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G = C2×Q8⋊D5order 160 = 25·5

Direct product of C2 and Q8⋊D5

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

Aliases: C2×Q8⋊D5, Q83D10, C103SD16, C20.18D4, C20.14C23, D20.9C22, (C2×Q8)⋊1D5, C54(C2×SD16), (Q8×C10)⋊1C2, C52C89C22, (C2×D20).9C2, (C2×C10).41D4, C10.53(C2×D4), (C2×C4).53D10, C4.8(C5⋊D4), (C5×Q8)⋊3C22, C4.14(C22×D5), (C2×C20).36C22, C22.23(C5⋊D4), (C2×C52C8)⋊6C2, C2.17(C2×C5⋊D4), SmallGroup(160,162)

Series: Derived Chief Lower central Upper central

C1C20 — C2×Q8⋊D5
C1C5C10C20D20C2×D20 — C2×Q8⋊D5
C5C10C20 — C2×Q8⋊D5
C1C22C2×C4C2×Q8

Generators and relations for C2×Q8⋊D5
 G = < a,b,c,d,e | a2=b4=d5=e2=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe=b-1, bd=db, cd=dc, ece=b-1c, ede=d-1 >

Subgroups: 248 in 68 conjugacy classes, 33 normal (17 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×2], C4 [×2], C22, C22 [×4], C5, C8 [×2], C2×C4, C2×C4, D4 [×3], Q8 [×2], Q8, C23, D5 [×2], C10, C10 [×2], C2×C8, SD16 [×4], C2×D4, C2×Q8, C20 [×2], C20 [×2], D10 [×4], C2×C10, C2×SD16, C52C8 [×2], D20 [×2], D20, C2×C20, C2×C20, C5×Q8 [×2], C5×Q8, C22×D5, C2×C52C8, Q8⋊D5 [×4], C2×D20, Q8×C10, C2×Q8⋊D5
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, SD16 [×2], C2×D4, D10 [×3], C2×SD16, C5⋊D4 [×2], C22×D5, Q8⋊D5 [×2], C2×C5⋊D4, C2×Q8⋊D5

Smallest permutation representation of C2×Q8⋊D5
On 80 points
Generators in S80
(1 41)(2 42)(3 43)(4 44)(5 45)(6 46)(7 47)(8 48)(9 49)(10 50)(11 51)(12 52)(13 53)(14 54)(15 55)(16 56)(17 57)(18 58)(19 59)(20 60)(21 61)(22 62)(23 63)(24 64)(25 65)(26 66)(27 67)(28 68)(29 69)(30 70)(31 71)(32 72)(33 73)(34 74)(35 75)(36 76)(37 77)(38 78)(39 79)(40 80)
(1 16 6 11)(2 17 7 12)(3 18 8 13)(4 19 9 14)(5 20 10 15)(21 31 26 36)(22 32 27 37)(23 33 28 38)(24 34 29 39)(25 35 30 40)(41 56 46 51)(42 57 47 52)(43 58 48 53)(44 59 49 54)(45 60 50 55)(61 71 66 76)(62 72 67 77)(63 73 68 78)(64 74 69 79)(65 75 70 80)
(1 66 6 61)(2 67 7 62)(3 68 8 63)(4 69 9 64)(5 70 10 65)(11 76 16 71)(12 77 17 72)(13 78 18 73)(14 79 19 74)(15 80 20 75)(21 41 26 46)(22 42 27 47)(23 43 28 48)(24 44 29 49)(25 45 30 50)(31 51 36 56)(32 52 37 57)(33 53 38 58)(34 54 39 59)(35 55 40 60)
(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 5)(2 4)(6 10)(7 9)(11 20)(12 19)(13 18)(14 17)(15 16)(21 35)(22 34)(23 33)(24 32)(25 31)(26 40)(27 39)(28 38)(29 37)(30 36)(41 45)(42 44)(46 50)(47 49)(51 60)(52 59)(53 58)(54 57)(55 56)(61 75)(62 74)(63 73)(64 72)(65 71)(66 80)(67 79)(68 78)(69 77)(70 76)

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

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

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

C2×Q8⋊D5 is a maximal subgroup of
D20.6D4  Dic57SD16  Q82D20  D102SD16  D204D4  C5⋊(C8⋊D4)  Q8⋊D56C4  Dic5⋊SD16  D20.12D4  C42.56D10  Q8⋊D20  Q8.1D20  D20.36D4  D20.37D4  C52C824D4  C22⋊Q8⋊D5  D20.23D4  C42.64D10  C42.214D10  C205SD16  C206SD16  C42.80D10  Dic55SD16  D106SD16  C4015D4  C409D4  (C2×Q16)⋊D5  D20.17D4  C40.37D4  C40.28D4  M4(2).15D10  (C5×Q8)⋊13D4  (C5×D4)⋊14D4  C2×D5×SD16  C40.C23  D20.34C23
C2×Q8⋊D5 is a maximal quotient of
C4⋊C4.228D10  C20.48SD16  Q8⋊D20  C22⋊Q8.D5  D20.36D4  C52C824D4  C20.SD16  C20.Q16  C205SD16  D205Q8  C206SD16  C20.D8  (C5×Q8)⋊13D4

34 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D5A5B8A8B8C8D10A···10F20A···20L
order122222444455888810···1020···20
size11112020224422101010102···24···4

34 irreducible representations

dim11111222222224
type+++++++++++
imageC1C2C2C2C2D4D4D5SD16D10D10C5⋊D4C5⋊D4Q8⋊D5
kernelC2×Q8⋊D5C2×C52C8Q8⋊D5C2×D20Q8×C10C20C2×C10C2×Q8C10C2×C4Q8C4C22C2
# reps11411112424444

Matrix representation of C2×Q8⋊D5 in GL4(𝔽41) generated by

1000
0100
00400
00040
,
403900
1100
00400
00040
,
03000
15000
00231
00518
,
1000
0100
003540
003640
,
1000
404000
00034
00350
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[40,1,0,0,39,1,0,0,0,0,40,0,0,0,0,40],[0,15,0,0,30,0,0,0,0,0,23,5,0,0,1,18],[1,0,0,0,0,1,0,0,0,0,35,36,0,0,40,40],[1,40,0,0,0,40,0,0,0,0,0,35,0,0,34,0] >;

C2×Q8⋊D5 in GAP, Magma, Sage, TeX

C_2\times Q_8\rtimes D_5
% in TeX

G:=Group("C2xQ8:D5");
// GroupNames label

G:=SmallGroup(160,162);
// by ID

G=gap.SmallGroup(160,162);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,218,86,579,159,69,4613]);
// Polycyclic

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

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