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G = C4⋊D20⋊C3  order 480 = 25·3·5

The semidirect product of C4⋊D20 and C3 acting faithfully

metabelian, soluble, monomial

Aliases: C4⋊D20⋊C3, (C4×C20)⋊3C6, C42⋊C3⋊3D5, C5⋊(C23.A4), C42⋊2(C3×D5), C22.1(D5×A4), (C22×D5).1A4, (C5×C42⋊C3)⋊3C2, (C2×C10).1(C2×A4), SmallGroup(480,262)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C4×C20 — C4⋊D20⋊C3
Chief series
C1 — C22 — C2×C10 — C4×C20 — C5×C42⋊C3 — C4⋊D20⋊C3
Lower central
C4×C20 — C4⋊D20⋊C3
Upper central
C1

Generators and relations for C4⋊D20⋊C3
 G = < a,b,c,d | a4=b20=c2=d3=1, ab=ba, cac=a-1, dad-1=a-1b15, cbc=b-1, dbd-1=ab16, dcd-1=ab15c >

Subgroups: 648 in 56 conjugacy classes, 11 normal (all characteristic)
C1, C2 [×3], C3, C4 [×2], C22, C22 [×4], C5, C6, C2×C4, D4 [×4], C23 [×2], D5 [×2], C10, A4, C15, C42, C2×D4 [×2], C20 [×2], D10 [×4], C2×C10, C2×A4, C3×D5, C4⋊1D4, D20 [×4], C2×C20, C22×D5, C22×D5, C42⋊C3, C5×A4, C4×C20, C2×D20 [×2], C23.A4, D5×A4, C4⋊D20, C5×C42⋊C3, C4⋊D20⋊C3
Quotients: C1, C2, C3, C6, D5, A4, C2×A4, C3×D5, C23.A4, D5×A4, C4⋊D20⋊C3

Character table of C4⋊D20⋊C3

 class 12A2B2C3A3B4A4B5A5B6A6B10A10B15A15B15C15D20A20B20C20D20E20F20G20H
 size 132060161666228080663232323266666666
ρ111111111111111111111111111    trivial
ρ211-1-1111111-1-111111111111111    linear of order 2
ρ311-1-1ζ3ζ321111ζ65ζ611ζ3ζ32ζ32ζ311111111    linear of order 6
ρ41111ζ3ζ321111ζ3ζ3211ζ3ζ32ζ32ζ311111111    linear of order 3
ρ51111ζ32ζ31111ζ32ζ311ζ32ζ3ζ3ζ3211111111    linear of order 3
ρ611-1-1ζ32ζ31111ζ6ζ6511ζ32ζ3ζ3ζ3211111111    linear of order 6
ρ722002222-1-√5/2-1+√5/200-1+√5/2-1-√5/2-1+√5/2-1+√5/2-1-√5/2-1-√5/2-1+√5/2-1+√5/2-1+√5/2-1+√5/2-1-√5/2-1-√5/2-1-√5/2-1-√5/2    orthogonal lifted from D5
ρ822002222-1+√5/2-1-√5/200-1-√5/2-1+√5/2-1-√5/2-1-√5/2-1+√5/2-1+√5/2-1-√5/2-1-√5/2-1-√5/2-1-√5/2-1+√5/2-1+√5/2-1+√5/2-1+√5/2    orthogonal lifted from D5
ρ92200-1+√-3-1-√-322-1-√5/2-1+√5/200-1+√5/2-1-√5/2ζ3ζ54+ζ3ζ5ζ32ζ54+ζ32ζ5ζ32ζ53+ζ32ζ52ζ3ζ53+ζ3ζ52-1+√5/2-1+√5/2-1+√5/2-1+√5/2-1-√5/2-1-√5/2-1-√5/2-1-√5/2    complex lifted from C3×D5
ρ102200-1-√-3-1+√-322-1+√5/2-1-√5/200-1-√5/2-1+√5/2ζ32ζ53+ζ32ζ52ζ3ζ53+ζ3ζ52ζ3ζ54+ζ3ζ5ζ32ζ54+ζ32ζ5-1-√5/2-1-√5/2-1-√5/2-1-√5/2-1+√5/2-1+√5/2-1+√5/2-1+√5/2    complex lifted from C3×D5
ρ112200-1+√-3-1-√-322-1+√5/2-1-√5/200-1-√5/2-1+√5/2ζ3ζ53+ζ3ζ52ζ32ζ53+ζ32ζ52ζ32ζ54+ζ32ζ5ζ3ζ54+ζ3ζ5-1-√5/2-1-√5/2-1-√5/2-1-√5/2-1+√5/2-1+√5/2-1+√5/2-1+√5/2    complex lifted from C3×D5
ρ122200-1-√-3-1+√-322-1-√5/2-1+√5/200-1+√5/2-1-√5/2ζ32ζ54+ζ32ζ5ζ3ζ54+ζ3ζ5ζ3ζ53+ζ3ζ52ζ32ζ53+ζ32ζ52-1+√5/2-1+√5/2-1+√5/2-1+√5/2-1-√5/2-1-√5/2-1-√5/2-1-√5/2    complex lifted from C3×D5
ρ13333-100-1-13300330000-1-1-1-1-1-1-1-1    orthogonal lifted from A4
ρ1433-3100-1-13300330000-1-1-1-1-1-1-1-1    orthogonal lifted from C2×A4
ρ156-20000-226600-2-20000-222-2-222-2    orthogonal lifted from C23.A4
ρ166-200002-26600-2-200002-2-222-2-22    orthogonal lifted from C23.A4
ρ17660000-2-2-3-3√5/2-3+3√5/200-3+3√5/2-3-3√5/200001-√5/21-√5/21-√5/21-√5/21+√5/21+√5/21+√5/21+√5/2    orthogonal lifted from D5×A4
ρ18660000-2-2-3+3√5/2-3-3√5/200-3-3√5/2-3+3√5/200001+√5/21+√5/21+√5/21+√5/21-√5/21-√5/21-√5/21-√5/2    orthogonal lifted from D5×A4
ρ196-200002-2-3-3√5/2-3+3√5/2001-√5/21+√5/20000-1+√5/22ζ43ζ54-2ζ43ζ5-ζ54-ζ5-2ζ43ζ54+2ζ43ζ5-ζ54-ζ5-1+√5/2-1-√5/22ζ4ζ53-2ζ4ζ52-ζ53-ζ52-2ζ4ζ53+2ζ4ζ52-ζ53-ζ52-1-√5/2    orthogonal faithful
ρ206-20000-22-3-3√5/2-3+3√5/2001-√5/21+√5/200002ζ43ζ54-2ζ43ζ5-ζ54-ζ5-1+√5/2-1+√5/2-2ζ43ζ54+2ζ43ζ5-ζ54-ζ52ζ4ζ53-2ζ4ζ52-ζ53-ζ52-1-√5/2-1-√5/2-2ζ4ζ53+2ζ4ζ52-ζ53-ζ52    orthogonal faithful
ρ216-20000-22-3-3√5/2-3+3√5/2001-√5/21+√5/20000-2ζ43ζ54+2ζ43ζ5-ζ54-ζ5-1+√5/2-1+√5/22ζ43ζ54-2ζ43ζ5-ζ54-ζ5-2ζ4ζ53+2ζ4ζ52-ζ53-ζ52-1-√5/2-1-√5/22ζ4ζ53-2ζ4ζ52-ζ53-ζ52    orthogonal faithful
ρ226-200002-2-3+3√5/2-3-3√5/2001+√5/21-√5/20000-1-√5/2-2ζ4ζ53+2ζ4ζ52-ζ53-ζ522ζ4ζ53-2ζ4ζ52-ζ53-ζ52-1-√5/2-1+√5/22ζ43ζ54-2ζ43ζ5-ζ54-ζ5-2ζ43ζ54+2ζ43ζ5-ζ54-ζ5-1+√5/2    orthogonal faithful
ρ236-20000-22-3+3√5/2-3-3√5/2001+√5/21-√5/20000-2ζ4ζ53+2ζ4ζ52-ζ53-ζ52-1-√5/2-1-√5/22ζ4ζ53-2ζ4ζ52-ζ53-ζ522ζ43ζ54-2ζ43ζ5-ζ54-ζ5-1+√5/2-1+√5/2-2ζ43ζ54+2ζ43ζ5-ζ54-ζ5    orthogonal faithful
ρ246-20000-22-3+3√5/2-3-3√5/2001+√5/21-√5/200002ζ4ζ53-2ζ4ζ52-ζ53-ζ52-1-√5/2-1-√5/2-2ζ4ζ53+2ζ4ζ52-ζ53-ζ52-2ζ43ζ54+2ζ43ζ5-ζ54-ζ5-1+√5/2-1+√5/22ζ43ζ54-2ζ43ζ5-ζ54-ζ5    orthogonal faithful
ρ256-200002-2-3-3√5/2-3+3√5/2001-√5/21+√5/20000-1+√5/2-2ζ43ζ54+2ζ43ζ5-ζ54-ζ52ζ43ζ54-2ζ43ζ5-ζ54-ζ5-1+√5/2-1-√5/2-2ζ4ζ53+2ζ4ζ52-ζ53-ζ522ζ4ζ53-2ζ4ζ52-ζ53-ζ52-1-√5/2    orthogonal faithful
ρ266-200002-2-3+3√5/2-3-3√5/2001+√5/21-√5/20000-1-√5/22ζ4ζ53-2ζ4ζ52-ζ53-ζ52-2ζ4ζ53+2ζ4ζ52-ζ53-ζ52-1-√5/2-1+√5/2-2ζ43ζ54+2ζ43ζ5-ζ54-ζ52ζ43ζ54-2ζ43ζ5-ζ54-ζ5-1+√5/2    orthogonal faithful

Smallest permutation representation of C4⋊D20⋊C3
►On 60 points
Generators in S60
(1 13 9 17)(2 14 10 18)(3 15 6 19)(4 11 7 20)(5 12 8 16)(41 56 51 46)(42 57 52 47)(43 58 53 48)(44 59 54 49)(45 60 55 50)

(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)

(1 5)(2 4)(7 10)(8 9)(11 18)(12 17)(13 16)(14 20)(15 19)(21 32)(22 31)(23 30)(24 29)(25 28)(26 27)(33 40)(34 39)(35 38)(36 37)(41 42)(43 60)(44 59)(45 58)(46 57)(47 56)(48 55)(49 54)(50 53)(51 52)

(1 52 37)(2 48 33)(3 44 29)(4 60 25)(5 56 21)(6 54 39)(7 50 35)(8 46 31)(9 42 27)(10 58 23)(11 45 40)(12 41 36)(13 57 32)(14 53 28)(15 49 24)(16 51 26)(17 47 22)(18 43 38)(19 59 34)(20 55 30)
 
G:=sub<Sym(60)| (1,13,9,17)(2,14,10,18)(3,15,6,19)(4,11,7,20)(5,12,8,16)(41,56,51,46)(42,57,52,47)(43,58,53,48)(44,59,54,49)(45,60,55,50), (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), (1,5)(2,4)(7,10)(8,9)(11,18)(12,17)(13,16)(14,20)(15,19)(21,32)(22,31)(23,30)(24,29)(25,28)(26,27)(33,40)(34,39)(35,38)(36,37)(41,42)(43,60)(44,59)(45,58)(46,57)(47,56)(48,55)(49,54)(50,53)(51,52), (1,52,37)(2,48,33)(3,44,29)(4,60,25)(5,56,21)(6,54,39)(7,50,35)(8,46,31)(9,42,27)(10,58,23)(11,45,40)(12,41,36)(13,57,32)(14,53,28)(15,49,24)(16,51,26)(17,47,22)(18,43,38)(19,59,34)(20,55,30)>;
 
G:=Group( (1,13,9,17)(2,14,10,18)(3,15,6,19)(4,11,7,20)(5,12,8,16)(41,56,51,46)(42,57,52,47)(43,58,53,48)(44,59,54,49)(45,60,55,50), (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), (1,5)(2,4)(7,10)(8,9)(11,18)(12,17)(13,16)(14,20)(15,19)(21,32)(22,31)(23,30)(24,29)(25,28)(26,27)(33,40)(34,39)(35,38)(36,37)(41,42)(43,60)(44,59)(45,58)(46,57)(47,56)(48,55)(49,54)(50,53)(51,52), (1,52,37)(2,48,33)(3,44,29)(4,60,25)(5,56,21)(6,54,39)(7,50,35)(8,46,31)(9,42,27)(10,58,23)(11,45,40)(12,41,36)(13,57,32)(14,53,28)(15,49,24)(16,51,26)(17,47,22)(18,43,38)(19,59,34)(20,55,30) );
 
G=PermutationGroup([(1,13,9,17),(2,14,10,18),(3,15,6,19),(4,11,7,20),(5,12,8,16),(41,56,51,46),(42,57,52,47),(43,58,53,48),(44,59,54,49),(45,60,55,50)], [(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)], [(1,5),(2,4),(7,10),(8,9),(11,18),(12,17),(13,16),(14,20),(15,19),(21,32),(22,31),(23,30),(24,29),(25,28),(26,27),(33,40),(34,39),(35,38),(36,37),(41,42),(43,60),(44,59),(45,58),(46,57),(47,56),(48,55),(49,54),(50,53),(51,52)], [(1,52,37),(2,48,33),(3,44,29),(4,60,25),(5,56,21),(6,54,39),(7,50,35),(8,46,31),(9,42,27),(10,58,23),(11,45,40),(12,41,36),(13,57,32),(14,53,28),(15,49,24),(16,51,26),(17,47,22),(18,43,38),(19,59,34),(20,55,30)])
 

Matrix representation of C4⋊D20⋊C3 ►in GL6(𝔽61)

3640000
57250000
001000
000100
00002557
0000436
,
43600000
100000
00343600
00255700
00002725
0000364
,
43600000
18180000
00343600
00342700
00002725
00002734
,
001000
000100
000010
000001
100000
010000

G:=sub<GL(6,GF(61))| [36,57,0,0,0,0,4,25,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,25,4,0,0,0,0,57,36],[43,1,0,0,0,0,60,0,0,0,0,0,0,0,34,25,0,0,0,0,36,57,0,0,0,0,0,0,27,36,0,0,0,0,25,4],[43,18,0,0,0,0,60,18,0,0,0,0,0,0,34,34,0,0,0,0,36,27,0,0,0,0,0,0,27,27,0,0,0,0,25,34],[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] >;
 

C4⋊D20⋊C3 in GAP, Magma, Sage, TeX 

C_4\rtimes D_{20}\rtimes C_3
 
% in TeX
 
G:=Group("C4:D20:C3");
 
// GroupNames label
 
G:=SmallGroup(480,262);
 
// by ID
 
G=gap.SmallGroup(480,262);
 
# by ID
 
G:=PCGroup([7,-2,-3,-2,2,-5,-2,2,7688,198,1276,7059,3454,584,3364,5052,8833]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^20=c^2=d^3=1,a*b=b*a,c*a*c=a^-1,d*a*d^-1=a^-1*b^15,c*b*c=b^-1,d*b*d^-1=a*b^16,d*c*d^-1=a*b^15*c>;
 
// generators/relations
 

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