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## G = (C2×D4)⋊8F5order 320 = 26·5

### 6th semidirect product of C2×D4 and F5 acting via F5/D5=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — (C2×D4)⋊8F5
 Chief series C1 — C5 — C10 — D10 — C22×D5 — C22⋊F5 — D10.C23 — (C2×D4)⋊8F5
 Lower central C5 — C10 — C2×C10 — (C2×D4)⋊8F5
 Upper central C1 — C2 — C2×C4 — C2×D4

Generators and relations for (C2×D4)⋊8F5
G = < a,b,c,d,e | a2=b4=c2=d5=e4=1, ab=ba, ac=ca, ad=da, eae-1=ab2, cbc=b-1, bd=db, be=eb, cd=dc, ece-1=ab2c, ede-1=d3 >

Subgroups: 682 in 158 conjugacy classes, 48 normal (18 characteristic)
C1, C2, C2 [×5], C4 [×2], C4 [×8], C22, C22 [×7], C5, C2×C4, C2×C4 [×15], D4 [×6], Q8 [×2], C23 [×2], C23, D5 [×2], C10, C10 [×3], C42 [×2], C22⋊C4 [×4], C4⋊C4 [×2], C22×C4 [×3], C2×D4, C2×D4 [×2], C2×Q8, C4○D4 [×4], Dic5 [×2], Dic5 [×2], C20 [×2], F5 [×4], D10 [×2], D10, C2×C10, C2×C10 [×4], C23⋊C4 [×4], C42⋊C2 [×2], C2×C4○D4, Dic10 [×2], C4×D5 [×4], C2×Dic5, C2×Dic5 [×2], C2×Dic5 [×4], C5⋊D4 [×4], C2×C20, C5×D4 [×2], C2×F5 [×4], C22×D5, C22×C10 [×2], C23.C23, C4×F5 [×2], C4⋊F5 [×2], C22⋊F5 [×4], C2×Dic10, C2×C4×D5, D42D5 [×4], C22×Dic5 [×2], C2×C5⋊D4 [×2], D4×C10, C23⋊F5 [×4], D10.C23 [×2], C2×D42D5, (C2×D4)⋊8F5
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], F5, C2×C22⋊C4, C2×F5 [×3], C23.C23, C22⋊F5 [×2], C22×F5, C2×C22⋊F5, (C2×D4)⋊8F5

Smallest permutation representation of (C2×D4)⋊8F5
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 46 6 41)(2 47 7 42)(3 48 8 43)(4 49 9 44)(5 50 10 45)(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)
(1 71)(2 72)(3 73)(4 74)(5 75)(6 76)(7 77)(8 78)(9 79)(10 80)(11 61)(12 62)(13 63)(14 64)(15 65)(16 66)(17 67)(18 68)(19 69)(20 70)(21 51)(22 52)(23 53)(24 54)(25 55)(26 56)(27 57)(28 58)(29 59)(30 60)(31 41)(32 42)(33 43)(34 44)(35 45)(36 46)(37 47)(38 48)(39 49)(40 50)
(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 61 11 76)(2 63 15 79)(3 65 14 77)(4 62 13 80)(5 64 12 78)(6 66 16 71)(7 68 20 74)(8 70 19 72)(9 67 18 75)(10 69 17 73)(21 56 36 46)(22 58 40 49)(23 60 39 47)(24 57 38 50)(25 59 37 48)(26 51 31 41)(27 53 35 44)(28 55 34 42)(29 52 33 45)(30 54 32 43)

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

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

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

32 conjugacy classes

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

32 irreducible representations

 dim 1 1 1 1 1 1 1 2 4 4 4 4 4 8 type + + + + + + + + + - image C1 C2 C2 C2 C4 C4 C4 D4 F5 C2×F5 C2×F5 C23.C23 C22⋊F5 (C2×D4)⋊8F5 kernel (C2×D4)⋊8F5 C23⋊F5 D10.C23 C2×D4⋊2D5 C2×Dic10 C22×Dic5 D4×C10 C4×D5 C2×D4 C2×C4 C23 C5 C4 C1 # reps 1 4 2 1 2 4 2 4 1 1 2 2 4 2

Matrix representation of (C2×D4)⋊8F5 in GL8(𝔽41)

 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 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
,
 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 32 0 0 0 0 0 0 32 0 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 9 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 0 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 9 0 0 0 0 32 0 0 0 0 0 0 0 0 32 0 0
,
 40 40 40 40 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 0 0 0 0 1
,
 40 0 0 0 0 0 0 0 0 0 0 40 0 0 0 0 0 40 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 32 0 0 0 0 0 0 0 0 9 0 0 0 0 0 9 0 0 0 0 0 0 32 0 0 0

G:=sub<GL(8,GF(41))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0],[40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,32,0,0,0,0,9,0,0,0,0,0,0,0,0,9,0,0],[40,1,0,0,0,0,0,0,40,0,1,0,0,0,0,0,40,0,0,1,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[40,0,0,1,0,0,0,0,0,0,40,1,0,0,0,0,0,0,0,1,0,0,0,0,0,40,0,1,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,9,0,0,0,0,0,32,0,0,0,0,0,0,0,0,9,0,0] >;

(C2×D4)⋊8F5 in GAP, Magma, Sage, TeX

(C_2\times D_4)\rtimes_8F_5
% in TeX

G:=Group("(C2xD4):8F5");
// GroupNames label

G:=SmallGroup(320,1109);
// by ID

G=gap.SmallGroup(320,1109);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,232,422,387,297,1684,6278,1595]);
// Polycyclic

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

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