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G = C2×D42Dic5order 320 = 26·5

Direct product of C2 and D42Dic5

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2×D42Dic5, C105C4≀C2, (D4×C10)⋊20C4, (Q8×C10)⋊18C4, C4○D43Dic5, (C2×D4)⋊8Dic5, D45(C2×Dic5), (C2×Q8)⋊6Dic5, Q85(C2×Dic5), C4○D4.36D10, C20.452(C2×D4), (C2×C20).197D4, C20.144(C22×C4), (C2×C20).481C23, (C4×Dic5)⋊63C22, (C22×C4).357D10, (C22×C10).113D4, C23.66(C5⋊D4), C4.Dic523C22, C4.33(C23.D5), C4.15(C22×Dic5), C20.146(C22⋊C4), C22.5(C23.D5), (C22×C20).207C22, C57(C2×C4≀C2), (C5×C4○D4)⋊9C4, (C2×C4×Dic5)⋊4C2, (C5×D4)⋊28(C2×C4), (C5×Q8)⋊26(C2×C4), (C2×C4○D4).4D5, (C10×C4○D4).4C2, (C2×C10).39(C2×D4), C4.143(C2×C5⋊D4), (C2×C20).298(C2×C4), (C2×C4.Dic5)⋊21C2, (C2×C4).54(C2×Dic5), C22.11(C2×C5⋊D4), C2.20(C2×C23.D5), (C2×C4).282(C5⋊D4), C10.125(C2×C22⋊C4), (C5×C4○D4).41C22, (C2×C4).566(C22×D5), (C2×C10).181(C22⋊C4), SmallGroup(320,862)

Series: Derived Chief Lower central Upper central

C1C20 — C2×D42Dic5
C1C5C10C20C2×C20C4.Dic5C2×C4.Dic5 — C2×D42Dic5
C5C10C20 — C2×D42Dic5
C1C2×C4C22×C4C2×C4○D4

Generators and relations for C2×D42Dic5
 G = < a,b,c,d,e | a2=b4=d10=1, c2=b2, e2=d5, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, dcd-1=b2c, ece-1=b-1c, ede-1=d-1 >

Subgroups: 446 in 170 conjugacy classes, 71 normal (43 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×4], C4 [×6], C22 [×3], C22 [×6], C5, C8 [×2], C2×C4 [×6], C2×C4 [×11], D4 [×2], D4 [×5], Q8 [×2], Q8, C23, C23, C10, C10 [×2], C10 [×4], C42 [×3], C2×C8, M4(2) [×3], C22×C4, C22×C4 [×2], C2×D4, C2×D4, C2×Q8, C4○D4 [×4], C4○D4 [×2], Dic5 [×4], C20 [×4], C20 [×2], C2×C10 [×3], C2×C10 [×6], C4≀C2 [×4], C2×C42, C2×M4(2), C2×C4○D4, C52C8 [×2], C2×Dic5 [×6], C2×C20 [×6], C2×C20 [×5], C5×D4 [×2], C5×D4 [×5], C5×Q8 [×2], C5×Q8, C22×C10, C22×C10, C2×C4≀C2, C2×C52C8, C4.Dic5 [×2], C4.Dic5, C4×Dic5 [×2], C4×Dic5, C22×Dic5, C22×C20, C22×C20, D4×C10, D4×C10, Q8×C10, C5×C4○D4 [×4], C5×C4○D4 [×2], D42Dic5 [×4], C2×C4.Dic5, C2×C4×Dic5, C10×C4○D4, C2×D42Dic5
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, D5, C22⋊C4 [×4], C22×C4, C2×D4 [×2], Dic5 [×4], D10 [×3], C4≀C2 [×2], C2×C22⋊C4, C2×Dic5 [×6], C5⋊D4 [×4], C22×D5, C2×C4≀C2, C23.D5 [×4], C22×Dic5, C2×C5⋊D4 [×2], D42Dic5 [×2], C2×C23.D5, C2×D42Dic5

Smallest permutation representation of C2×D42Dic5
On 80 points
Generators in S80
(1 30)(2 26)(3 27)(4 28)(5 29)(6 37)(7 38)(8 39)(9 40)(10 36)(11 33)(12 34)(13 35)(14 31)(15 32)(16 23)(17 24)(18 25)(19 21)(20 22)(41 56)(42 57)(43 58)(44 59)(45 60)(46 51)(47 52)(48 53)(49 54)(50 55)(61 71)(62 72)(63 73)(64 74)(65 75)(66 76)(67 77)(68 78)(69 79)(70 80)
(1 9 18 13)(2 10 19 14)(3 6 20 15)(4 7 16 11)(5 8 17 12)(21 31 26 36)(22 32 27 37)(23 33 28 38)(24 34 29 39)(25 35 30 40)(41 61 46 66)(42 62 47 67)(43 63 48 68)(44 64 49 69)(45 65 50 70)(51 76 56 71)(52 77 57 72)(53 78 58 73)(54 79 59 74)(55 80 60 75)
(1 64 18 69)(2 70 19 65)(3 66 20 61)(4 62 16 67)(5 68 17 63)(6 46 15 41)(7 42 11 47)(8 48 12 43)(9 44 13 49)(10 50 14 45)(21 75 26 80)(22 71 27 76)(23 77 28 72)(24 73 29 78)(25 79 30 74)(31 60 36 55)(32 56 37 51)(33 52 38 57)(34 58 39 53)(35 54 40 59)
(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 3)(4 5)(6 9)(7 8)(11 12)(13 15)(16 17)(18 20)(22 25)(23 24)(27 30)(28 29)(32 35)(33 34)(37 40)(38 39)(41 64 46 69)(42 63 47 68)(43 62 48 67)(44 61 49 66)(45 70 50 65)(51 79 56 74)(52 78 57 73)(53 77 58 72)(54 76 59 71)(55 75 60 80)

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

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

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

68 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I···4P5A5B8A8B8C8D10A···10F10G···10R20A···20H20I···20T
order12222222444444444···455888810···1010···1020···2020···20
size111122441111224410···1022202020202···24···42···24···4

68 irreducible representations

dim11111111222222222224
type+++++++++---+
imageC1C2C2C2C2C4C4C4D4D4D5D10Dic5Dic5Dic5D10C4≀C2C5⋊D4C5⋊D4D42Dic5
kernelC2×D42Dic5D42Dic5C2×C4.Dic5C2×C4×Dic5C10×C4○D4D4×C10Q8×C10C5×C4○D4C2×C20C22×C10C2×C4○D4C22×C4C2×D4C2×Q8C4○D4C4○D4C10C2×C4C23C2
# reps141112243122224481248

Matrix representation of C2×D42Dic5 in GL4(𝔽41) generated by

40000
04000
0010
0001
,
40000
04000
0090
00032
,
173500
72400
0009
0090
,
404000
8700
00400
0001
,
7600
333400
00320
0001
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[40,0,0,0,0,40,0,0,0,0,9,0,0,0,0,32],[17,7,0,0,35,24,0,0,0,0,0,9,0,0,9,0],[40,8,0,0,40,7,0,0,0,0,40,0,0,0,0,1],[7,33,0,0,6,34,0,0,0,0,32,0,0,0,0,1] >;

C2×D42Dic5 in GAP, Magma, Sage, TeX

C_2\times D_4\rtimes_2{\rm Dic}_5
% in TeX

G:=Group("C2xD4:2Dic5");
// GroupNames label

G:=SmallGroup(320,862);
// by ID

G=gap.SmallGroup(320,862);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,422,136,1684,438,102,12550]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^4=d^10=1,c^2=b^2,e^2=d^5,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,d*c*d^-1=b^2*c,e*c*e^-1=b^-1*c,e*d*e^-1=d^-1>;
// generators/relations

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