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## G = (C2×Q8)⋊F5order 320 = 26·5

### 2nd semidirect product of C2×Q8 and F5 acting via F5/C5=C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — (C2×Q8)⋊F5
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×D20 — D10.D4 — (C2×Q8)⋊F5
 Lower central C5 — C10 — C2×C10 — C2×C20 — (C2×Q8)⋊F5
 Upper central C1 — C2 — C22 — C2×C4 — C2×Q8

Generators and relations for (C2×Q8)⋊F5
G = < a,b,c,d,e | a2=b4=d5=e4=1, c2=b2, ebe-1=ab=ba, ac=ca, ad=da, eae-1=ab2, cbc-1=b-1, bd=db, cd=dc, ece-1=b-1c, ede-1=d3 >

Subgroups: 458 in 70 conjugacy classes, 18 normal (14 characteristic)
C1, C2, C2 [×3], C4 [×5], C22, C22 [×4], C5, C2×C4, C2×C4 [×4], D4, Q8, C23 [×2], D5 [×2], C10, C10, C42, C22⋊C4 [×4], C2×D4, C2×Q8, Dic5, C20 [×2], F5 [×2], D10 [×4], C2×C10, C23⋊C4 [×2], C4.4D4, D20, C2×Dic5, C2×C20, C2×C20, C5×Q8, C2×F5 [×2], C22×D5 [×2], C423C4, C4×Dic5, D10⋊C4 [×2], C22⋊F5 [×2], C2×D20, Q8×C10, D10.D4 [×2], C20.23D4, (C2×Q8)⋊F5
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], C22⋊C4, F5, C23⋊C4, C2×F5, C423C4, C22⋊F5, C23⋊F5, (C2×Q8)⋊F5

Character table of (C2×Q8)⋊F5

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 4F 4G 4H 5 10A 10B 10C 20A 20B 20C 20D 20E 20F size 1 1 2 20 20 4 8 20 20 40 40 40 40 4 4 4 4 8 8 8 8 8 8 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ3 1 1 1 1 1 1 -1 -1 -1 -1 1 1 -1 1 1 1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ4 1 1 1 1 1 1 -1 -1 -1 1 -1 -1 1 1 1 1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ5 1 1 1 -1 -1 1 -1 1 1 -i -i i i 1 1 1 1 1 1 -1 -1 -1 -1 linear of order 4 ρ6 1 1 1 -1 -1 1 -1 1 1 i i -i -i 1 1 1 1 1 1 -1 -1 -1 -1 linear of order 4 ρ7 1 1 1 -1 -1 1 1 -1 -1 i -i i -i 1 1 1 1 1 1 1 1 1 1 linear of order 4 ρ8 1 1 1 -1 -1 1 1 -1 -1 -i i -i i 1 1 1 1 1 1 1 1 1 1 linear of order 4 ρ9 2 2 2 2 -2 -2 0 0 0 0 0 0 0 2 2 2 2 -2 -2 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 -2 2 -2 0 0 0 0 0 0 0 2 2 2 2 -2 -2 0 0 0 0 orthogonal lifted from D4 ρ11 4 4 -4 0 0 0 0 0 0 0 0 0 0 4 -4 -4 4 0 0 0 0 0 0 orthogonal lifted from C23⋊C4 ρ12 4 4 4 0 0 4 -4 0 0 0 0 0 0 -1 -1 -1 -1 -1 -1 1 1 1 1 orthogonal lifted from C2×F5 ρ13 4 4 4 0 0 4 4 0 0 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from F5 ρ14 4 4 4 0 0 -4 0 0 0 0 0 0 0 -1 -1 -1 -1 1 1 -√5 √5 √5 -√5 orthogonal lifted from C22⋊F5 ρ15 4 4 4 0 0 -4 0 0 0 0 0 0 0 -1 -1 -1 -1 1 1 √5 -√5 -√5 √5 orthogonal lifted from C22⋊F5 ρ16 4 -4 0 0 0 0 0 2i -2i 0 0 0 0 4 0 0 -4 0 0 0 0 0 0 complex lifted from C42⋊3C4 ρ17 4 4 -4 0 0 0 0 0 0 0 0 0 0 -1 1 1 -1 √5 -√5 2ζ54+2ζ52+1 2ζ54+2ζ53+1 2ζ52+2ζ5+1 2ζ53+2ζ5+1 complex lifted from C23⋊F5 ρ18 4 -4 0 0 0 0 0 -2i 2i 0 0 0 0 4 0 0 -4 0 0 0 0 0 0 complex lifted from C42⋊3C4 ρ19 4 4 -4 0 0 0 0 0 0 0 0 0 0 -1 1 1 -1 √5 -√5 2ζ53+2ζ5+1 2ζ52+2ζ5+1 2ζ54+2ζ53+1 2ζ54+2ζ52+1 complex lifted from C23⋊F5 ρ20 4 4 -4 0 0 0 0 0 0 0 0 0 0 -1 1 1 -1 -√5 √5 2ζ54+2ζ53+1 2ζ53+2ζ5+1 2ζ54+2ζ52+1 2ζ52+2ζ5+1 complex lifted from C23⋊F5 ρ21 4 4 -4 0 0 0 0 0 0 0 0 0 0 -1 1 1 -1 -√5 √5 2ζ52+2ζ5+1 2ζ54+2ζ52+1 2ζ53+2ζ5+1 2ζ54+2ζ53+1 complex lifted from C23⋊F5 ρ22 8 -8 0 0 0 0 0 0 0 0 0 0 0 -2 -2√5 2√5 2 0 0 0 0 0 0 orthogonal faithful ρ23 8 -8 0 0 0 0 0 0 0 0 0 0 0 -2 2√5 -2√5 2 0 0 0 0 0 0 orthogonal faithful

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

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

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

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

Matrix representation of (C2×Q8)⋊F5 in GL8(𝔽41)

 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40
,
 35 7 0 5 0 0 0 0 7 35 5 0 0 0 0 0 25 32 6 34 0 0 0 0 32 25 34 6 0 0 0 0 0 0 0 0 22 38 0 3 0 0 0 0 0 19 38 3 0 0 0 0 3 38 19 0 0 0 0 0 3 0 38 22
,
 28 22 39 2 0 0 0 0 22 28 2 39 0 0 0 0 10 11 13 19 0 0 0 0 11 10 19 13 0 0 0 0 0 0 0 0 14 32 31 32 0 0 0 0 28 5 22 22 0 0 0 0 19 19 36 13 0 0 0 0 9 10 9 27
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 40 0 0 0 0 1 0 0 40 0 0 0 0 0 1 0 40 0 0 0 0 0 0 1 40
,
 1 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 22 33 0 1 0 0 0 0 8 19 40 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0

G:=sub<GL(8,GF(41))| [0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[35,7,25,32,0,0,0,0,7,35,32,25,0,0,0,0,0,5,6,34,0,0,0,0,5,0,34,6,0,0,0,0,0,0,0,0,22,0,3,3,0,0,0,0,38,19,38,0,0,0,0,0,0,38,19,38,0,0,0,0,3,3,0,22],[28,22,10,11,0,0,0,0,22,28,11,10,0,0,0,0,39,2,13,19,0,0,0,0,2,39,19,13,0,0,0,0,0,0,0,0,14,28,19,9,0,0,0,0,32,5,19,10,0,0,0,0,31,22,36,9,0,0,0,0,32,22,13,27],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,40,40,40,40],[1,0,22,8,0,0,0,0,0,40,33,19,0,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0] >;

(C2×Q8)⋊F5 in GAP, Magma, Sage, TeX

(C_2\times Q_8)\rtimes F_5
% in TeX

G:=Group("(C2xQ8):F5");
// GroupNames label

G:=SmallGroup(320,266);
// by ID

G=gap.SmallGroup(320,266);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,232,219,1571,570,297,136,1684,6278,3156]);
// Polycyclic

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

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