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G = C204D4order 160 = 25·5

1st semidirect product of C20 and D4 acting via D4/C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C204D4, C41D20, C425D5, (C4×C20)⋊4C2, (C2×D20)⋊1C2, C51(C41D4), C2.5(C2×D20), C10.3(C2×D4), (C2×C4).76D10, (C2×C10).15C23, (C2×C20).87C22, (C22×D5).1C22, C22.36(C22×D5), SmallGroup(160,95)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C204D4
C1C5C10C2×C10C22×D5C2×D20 — C204D4
C5C2×C10 — C204D4
C1C22C42

Generators and relations for C204D4
 G = < a,b,c | a4=b20=c2=1, ab=ba, cac=a-1, cbc=b-1 >

Subgroups: 480 in 108 conjugacy classes, 41 normal (7 characteristic)
C1, C2 [×3], C2 [×4], C4 [×6], C22, C22 [×12], C5, C2×C4 [×3], D4 [×12], C23 [×4], D5 [×4], C10 [×3], C42, C2×D4 [×6], C20 [×6], D10 [×12], C2×C10, C41D4, D20 [×12], C2×C20 [×3], C22×D5 [×4], C4×C20, C2×D20 [×6], C204D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D5, C2×D4 [×3], D10 [×3], C41D4, D20 [×6], C22×D5, C2×D20 [×3], C204D4

Smallest permutation representation of C204D4
On 80 points
Generators in S80
(1 70 31 46)(2 71 32 47)(3 72 33 48)(4 73 34 49)(5 74 35 50)(6 75 36 51)(7 76 37 52)(8 77 38 53)(9 78 39 54)(10 79 40 55)(11 80 21 56)(12 61 22 57)(13 62 23 58)(14 63 24 59)(15 64 25 60)(16 65 26 41)(17 66 27 42)(18 67 28 43)(19 68 29 44)(20 69 30 45)
(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 10)(7 9)(16 20)(17 19)(21 35)(22 34)(23 33)(24 32)(25 31)(26 30)(27 29)(36 40)(37 39)(41 69)(42 68)(43 67)(44 66)(45 65)(46 64)(47 63)(48 62)(49 61)(50 80)(51 79)(52 78)(53 77)(54 76)(55 75)(56 74)(57 73)(58 72)(59 71)(60 70)

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

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

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

C204D4 is a maximal subgroup of
C4.D40  C42⋊F5  C85D20  C4.5D40  C204D8  C8⋊D20  C42.19D10  D44D20  C4⋊D40  Dic108D4  C207D8  Q8⋊D20  C42.64D10  C42.70D10  C20⋊D8  C206SD16  C20.D8  C42.276D10  C429D10  C42.100D10  D4×D20  Dic1024D4  Q86D20  C42.136D10  C4218D10  C42.156D10  C4225D10  D5×C41D4  C42.240D10  C204D4⋊C3  C12⋊D20  C426D15
C204D4 is a maximal quotient of
(C2×C20)⋊5D4  (C2×C20).33D4  C85D20  C204D8  C8.8D20  C204Q16  C8⋊D20  C8.D20  C428Dic5  (C2×C4)⋊6D20  C12⋊D20  C426D15

46 conjugacy classes

class 1 2A2B2C2D2E2F2G4A···4F5A5B10A···10F20A···20X
order122222224···45510···1020···20
size1111202020202···2222···22···2

46 irreducible representations

dim1112222
type+++++++
imageC1C2C2D4D5D10D20
kernelC204D4C4×C20C2×D20C20C42C2×C4C4
# reps11662624

Matrix representation of C204D4 in GL4(𝔽41) generated by

40200
40100
00400
00040
,
1000
0100
003230
001127
,
1000
14000
00040
00400
G:=sub<GL(4,GF(41))| [40,40,0,0,2,1,0,0,0,0,40,0,0,0,0,40],[1,0,0,0,0,1,0,0,0,0,32,11,0,0,30,27],[1,1,0,0,0,40,0,0,0,0,0,40,0,0,40,0] >;

C204D4 in GAP, Magma, Sage, TeX

C_{20}\rtimes_4D_4
% in TeX

G:=Group("C20:4D4");
// GroupNames label

G:=SmallGroup(160,95);
// by ID

G=gap.SmallGroup(160,95);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,217,103,218,50,4613]);
// Polycyclic

G:=Group<a,b,c|a^4=b^20=c^2=1,a*b=b*a,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations

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