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G = C2×D4.D5order 160 = 25·5

Direct product of C2 and D4.D5

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

Aliases: C2×D4.D5, D4.7D10, C102SD16, C20.16D4, C20.13C23, Dic106C22, C53(C2×SD16), (C2×D4).4D5, C52C88C22, (D4×C10).3C2, (C2×C4).48D10, C10.46(C2×D4), (C2×C10).40D4, C4.6(C5⋊D4), (C2×Dic10)⋊9C2, (C5×D4).7C22, C4.13(C22×D5), (C2×C20).31C22, C22.22(C5⋊D4), (C2×C52C8)⋊5C2, C2.10(C2×C5⋊D4), SmallGroup(160,154)

Series: Derived Chief Lower central Upper central

C1C20 — C2×D4.D5
C1C5C10C20Dic10C2×Dic10 — C2×D4.D5
C5C10C20 — C2×D4.D5
C1C22C2×C4C2×D4

Generators and relations for C2×D4.D5
 G = < a,b,c,d,e | a2=b4=c2=d5=1, e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b-1, bd=db, cd=dc, ece-1=bc, ede-1=d-1 >

Subgroups: 184 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 [×2], D4, Q8 [×3], C23, C10, C10 [×2], C10 [×2], C2×C8, SD16 [×4], C2×D4, C2×Q8, Dic5 [×2], C20 [×2], C2×C10, C2×C10 [×4], C2×SD16, C52C8 [×2], Dic10 [×2], Dic10, C2×Dic5, C2×C20, C5×D4 [×2], C5×D4, C22×C10, C2×C52C8, D4.D5 [×4], C2×Dic10, D4×C10, C2×D4.D5
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, SD16 [×2], C2×D4, D10 [×3], C2×SD16, C5⋊D4 [×2], C22×D5, D4.D5 [×2], C2×C5⋊D4, C2×D4.D5

Smallest permutation representation of C2×D4.D5
On 80 points
Generators in S80
(1 21)(2 22)(3 23)(4 24)(5 25)(6 26)(7 27)(8 28)(9 29)(10 30)(11 36)(12 37)(13 38)(14 39)(15 40)(16 31)(17 32)(18 33)(19 34)(20 35)(41 61)(42 62)(43 63)(44 64)(45 65)(46 66)(47 67)(48 68)(49 69)(50 70)(51 76)(52 77)(53 78)(54 79)(55 80)(56 71)(57 72)(58 73)(59 74)(60 75)
(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 51 46 56)(42 52 47 57)(43 53 48 58)(44 54 49 59)(45 55 50 60)(61 76 66 71)(62 77 67 72)(63 78 68 73)(64 79 69 74)(65 80 70 75)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(21 36)(22 37)(23 38)(24 39)(25 40)(26 31)(27 32)(28 33)(29 34)(30 35)(41 46)(42 47)(43 48)(44 49)(45 50)(61 66)(62 67)(63 68)(64 69)(65 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 68 6 63)(2 67 7 62)(3 66 8 61)(4 70 9 65)(5 69 10 64)(11 73 16 78)(12 72 17 77)(13 71 18 76)(14 75 19 80)(15 74 20 79)(21 48 26 43)(22 47 27 42)(23 46 28 41)(24 50 29 45)(25 49 30 44)(31 53 36 58)(32 52 37 57)(33 51 38 56)(34 55 39 60)(35 54 40 59)

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

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

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

C2×D4.D5 is a maximal subgroup of
D20.2D4  D4.D55C4  Dic56SD16  Dic102D4  Dic10.D4  D20.8D4  D10⋊SD16  C52C8⋊D4  D4.D20  D4.1D20  C42.51D10  D4.2D20  D2017D4  Dic1017D4  C52C823D4  C4.(D4×D5)  C42.61D10  C42.214D10  C42.65D10  C42.74D10  Dic109D4  C204SD16  (C2×D8).D5  C4011D4  C40.22D4  Dic10⋊D4  Dic53SD16  C40.31D4  D108SD16  C4015D4  M4(2).13D10  (C5×D4).31D4  (C5×D4).32D4  C2×D5×SD16  D86D10  D20.33C23
C2×D4.D5 is a maximal quotient of
C4⋊C4.231D10  C20.38SD16  D4.2D20  C4⋊D4.D5  Dic1017D4  C52C823D4  C20.16D8  Dic109D4  C204SD16  C20.SD16  C20.11Q16  Dic106Q8  (C5×D4).31D4

34 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D5A5B8A8B8C8D10A···10F10G···10N20A20B20C20D
order122222444455888810···1010···1020202020
size11114422202022101010102···24···44444

34 irreducible representations

dim11111222222224
type++++++++++-
imageC1C2C2C2C2D4D4D5SD16D10D10C5⋊D4C5⋊D4D4.D5
kernelC2×D4.D5C2×C52C8D4.D5C2×Dic10D4×C10C20C2×C10C2×D4C10C2×C4D4C4C22C2
# reps11411112424444

Matrix representation of C2×D4.D5 in GL6(𝔽41)

100000
010000
0040000
0004000
000010
000001
,
010000
4000000
0004000
001000
000010
000001
,
010000
100000
0004000
0040000
000010
000001
,
100000
010000
001000
000100
0000160
00002218
,
15260000
26260000
00151500
00152600
0000723
00003034

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,22,0,0,0,0,0,18],[15,26,0,0,0,0,26,26,0,0,0,0,0,0,15,15,0,0,0,0,15,26,0,0,0,0,0,0,7,30,0,0,0,0,23,34] >;

C2×D4.D5 in GAP, Magma, Sage, TeX

C_2\times D_4.D_5
% in TeX

G:=Group("C2xD4.D5");
// GroupNames label

G:=SmallGroup(160,154);
// by ID

G=gap.SmallGroup(160,154);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,96,218,579,159,69,4613]);
// Polycyclic

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

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