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G = C204D4⋊C3order 480 = 25·3·5

The semidirect product of C204D4 and C3 acting faithfully

metabelian, soluble, monomial

Aliases: C204D4⋊C3, (C4×C20)⋊3C6, C42⋊C33D5, C5⋊(C23.A4), C422(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

C1C4×C20 — C204D4⋊C3
C1C22C2×C10C4×C20C5×C42⋊C3 — C204D4⋊C3
C4×C20 — C204D4⋊C3
C1

Generators and relations for C204D4⋊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, C3, C4, C22, C22, C5, C6, C2×C4, D4, C23, D5, C10, A4, C15, C42, C2×D4, C20, D10, C2×C10, C2×A4, C3×D5, C41D4, D20, C2×C20, C22×D5, C22×D5, C42⋊C3, C5×A4, C4×C20, C2×D20, C23.A4, D5×A4, C204D4, C5×C42⋊C3, C204D4⋊C3
Quotients: C1, C2, C3, C6, D5, A4, C2×A4, C3×D5, C23.A4, D5×A4, C204D4⋊C3

Character table of C204D4⋊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+5/2-1-5/2ζ3ζ543ζ5ζ32ζ5432ζ5ζ32ζ5332ζ52ζ3ζ533ζ52-1+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+5/2ζ32ζ5332ζ52ζ3ζ533ζ52ζ3ζ543ζ5ζ32ζ5432ζ5-1-5/2-1-5/2-1-5/2-1-5/2-1+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+5/2ζ3ζ533ζ52ζ32ζ5332ζ52ζ32ζ5432ζ5ζ3ζ543ζ5-1-5/2-1-5/2-1-5/2-1-5/2-1+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+5/2-1-5/2ζ32ζ5432ζ5ζ3ζ543ζ5ζ3ζ533ζ52ζ32ζ5332ζ52-1+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
ρ17660000-2-2-3-35/2-3+35/200-3+35/2-3-35/200001-5/21-5/21-5/21-5/21+5/21+5/21+5/21+5/2    orthogonal lifted from D5×A4
ρ18660000-2-2-3+35/2-3-35/200-3-35/2-3+35/200001+5/21+5/21+5/21+5/21-5/21-5/21-5/21-5/2    orthogonal lifted from D5×A4
ρ196-200002-2-3-35/2-3+35/2001-5/21+5/20000-1+5/243ζ54-2ζ43ζ5545-2ζ43ζ54+2ζ43ζ5545-1+5/2-1-5/24ζ53-2ζ4ζ525352-2ζ4ζ53+2ζ4ζ525352-1-5/2    orthogonal faithful
ρ206-20000-22-3-35/2-3+35/2001-5/21+5/2000043ζ54-2ζ43ζ5545-1+5/2-1+5/2-2ζ43ζ54+2ζ43ζ55454ζ53-2ζ4ζ525352-1-5/2-1-5/2-2ζ4ζ53+2ζ4ζ525352    orthogonal faithful
ρ216-20000-22-3-35/2-3+35/2001-5/21+5/20000-2ζ43ζ54+2ζ43ζ5545-1+5/2-1+5/243ζ54-2ζ43ζ5545-2ζ4ζ53+2ζ4ζ525352-1-5/2-1-5/24ζ53-2ζ4ζ525352    orthogonal faithful
ρ226-200002-2-3+35/2-3-35/2001+5/21-5/20000-1-5/2-2ζ4ζ53+2ζ4ζ5253524ζ53-2ζ4ζ525352-1-5/2-1+5/243ζ54-2ζ43ζ5545-2ζ43ζ54+2ζ43ζ5545-1+5/2    orthogonal faithful
ρ236-20000-22-3+35/2-3-35/2001+5/21-5/20000-2ζ4ζ53+2ζ4ζ525352-1-5/2-1-5/24ζ53-2ζ4ζ52535243ζ54-2ζ43ζ5545-1+5/2-1+5/2-2ζ43ζ54+2ζ43ζ5545    orthogonal faithful
ρ246-20000-22-3+35/2-3-35/2001+5/21-5/200004ζ53-2ζ4ζ525352-1-5/2-1-5/2-2ζ4ζ53+2ζ4ζ525352-2ζ43ζ54+2ζ43ζ5545-1+5/2-1+5/243ζ54-2ζ43ζ5545    orthogonal faithful
ρ256-200002-2-3-35/2-3+35/2001-5/21+5/20000-1+5/2-2ζ43ζ54+2ζ43ζ554543ζ54-2ζ43ζ5545-1+5/2-1-5/2-2ζ4ζ53+2ζ4ζ5253524ζ53-2ζ4ζ525352-1-5/2    orthogonal faithful
ρ266-200002-2-3+35/2-3-35/2001+5/21-5/20000-1-5/24ζ53-2ζ4ζ525352-2ζ4ζ53+2ζ4ζ525352-1-5/2-1+5/2-2ζ43ζ54+2ζ43ζ554543ζ54-2ζ43ζ5545-1+5/2    orthogonal faithful

Smallest permutation representation of C204D4⋊C3
On 60 points
Generators in S60
(1 9 16 15)(2 10 17 11)(3 6 18 12)(4 7 19 13)(5 8 20 14)(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 19)(2 18)(3 17)(4 16)(5 20)(6 10)(7 9)(11 12)(13 15)(21 34)(22 33)(23 32)(24 31)(25 30)(26 29)(27 28)(35 40)(36 39)(37 38)(41 54)(42 53)(43 52)(44 51)(45 50)(46 49)(47 48)(55 60)(56 59)(57 58)
(1 41 21)(2 57 37)(3 53 33)(4 49 29)(5 45 25)(6 58 28)(7 54 24)(8 50 40)(9 46 36)(10 42 32)(11 52 22)(12 48 38)(13 44 34)(14 60 30)(15 56 26)(16 51 31)(17 47 27)(18 43 23)(19 59 39)(20 55 35)

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

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

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

Matrix representation of C204D4⋊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] >;

C204D4⋊C3 in GAP, Magma, Sage, TeX

C_{20}\rtimes_4D_4\rtimes C_3
% in TeX

G:=Group("C20:4D4: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

Export

Character table of C204D4⋊C3 in TeX

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