,0,1,1,0,0,0,0,1,0,0] >;

C5×C8.C22 in GAP, Magma, Sage, TeX

C_5\times C_8.C_2^2
% in TeX

G:=Group("C5xC8.C2^2");
// GroupNames label

G:=SmallGroup(160,198);
// by ID

G=gap.SmallGroup(160,198);
# by ID

G:=PCGroup([6,-2,-2,-2,-5,-2,-2,505,487,1514,3604,1810,88]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^8=c^2=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^3,d*b*d=b^5,d*c*d=b^4*c>;
// generators/relations

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