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G = (C2×D4)⋊8F5order 320 = 26·5

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C2×D4)⋊8F5, (D4×C10)⋊9C4, C23⋊F54C2, D10.6(C2×D4), (C4×D5).36D4, C23.3(C2×F5), (C2×Dic10)⋊10C4, (C22×Dic5)⋊8C4, C4.13(C22⋊F5), C20.13(C22⋊C4), C22⋊F5.2C22, C22.12(C22×F5), C52(C23.C23), D10.C233C2, Dic5.44(C22⋊C4), (C22×D5).148C23, (C2×C4).35(C2×F5), (C2×C20).53(C2×C4), C2.19(C2×C22⋊F5), C10.18(C2×C22⋊C4), (C2×C4×D5).200C22, (C2×D42D5).15C2, (C2×C10).74(C22×C4), (C22×C10).22(C2×C4), (C2×C5⋊D4).88C22, (C2×Dic5).192(C2×C4), SmallGroup(320,1109)

Series: Derived Chief Lower central Upper central

C1C2×C10 — (C2×D4)⋊8F5
C1C5C10D10C22×D5C22⋊F5D10.C23 — (C2×D4)⋊8F5
C5C10C2×C10 — (C2×D4)⋊8F5
C1C2C2×C4C2×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 2A2B2C2D2E2F4A4B4C4D4E4F···4O 5 10A10B10C10D10E10F10G20A20B
order1222222444444···45101010101010102020
size11244101022551020···204444888888

32 irreducible representations

dim11111112444448
type+++++++++-
imageC1C2C2C2C4C4C4D4F5C2×F5C2×F5C23.C23C22⋊F5(C2×D4)⋊8F5
kernel(C2×D4)⋊8F5C23⋊F5D10.C23C2×D42D5C2×Dic10C22×Dic5D4×C10C4×D5C2×D4C2×C4C23C5C4C1
# reps14212424112242

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

10000000
01000000
00100000
00010000
000004000
000040000
000000040
000000400
,
10000000
01000000
00100000
00010000
000003200
000032000
00000009
00000090
,
400000000
040000000
004000000
000400000
00000090
00000009
000032000
000003200
,
404040400000
10000000
01000000
00100000
00001000
00000100
00000010
00000001
,
400000000
000400000
040000000
11110000
000000320
00000009
00000900
000032000

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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