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## G = (C2×F5)⋊Q8order 320 = 26·5

### The semidirect product of C2×F5 and Q8 acting via Q8/C4=C2

Series: Derived Chief Lower central Upper central

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

Generators and relations for (C2×F5)⋊Q8
G = < a,b,c,d,e | a2=b5=c4=d4=1, e2=d2, ab=ba, dcd-1=ac=ca, ad=da, ae=ea, cbc-1=b3, bd=db, be=eb, ce=ec, ede-1=d-1 >

Subgroups: 746 in 186 conjugacy classes, 64 normal (22 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×12], C22, C22 [×6], C5, C2×C4, C2×C4 [×2], C2×C4 [×25], Q8 [×8], C23, D5 [×2], D5 [×2], C10, C10 [×2], C42 [×2], C4⋊C4 [×2], C22×C4 [×7], C2×Q8, C2×Q8 [×7], Dic5 [×2], Dic5 [×2], C20 [×2], C20 [×2], F5 [×6], D10 [×2], D10 [×4], C2×C10, C2.C42 [×4], C2×C42, C2×C4⋊C4, C22×Q8, Dic10 [×6], C4×D5 [×4], C4×D5 [×4], C2×Dic5, C2×Dic5 [×2], C2×C20, C2×C20 [×2], C5×Q8 [×2], C2×F5 [×4], C2×F5 [×10], C22×D5, C23.67C23, C4×F5 [×2], C4⋊F5 [×2], C2×Dic10, C2×Dic10 [×2], C2×C4×D5, C2×C4×D5 [×2], Q8×D5 [×4], Q8×C10, C22×F5 [×4], D10.3Q8 [×4], C2×C4×F5, C2×C4⋊F5, C2×Q8×D5, (C2×F5)⋊Q8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], Q8 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], C2×Q8 [×2], C4○D4 [×2], F5, C2×C22⋊C4, C4×Q8 [×2], C22⋊Q8 [×2], C4.4D4, C4⋊Q8, C2×F5 [×3], C23.67C23, C22⋊F5 [×2], C22×F5, Q8×F5 [×2], C2×C22⋊F5, (C2×F5)⋊Q8

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

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

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

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

38 conjugacy classes

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

38 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 4 4 4 8 type + + + + + + - + + + - image C1 C2 C2 C2 C2 C4 C4 D4 Q8 C4○D4 F5 C2×F5 C22⋊F5 Q8×F5 kernel (C2×F5)⋊Q8 D10.3Q8 C2×C4×F5 C2×C4⋊F5 C2×Q8×D5 C2×Dic10 Q8×C10 C4×D5 C2×F5 D10 C2×Q8 C2×C4 C4 C2 # reps 1 4 1 1 1 6 2 4 4 4 1 3 4 2

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

 1 0 0 0 0 0 0 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
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 1 0 0 0 0 40 0 1 0 0 0 40 0 0 1 0 0 40 0 0 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 38 3 22 0 0 0 19 3 0 38 0 0 38 0 3 19 0 0 0 22 3 38
,
 32 39 0 0 0 0 0 9 0 0 0 0 0 0 22 38 0 3 0 0 0 19 38 3 0 0 3 38 19 0 0 0 3 0 38 22
,
 40 18 0 0 0 0 9 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

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,40,40,40,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,38,19,38,0,0,0,3,3,0,22,0,0,22,0,3,3,0,0,0,38,19,38],[32,0,0,0,0,0,39,9,0,0,0,0,0,0,22,0,3,3,0,0,38,19,38,0,0,0,0,38,19,38,0,0,3,3,0,22],[40,9,0,0,0,0,18,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40] >;

(C2×F5)⋊Q8 in GAP, Magma, Sage, TeX

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

G:=Group("(C2xF5):Q8");
// GroupNames label

G:=SmallGroup(320,1128);
// by ID

G=gap.SmallGroup(320,1128);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,232,422,387,184,6278,1595]);
// Polycyclic

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

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