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## G = C5×Q8○M4(2)  order 320 = 26·5

### Direct product of C5 and Q8○M4(2)

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C5×Q8○M4(2)
 Chief series C1 — C2 — C4 — C20 — C40 — C2×C40 — C5×C8○D4 — C5×Q8○M4(2)
 Lower central C1 — C2 — C5×Q8○M4(2)
 Upper central C1 — C20 — C5×Q8○M4(2)

Generators and relations for C5×Q8○M4(2)
G = < a,b,c,d,e | a5=b4=e2=1, c2=d4=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=b2d >

Subgroups: 290 in 258 conjugacy classes, 238 normal (18 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, C23, C10, C10, C2×C8, M4(2), C22×C4, C2×D4, C2×Q8, C4○D4, C20, C20, C2×C10, C2×C10, C2×C10, C2×M4(2), C8○D4, C2×C4○D4, C40, C2×C20, C2×C20, C5×D4, C5×Q8, C22×C10, Q8○M4(2), C2×C40, C5×M4(2), C22×C20, D4×C10, Q8×C10, C5×C4○D4, C10×M4(2), C5×C8○D4, C10×C4○D4, C5×Q8○M4(2)
Quotients: C1, C2, C4, C22, C5, C2×C4, C23, C10, C22×C4, C24, C20, C2×C10, C23×C4, C2×C20, C22×C10, Q8○M4(2), C22×C20, C23×C10, C23×C20, C5×Q8○M4(2)

Smallest permutation representation of C5×Q8○M4(2)
On 80 points
Generators in S80
(1 58 17 75 25)(2 59 18 76 26)(3 60 19 77 27)(4 61 20 78 28)(5 62 21 79 29)(6 63 22 80 30)(7 64 23 73 31)(8 57 24 74 32)(9 44 52 69 40)(10 45 53 70 33)(11 46 54 71 34)(12 47 55 72 35)(13 48 56 65 36)(14 41 49 66 37)(15 42 50 67 38)(16 43 51 68 39)
(1 44 5 48)(2 45 6 41)(3 46 7 42)(4 47 8 43)(9 29 13 25)(10 30 14 26)(11 31 15 27)(12 32 16 28)(17 69 21 65)(18 70 22 66)(19 71 23 67)(20 72 24 68)(33 80 37 76)(34 73 38 77)(35 74 39 78)(36 75 40 79)(49 59 53 63)(50 60 54 64)(51 61 55 57)(52 62 56 58)
(1 3 5 7)(2 4 6 8)(9 15 13 11)(10 16 14 12)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 32)(33 39 37 35)(34 40 38 36)(41 47 45 43)(42 48 46 44)(49 55 53 51)(50 56 54 52)(57 59 61 63)(58 60 62 64)(65 71 69 67)(66 72 70 68)(73 75 77 79)(74 76 78 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)
(2 6)(4 8)(10 14)(12 16)(18 22)(20 24)(26 30)(28 32)(33 37)(35 39)(41 45)(43 47)(49 53)(51 55)(57 61)(59 63)(66 70)(68 72)(74 78)(76 80)

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

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

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

170 conjugacy classes

 class 1 2A 2B ··· 2H 4A 4B 4C ··· 4I 5A 5B 5C 5D 8A ··· 8P 10A 10B 10C 10D 10E ··· 10AF 20A ··· 20H 20I ··· 20AJ 40A ··· 40BL order 1 2 2 ··· 2 4 4 4 ··· 4 5 5 5 5 8 ··· 8 10 10 10 10 10 ··· 10 20 ··· 20 20 ··· 20 40 ··· 40 size 1 1 2 ··· 2 1 1 2 ··· 2 1 1 1 1 2 ··· 2 1 1 1 1 2 ··· 2 1 ··· 1 2 ··· 2 2 ··· 2

170 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4 4 type + + + + image C1 C2 C2 C2 C4 C4 C4 C5 C10 C10 C10 C20 C20 C20 Q8○M4(2) C5×Q8○M4(2) kernel C5×Q8○M4(2) C10×M4(2) C5×C8○D4 C10×C4○D4 D4×C10 Q8×C10 C5×C4○D4 Q8○M4(2) C2×M4(2) C8○D4 C2×C4○D4 C2×D4 C2×Q8 C4○D4 C5 C1 # reps 1 6 8 1 6 2 8 4 24 32 4 24 8 32 2 8

Matrix representation of C5×Q8○M4(2) in GL4(𝔽41) generated by

 18 0 0 0 0 18 0 0 0 0 18 0 0 0 0 18
,
 40 18 0 0 9 1 0 0 25 37 0 32 16 4 32 0
,
 32 0 0 0 40 9 0 0 4 0 9 0 0 0 0 32
,
 5 0 0 39 21 0 1 9 29 32 0 5 17 0 0 36
,
 1 0 0 0 0 1 0 0 36 0 40 0 5 0 0 40
G:=sub<GL(4,GF(41))| [18,0,0,0,0,18,0,0,0,0,18,0,0,0,0,18],[40,9,25,16,18,1,37,4,0,0,0,32,0,0,32,0],[32,40,4,0,0,9,0,0,0,0,9,0,0,0,0,32],[5,21,29,17,0,0,32,0,0,1,0,0,39,9,5,36],[1,0,36,5,0,1,0,0,0,0,40,0,0,0,0,40] >;

C5×Q8○M4(2) in GAP, Magma, Sage, TeX

C_5\times Q_8\circ M_4(2)
% in TeX

G:=Group("C5xQ8oM4(2)");
// GroupNames label

G:=SmallGroup(320,1570);
// by ID

G=gap.SmallGroup(320,1570);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,560,1731,4707,124]);
// Polycyclic

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

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