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G = C2×D4⋊D10order 320 = 26·5

Direct product of C2 and D4⋊D10

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

Aliases: C2×D4⋊D10, C20.33C24, D20.29C23, C4○D414D10, (C2×D4)⋊41D10, C52C85C23, (C2×Q8)⋊30D10, D45(C22×D5), (C5×D4)⋊5C23, (C5×Q8)⋊5C23, Q85(C22×D5), C105(C8⋊C22), D4⋊D518C22, C20.426(C2×D4), (C2×C20).217D4, Q8⋊D517C22, C4.33(C23×D5), (C2×D20)⋊58C22, (D4×C10)⋊45C22, (C22×D20)⋊20C2, (Q8×C10)⋊37C22, (C2×C20).555C23, C10.158(C22×D4), (C22×C10).122D4, (C22×C4).281D10, C23.68(C5⋊D4), C4.Dic536C22, (C22×C20).290C22, C56(C2×C8⋊C22), (C2×C4○D4)⋊2D5, (C2×D4⋊D5)⋊31C2, (C10×C4○D4)⋊2C2, (C2×Q8⋊D5)⋊31C2, C4.29(C2×C5⋊D4), (C2×C10).75(C2×D4), (C2×C52C8)⋊22C22, (C5×C4○D4)⋊16C22, (C2×C4).95(C5⋊D4), (C2×C4.Dic5)⋊30C2, C2.31(C22×C5⋊D4), (C2×C4).245(C22×D5), C22.118(C2×C5⋊D4), SmallGroup(320,1492)

Series: Derived Chief Lower central Upper central

C1C20 — C2×D4⋊D10
C1C5C10C20D20C2×D20C22×D20 — C2×D4⋊D10
C5C10C20 — C2×D4⋊D10
C1C22C22×C4C2×C4○D4

Generators and relations for C2×D4⋊D10
 G = < a,b,c,d,e | a2=b4=c2=d10=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe=b-1, bd=db, dcd-1=b2c, ece=b-1c, ede=d-1 >

Subgroups: 1214 in 298 conjugacy classes, 111 normal (27 characteristic)
C1, C2, C2 [×2], C2 [×8], C4 [×2], C4 [×2], C4 [×2], C22, C22 [×2], C22 [×22], C5, C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×5], D4 [×2], D4 [×15], Q8 [×2], Q8, C23, C23 [×11], D5 [×4], C10, C10 [×2], C10 [×4], C2×C8 [×2], M4(2) [×4], D8 [×8], SD16 [×8], C22×C4, C22×C4, C2×D4, C2×D4 [×10], C2×Q8, C4○D4 [×4], C4○D4 [×2], C24, C20 [×2], C20 [×2], C20 [×2], D10 [×16], C2×C10, C2×C10 [×2], C2×C10 [×6], C2×M4(2), C2×D8 [×2], C2×SD16 [×2], C8⋊C22 [×8], C22×D4, C2×C4○D4, C52C8 [×4], D20 [×4], D20 [×6], C2×C20 [×2], C2×C20 [×4], C2×C20 [×5], C5×D4 [×2], C5×D4 [×5], C5×Q8 [×2], C5×Q8, C22×D5 [×10], C22×C10, C22×C10, C2×C8⋊C22, C2×C52C8 [×2], C4.Dic5 [×4], D4⋊D5 [×8], Q8⋊D5 [×8], C2×D20 [×6], C2×D20 [×3], C22×C20, C22×C20, D4×C10, D4×C10, Q8×C10, C5×C4○D4 [×4], C5×C4○D4 [×2], C23×D5, C2×C4.Dic5, C2×D4⋊D5 [×2], C2×Q8⋊D5 [×2], D4⋊D10 [×8], C22×D20, C10×C4○D4, C2×D4⋊D10
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C8⋊C22 [×2], C22×D4, C5⋊D4 [×4], C22×D5 [×7], C2×C8⋊C22, C2×C5⋊D4 [×6], C23×D5, D4⋊D10 [×2], C22×C5⋊D4, C2×D4⋊D10

Smallest permutation representation of C2×D4⋊D10
On 80 points
Generators in S80
(1 21)(2 22)(3 23)(4 24)(5 25)(6 34)(7 35)(8 31)(9 32)(10 33)(11 36)(12 37)(13 38)(14 39)(15 40)(16 30)(17 26)(18 27)(19 28)(20 29)(41 76)(42 77)(43 78)(44 79)(45 80)(46 71)(47 72)(48 73)(49 74)(50 75)(51 63)(52 64)(53 65)(54 66)(55 67)(56 68)(57 69)(58 70)(59 61)(60 62)
(1 28 32 39)(2 29 33 40)(3 30 34 36)(4 26 35 37)(5 27 31 38)(6 11 23 16)(7 12 24 17)(8 13 25 18)(9 14 21 19)(10 15 22 20)(41 59 46 54)(42 60 47 55)(43 51 48 56)(44 52 49 57)(45 53 50 58)(61 71 66 76)(62 72 67 77)(63 73 68 78)(64 74 69 79)(65 75 70 80)
(1 79)(2 75)(3 71)(4 77)(5 73)(6 41)(7 47)(8 43)(9 49)(10 45)(11 54)(12 60)(13 56)(14 52)(15 58)(16 59)(17 55)(18 51)(19 57)(20 53)(21 44)(22 50)(23 46)(24 42)(25 48)(26 67)(27 63)(28 69)(29 65)(30 61)(31 78)(32 74)(33 80)(34 76)(35 72)(36 66)(37 62)(38 68)(39 64)(40 70)
(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 21)(2 25)(3 24)(4 23)(5 22)(6 35)(7 34)(8 33)(9 32)(10 31)(11 26)(12 30)(13 29)(14 28)(15 27)(16 37)(17 36)(18 40)(19 39)(20 38)(41 67)(42 66)(43 65)(44 64)(45 63)(46 62)(47 61)(48 70)(49 69)(50 68)(51 80)(52 79)(53 78)(54 77)(55 76)(56 75)(57 74)(58 73)(59 72)(60 71)

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

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

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

62 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J2K4A4B4C4D4E4F5A5B8A8B8C8D10A···10F10G···10R20A···20H20I···20T
order12222222222244444455888810···1010···1020···2020···20
size111122442020202022224422202020202···24···42···24···4

62 irreducible representations

dim111111122222222244
type++++++++++++++++
imageC1C2C2C2C2C2C2D4D4D5D10D10D10D10C5⋊D4C5⋊D4C8⋊C22D4⋊D10
kernelC2×D4⋊D10C2×C4.Dic5C2×D4⋊D5C2×Q8⋊D5D4⋊D10C22×D20C10×C4○D4C2×C20C22×C10C2×C4○D4C22×C4C2×D4C2×Q8C4○D4C2×C4C23C10C2
# reps1122811312222812428

Matrix representation of C2×D4⋊D10 in GL6(𝔽41)

4000000
0400000
001000
000100
000010
000001
,
4000000
0400000
00302800
00221100
0019241113
0013321930
,
1160000
0400000
0022171915
00289322
0010271924
0030401332
,
4000000
0400000
000600
00343400
00379035
0010677
,
4000000
3610000
001000
00334000
00001113
00001630

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,30,22,19,13,0,0,28,11,24,32,0,0,0,0,11,19,0,0,0,0,13,30],[1,0,0,0,0,0,16,40,0,0,0,0,0,0,22,28,10,30,0,0,17,9,27,40,0,0,19,3,19,13,0,0,15,22,24,32],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,34,37,10,0,0,6,34,9,6,0,0,0,0,0,7,0,0,0,0,35,7],[40,36,0,0,0,0,0,1,0,0,0,0,0,0,1,33,0,0,0,0,0,40,0,0,0,0,0,0,11,16,0,0,0,0,13,30] >;

C2×D4⋊D10 in GAP, Magma, Sage, TeX

C_2\times D_4\rtimes D_{10}
% in TeX

G:=Group("C2xD4:D10");
// GroupNames label

G:=SmallGroup(320,1492);
// by ID

G=gap.SmallGroup(320,1492);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,675,297,1684,235,102,12550]);
// Polycyclic

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

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