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## G = C5×M4(2).8C22order 320 = 26·5

### Direct product of C5 and M4(2).8C22

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C5×M4(2).8C22
 Chief series C1 — C2 — C4 — C2×C4 — C2×C20 — C5×M4(2) — C5×C4.D4 — C5×M4(2).8C22
 Lower central C1 — C2 — C22 — C5×M4(2).8C22
 Upper central C1 — C20 — C22×C20 — C5×M4(2).8C22

Generators and relations for C5×M4(2).8C22
G = < a,b,c,d,e | a5=b8=c2=d2=1, e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b5, dbd=bc, cd=dc, ece-1=b4c, ede-1=b4cd >

Subgroups: 242 in 150 conjugacy classes, 78 normal (26 characteristic)
C1, C2, C2 [×5], C4 [×2], C4 [×2], C4 [×2], C22, C22 [×2], C22 [×5], C5, C8 [×4], C2×C4 [×2], C2×C4 [×6], C2×C4 [×4], D4 [×6], Q8 [×2], C23, C23 [×2], C10, C10 [×5], C2×C8 [×2], M4(2) [×4], M4(2) [×2], C22×C4, C22×C4 [×2], C2×D4, C2×D4 [×2], C2×Q8, C4○D4 [×4], C20 [×2], C20 [×2], C20 [×2], C2×C10, C2×C10 [×2], C2×C10 [×5], C4.D4 [×2], C4.10D4 [×2], C2×M4(2) [×2], C2×C4○D4, C40 [×4], C2×C20 [×2], C2×C20 [×6], C2×C20 [×4], C5×D4 [×6], C5×Q8 [×2], C22×C10, C22×C10 [×2], M4(2).8C22, C2×C40 [×2], C5×M4(2) [×4], C5×M4(2) [×2], C22×C20, C22×C20 [×2], D4×C10, D4×C10 [×2], Q8×C10, C5×C4○D4 [×4], C5×C4.D4 [×2], C5×C4.10D4 [×2], C10×M4(2) [×2], C10×C4○D4, C5×M4(2).8C22
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C5, C2×C4 [×6], D4 [×4], C23, C10 [×7], C22⋊C4 [×4], C22×C4, C2×D4 [×2], C20 [×4], C2×C10 [×7], C2×C22⋊C4, C2×C20 [×6], C5×D4 [×4], C22×C10, M4(2).8C22, C5×C22⋊C4 [×4], C22×C20, D4×C10 [×2], C10×C22⋊C4, C5×M4(2).8C22

Smallest permutation representation of C5×M4(2).8C22
On 80 points
Generators in S80
(1 20 65 55 30)(2 21 66 56 31)(3 22 67 49 32)(4 23 68 50 25)(5 24 69 51 26)(6 17 70 52 27)(7 18 71 53 28)(8 19 72 54 29)(9 80 58 37 42)(10 73 59 38 43)(11 74 60 39 44)(12 75 61 40 45)(13 76 62 33 46)(14 77 63 34 47)(15 78 64 35 48)(16 79 57 36 41)
(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 6)(4 8)(9 13)(11 15)(17 21)(19 23)(25 29)(27 31)(33 37)(35 39)(42 46)(44 48)(50 54)(52 56)(58 62)(60 64)(66 70)(68 72)(74 78)(76 80)
(1 38)(2 35)(3 36)(4 33)(5 34)(6 39)(7 40)(8 37)(9 72)(10 65)(11 70)(12 71)(13 68)(14 69)(15 66)(16 67)(17 44)(18 45)(19 42)(20 43)(21 48)(22 41)(23 46)(24 47)(25 62)(26 63)(27 60)(28 61)(29 58)(30 59)(31 64)(32 57)(49 79)(50 76)(51 77)(52 74)(53 75)(54 80)(55 73)(56 78)
(1 8 3 2 5 4 7 6)(9 16 11 10 13 12 15 14)(17 20 19 22 21 24 23 18)(25 28 27 30 29 32 31 26)(33 40 35 34 37 36 39 38)(41 44 43 46 45 48 47 42)(49 56 51 50 53 52 55 54)(57 60 59 62 61 64 63 58)(65 72 67 66 69 68 71 70)(73 76 75 78 77 80 79 74)

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

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

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

110 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 4E 4F 4G 5A 5B 5C 5D 8A ··· 8H 10A 10B 10C 10D 10E ··· 10P 10Q ··· 10X 20A ··· 20H 20I ··· 20T 20U ··· 20AB 40A ··· 40AF order 1 2 2 2 2 2 2 4 4 4 4 4 4 4 5 5 5 5 8 ··· 8 10 10 10 10 10 ··· 10 10 ··· 10 20 ··· 20 20 ··· 20 20 ··· 20 40 ··· 40 size 1 1 2 2 2 4 4 1 1 2 2 2 4 4 1 1 1 1 4 ··· 4 1 1 1 1 2 ··· 2 4 ··· 4 1 ··· 1 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 4 4 type + + + + + + image C1 C2 C2 C2 C2 C4 C4 C5 C10 C10 C10 C10 C20 C20 D4 C5×D4 M4(2).8C22 C5×M4(2).8C22 kernel C5×M4(2).8C22 C5×C4.D4 C5×C4.10D4 C10×M4(2) C10×C4○D4 C22×C20 D4×C10 M4(2).8C22 C4.D4 C4.10D4 C2×M4(2) C2×C4○D4 C22×C4 C2×D4 C2×C20 C2×C4 C5 C1 # reps 1 2 2 2 1 4 4 4 8 8 8 4 16 16 4 16 2 8

Matrix representation of C5×M4(2).8C22 in GL4(𝔽41) generated by

 37 0 0 0 0 37 0 0 0 0 37 0 0 0 0 37
,
 20 6 21 1 0 0 0 9 2 11 21 3 0 1 0 0
,
 1 0 21 3 0 1 0 0 0 0 40 0 0 0 0 40
,
 11 19 32 6 39 30 20 38 0 0 0 1 0 0 1 0
,
 16 13 24 39 0 0 0 1 18 17 25 27 0 9 0 0
G:=sub<GL(4,GF(41))| [37,0,0,0,0,37,0,0,0,0,37,0,0,0,0,37],[20,0,2,0,6,0,11,1,21,0,21,0,1,9,3,0],[1,0,0,0,0,1,0,0,21,0,40,0,3,0,0,40],[11,39,0,0,19,30,0,0,32,20,0,1,6,38,1,0],[16,0,18,0,13,0,17,9,24,0,25,0,39,1,27,0] >;

C5×M4(2).8C22 in GAP, Magma, Sage, TeX

C_5\times M_4(2)._8C_2^2
% in TeX

G:=Group("C5xM4(2).8C2^2");
// GroupNames label

G:=SmallGroup(320,914);
// by ID

G=gap.SmallGroup(320,914);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,560,589,856,7004,5052,124]);
// Polycyclic

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

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