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G = C2×Q8×F5order 320 = 26·5

Direct product of C2, Q8 and F5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C2×Q8×F5
 Chief series C1 — C5 — D5 — D10 — C2×F5 — C22×F5 — C2×C4×F5 — C2×Q8×F5
 Lower central C5 — C10 — C2×Q8×F5
 Upper central C1 — C22 — C2×Q8

Generators and relations for C2×Q8×F5
G = < a,b,c,d,e | a2=b4=d5=e4=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d3 >

Subgroups: 890 in 298 conjugacy classes, 156 normal (16 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×6], C4 [×16], C22, C22 [×6], C5, C2×C4 [×3], C2×C4 [×33], Q8 [×4], Q8 [×12], C23, D5 [×2], D5 [×2], C10, C10 [×2], C42 [×12], C4⋊C4 [×12], C22×C4 [×7], C2×Q8, C2×Q8 [×11], Dic5 [×6], C20 [×6], F5 [×4], F5 [×6], D10 [×2], D10 [×4], C2×C10, C2×C42 [×3], C2×C4⋊C4 [×3], C4×Q8 [×8], C22×Q8, Dic10 [×12], C4×D5 [×12], C2×Dic5 [×3], C2×C20 [×3], C5×Q8 [×4], C2×F5 [×12], C2×F5 [×6], C22×D5, C2×C4×Q8, C4×F5 [×12], C4⋊F5 [×12], C2×Dic10 [×3], C2×C4×D5 [×3], Q8×D5 [×8], Q8×C10, C22×F5, C22×F5 [×3], C2×C4×F5 [×3], C2×C4⋊F5 [×3], Q8×F5 [×8], C2×Q8×D5, C2×Q8×F5
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], Q8 [×4], C23 [×15], C22×C4 [×14], C2×Q8 [×6], C4○D4 [×2], C24, F5, C4×Q8 [×4], C23×C4, C22×Q8, C2×C4○D4, C2×F5 [×7], C2×C4×Q8, C22×F5 [×7], Q8×F5 [×2], C23×F5, C2×Q8×F5

Smallest permutation representation of C2×Q8×F5
On 80 points
Generators in S80
(1 41)(2 42)(3 43)(4 44)(5 45)(6 46)(7 47)(8 48)(9 49)(10 50)(11 51)(12 52)(13 53)(14 54)(15 55)(16 56)(17 57)(18 58)(19 59)(20 60)(21 61)(22 62)(23 63)(24 64)(25 65)(26 66)(27 67)(28 68)(29 69)(30 70)(31 71)(32 72)(33 73)(34 74)(35 75)(36 76)(37 77)(38 78)(39 79)(40 80)
(1 16 6 11)(2 17 7 12)(3 18 8 13)(4 19 9 14)(5 20 10 15)(21 31 26 36)(22 32 27 37)(23 33 28 38)(24 34 29 39)(25 35 30 40)(41 56 46 51)(42 57 47 52)(43 58 48 53)(44 59 49 54)(45 60 50 55)(61 71 66 76)(62 72 67 77)(63 73 68 78)(64 74 69 79)(65 75 70 80)
(1 66 6 61)(2 67 7 62)(3 68 8 63)(4 69 9 64)(5 70 10 65)(11 76 16 71)(12 77 17 72)(13 78 18 73)(14 79 19 74)(15 80 20 75)(21 41 26 46)(22 42 27 47)(23 43 28 48)(24 44 29 49)(25 45 30 50)(31 51 36 56)(32 52 37 57)(33 53 38 58)(34 54 39 59)(35 55 40 60)
(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)
(1 36 6 31)(2 38 10 34)(3 40 9 32)(4 37 8 35)(5 39 7 33)(11 26 16 21)(12 28 20 24)(13 30 19 22)(14 27 18 25)(15 29 17 23)(41 76 46 71)(42 78 50 74)(43 80 49 72)(44 77 48 75)(45 79 47 73)(51 66 56 61)(52 68 60 64)(53 70 59 62)(54 67 58 65)(55 69 57 63)

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

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

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

50 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A ··· 4F 4G ··· 4N 4O ··· 4AF 5 10A 10B 10C 20A ··· 20F order 1 2 2 2 2 2 2 2 4 ··· 4 4 ··· 4 4 ··· 4 5 10 10 10 20 ··· 20 size 1 1 1 1 5 5 5 5 2 ··· 2 5 ··· 5 10 ··· 10 4 4 4 4 8 ··· 8

50 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 4 4 4 8 type + + + + + - + + + - image C1 C2 C2 C2 C2 C4 C4 C4 Q8 C4○D4 F5 C2×F5 C2×F5 Q8×F5 kernel C2×Q8×F5 C2×C4×F5 C2×C4⋊F5 Q8×F5 C2×Q8×D5 C2×Dic10 Q8×D5 Q8×C10 C2×F5 D10 C2×Q8 C2×C4 Q8 C2 # reps 1 3 3 8 1 6 8 2 4 4 1 3 4 2

Matrix representation of C2×Q8×F5 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 40 2 0 0 0 0 40 1 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 9 23 0 0 0 0 0 32 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
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 40 0 0 1 0 0 40 0 0 0 1 0 40 0 0 0 0 1 40
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,40,0,0,0,0,2,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[9,0,0,0,0,0,23,32,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,40,40,40,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,1,0] >;

C2×Q8×F5 in GAP, Magma, Sage, TeX

C_2\times Q_8\times F_5
% in TeX

G:=Group("C2xQ8xF5");
// GroupNames label

G:=SmallGroup(320,1599);
// by ID

G=gap.SmallGroup(320,1599);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,184,297,136,6278,818]);
// Polycyclic

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

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