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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

Aliases: C5×Q8○M4(2), C40.81C23, C20.94C24, C8○D48C10, C4○D4.2C20, D4.9(C2×C20), (C2×Q8).8C20, (C2×C40)⋊39C22, (C2×D4).10C20, (D4×C10).35C4, Q8.10(C2×C20), (Q8×C10).28C4, C4.23(C22×C20), C2.12(C23×C20), C23.13(C2×C20), C4.18(C23×C10), C10.85(C23×C4), C8.14(C22×C10), (C2×M4(2))⋊16C10, (C10×M4(2))⋊34C2, M4(2)⋊12(C2×C10), C20.227(C22×C4), (C2×C20).970C23, C22.5(C22×C20), (C5×M4(2))⋊41C22, (C22×C20).464C22, (C2×C8)⋊9(C2×C10), (C5×C8○D4)⋊17C2, (C2×C4).32(C2×C20), (C5×C4○D4).10C4, (C5×D4).45(C2×C4), (C5×Q8).49(C2×C4), (C2×C20).377(C2×C4), (C2×C4○D4).11C10, C4○D4.15(C2×C10), (C10×C4○D4).25C2, (C22×C10).93(C2×C4), (C22×C4).75(C2×C10), (C5×C4○D4).60C22, (C2×C4).140(C22×C10), (C2×C10).137(C22×C4), SmallGroup(320,1570)

Series: Derived Chief Lower central Upper central

C1C2 — C5×Q8○M4(2)
C1C2C4C20C40C2×C40C5×C8○D4 — C5×Q8○M4(2)
C1C2 — C5×Q8○M4(2)
C1C20 — 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 [×7], C4 [×2], C4 [×6], C22, C22 [×6], C22 [×3], C5, C8 [×8], C2×C4, C2×C4 [×15], D4 [×12], Q8 [×4], C23 [×3], C10, C10 [×7], C2×C8 [×12], M4(2) [×16], C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4 [×8], C20 [×2], C20 [×6], C2×C10, C2×C10 [×6], C2×C10 [×3], C2×M4(2) [×6], C8○D4 [×8], C2×C4○D4, C40 [×8], C2×C20, C2×C20 [×15], C5×D4 [×12], C5×Q8 [×4], C22×C10 [×3], Q8○M4(2), C2×C40 [×12], C5×M4(2) [×16], C22×C20 [×3], D4×C10 [×3], Q8×C10, C5×C4○D4 [×8], C10×M4(2) [×6], C5×C8○D4 [×8], C10×C4○D4, C5×Q8○M4(2)
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C5, C2×C4 [×28], C23 [×15], C10 [×15], C22×C4 [×14], C24, C20 [×8], C2×C10 [×35], C23×C4, C2×C20 [×28], C22×C10 [×15], Q8○M4(2), C22×C20 [×14], 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 2A2B···2H4A4B4C···4I5A5B5C5D8A···8P10A10B10C10D10E···10AF20A···20H20I···20AJ40A···40BL
order122···2444···455558···81010101010···1020···2020···2040···40
size112···2112···211112···211112···21···12···22···2

170 irreducible representations

dim1111111111111144
type++++
imageC1C2C2C2C4C4C4C5C10C10C10C20C20C20Q8○M4(2)C5×Q8○M4(2)
kernelC5×Q8○M4(2)C10×M4(2)C5×C8○D4C10×C4○D4D4×C10Q8×C10C5×C4○D4Q8○M4(2)C2×M4(2)C8○D4C2×C4○D4C2×D4C2×Q8C4○D4C5C1
# reps16816284243242483228

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

18000
01800
00180
00018
,
401800
9100
2537032
164320
,
32000
40900
4090
00032
,
50039
21019
293205
170036
,
1000
0100
360400
50040
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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