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G = C2×D4⋊D10  order 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○D4⋊14D10, (C2×D4)⋊41D10, C5⋊2C8⋊5C23, (C2×Q8)⋊30D10, D4⋊5(C22×D5), (C5×D4)⋊5C23, Q8⋊5(C22×D5), (C5×Q8)⋊5C23, C10⋊5(C8⋊C22), D4⋊D5⋊18C22, C20.426(C2×D4), (C2×C20).217D4, Q8⋊D5⋊17C22, 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.Dic5⋊36C22, (C22×C20).290C22, C5⋊6(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×C5⋊2C8)⋊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

Derived series
C1 — C20 — C2×D4⋊D10
Chief series
C1 — C5 — C10 — C20 — D20 — C2×D20 — C22×D20 — C2×D4⋊D10
Lower central
C5 — C10 — C20 — C2×D4⋊D10

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, C5⋊2C8 [×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×C5⋊2C8 [×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

Generators and relations
 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 >

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

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

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

Matrix representation ►G ⊆ 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] >;
 

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

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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