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G = (C4×C20)⋊C6order 480 = 25·3·5

1st semidirect product of C4×C20 and C6 acting faithfully

metabelian, soluble, monomial

Aliases: (C4×C20)⋊1C6, C422D5⋊C3, C42⋊C31D5, C5⋊(C42⋊C6), C421(C3×D5), C22.2(D5×A4), (C22×D5).2A4, (C5×C42⋊C3)⋊1C2, (C2×C10).2(C2×A4), SmallGroup(480,263)

Series: Derived Chief Lower central Upper central

C1C4×C20 — (C4×C20)⋊C6
C1C22C2×C10C4×C20C5×C42⋊C3 — (C4×C20)⋊C6
C4×C20 — (C4×C20)⋊C6
C1

Generators and relations for (C4×C20)⋊C6
 G = < a,b,c | a4=b20=c6=1, ab=ba, cac-1=a-1b5, cbc-1=a-1b14 >

3C2
20C2
16C3
6C4
30C22
30C4
80C6
3C10
4D5
16C15
3C2×C4
5C23
15C2×C4
4A4
6Dic5
6D10
6C20
16C3×D5
15C4⋊C4
15C22⋊C4
20C2×A4
3C2×C20
3C2×Dic5
4C5×A4
5C422C2
3D10⋊C4
3C10.D4
4D5×A4
5C42⋊C6

Character table of (C4×C20)⋊C6

 class 12A2B3A3B4A4B4C5A5B6A6B10A10B15A15B15C15D20A20B20C20D20E20F20G20H
 size 132016166660228080663232323266666666
ρ111111111111111111111111111    trivial
ρ211-11111-111-1-111111111111111    linear of order 2
ρ311-1ζ32ζ311-111ζ65ζ611ζ3ζ32ζ32ζ311111111    linear of order 6
ρ4111ζ32ζ311111ζ3ζ3211ζ3ζ32ζ32ζ311111111    linear of order 3
ρ511-1ζ3ζ3211-111ζ6ζ6511ζ32ζ3ζ3ζ3211111111    linear of order 6
ρ6111ζ3ζ3211111ζ32ζ311ζ32ζ3ζ3ζ3211111111    linear of order 3
ρ722022220-1-5/2-1+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
ρ822022220-1+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
ρ9220-1--3-1+-3220-1-5/2-1+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
ρ10220-1+-3-1--3220-1-5/2-1+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
ρ11220-1--3-1+-3220-1+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
ρ12220-1+-3-1--3220-1+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
ρ1333300-1-1-13300330000-1-1-1-1-1-1-1-1    orthogonal lifted from A4
ρ1433-300-1-113300330000-1-1-1-1-1-1-1-1    orthogonal lifted from C2×A4
ρ1566000-2-20-3+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
ρ1666000-2-20-3-35/2-3+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
ρ176-2000-2i2i06600-2-200002i-2i-2i2i2i-2i-2i2i    complex lifted from C42⋊C6
ρ186-20002i-2i06600-2-20000-2i2i2i-2i-2i2i2i-2i    complex lifted from C42⋊C6
ρ196-2000-2i2i0-3-35/2-3+35/2001-5/21+5/200004ζ5454543ζ554543ζ545454ζ55454ζ52535243ζ53535243ζ5253524ζ535352    complex faithful
ρ206-2000-2i2i0-3+35/2-3-35/2001+5/21-5/200004ζ52535243ζ53535243ζ5253524ζ5353524ζ554543ζ5454543ζ55454ζ54545    complex faithful
ρ216-20002i-2i0-3+35/2-3-35/2001+5/21-5/2000043ζ5353524ζ5253524ζ53535243ζ52535243ζ545454ζ55454ζ5454543ζ5545    complex faithful
ρ226-2000-2i2i0-3-35/2-3+35/2001-5/21+5/200004ζ554543ζ5454543ζ55454ζ545454ζ53535243ζ52535243ζ5353524ζ525352    complex faithful
ρ236-20002i-2i0-3-35/2-3+35/2001-5/21+5/2000043ζ55454ζ545454ζ554543ζ5454543ζ5353524ζ5253524ζ53535243ζ525352    complex faithful
ρ246-20002i-2i0-3-35/2-3+35/2001-5/21+5/2000043ζ545454ζ55454ζ5454543ζ554543ζ5253524ζ5353524ζ52535243ζ535352    complex faithful
ρ256-2000-2i2i0-3+35/2-3-35/2001+5/21-5/200004ζ53535243ζ52535243ζ5353524ζ5253524ζ5454543ζ554543ζ545454ζ5545    complex faithful
ρ266-20002i-2i0-3+35/2-3-35/2001+5/21-5/2000043ζ5253524ζ5353524ζ52535243ζ53535243ζ55454ζ545454ζ554543ζ54545    complex faithful

Smallest permutation representation of (C4×C20)⋊C6
On 80 points
Generators in S80
(1 77 30 44)(2 78 31 45)(3 79 32 46)(4 80 33 47)(5 61 34 48)(6 62 35 49)(7 63 36 50)(8 64 37 51)(9 65 38 52)(10 66 39 53)(11 67 40 54)(12 68 21 55)(13 69 22 56)(14 70 23 57)(15 71 24 58)(16 72 25 59)(17 73 26 60)(18 74 27 41)(19 75 28 42)(20 76 29 43)
(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)
(2 53 73 39 55 58)(3 28 22 19 32 38)(4 64 52 37 70 79)(5 17)(6 49 77 35 59 54)(7 24 26 15 36 34)(8 80 56 33 74 75)(9 13)(10 45 61 31 43 50)(11 40 30)(12 76 60 29 78 71)(14 41 65 27 47 46)(16 72 44 25 62 67)(18 57 69 23 51 42)(20 68 48 21 66 63)

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

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

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

Matrix representation of (C4×C20)⋊C6 in GL8(𝔽61)

10000000
01000000
003153591138
00325826525038
0029532491123
000005000
00254935909
00121200059
,
060000000
117000000
00000010
00505000600
001133224610
00012123900
00001000
000000050
,
1338000000
048000000
00100000
00606060000
00010000
0023491911270
0030637525023
00045451100

G:=sub<GL(8,GF(61))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,31,32,29,0,25,12,0,0,5,58,53,0,49,12,0,0,35,26,24,0,35,0,0,0,9,52,9,50,9,0,0,0,11,50,11,0,0,0,0,0,38,38,23,0,9,59],[0,1,0,0,0,0,0,0,60,17,0,0,0,0,0,0,0,0,0,50,11,0,0,0,0,0,0,50,33,12,0,0,0,0,0,0,22,12,1,0,0,0,0,0,46,39,0,0,0,0,1,60,1,0,0,0,0,0,0,0,0,0,0,50],[13,0,0,0,0,0,0,0,38,48,0,0,0,0,0,0,0,0,1,60,0,23,30,0,0,0,0,60,1,49,6,45,0,0,0,60,0,19,37,45,0,0,0,0,0,11,52,11,0,0,0,0,0,27,50,0,0,0,0,0,0,0,23,0] >;

(C4×C20)⋊C6 in GAP, Magma, Sage, TeX

(C_4\times C_{20})\rtimes C_6
% in TeX

G:=Group("(C4xC20):C6");
// GroupNames label

G:=SmallGroup(480,263);
// by ID

G=gap.SmallGroup(480,263);
# by ID

G:=PCGroup([7,-2,-3,-2,2,-5,-2,2,7688,198,856,7059,1774,304,3364,5052,8833]);
// Polycyclic

G:=Group<a,b,c|a^4=b^20=c^6=1,a*b=b*a,c*a*c^-1=a^-1*b^5,c*b*c^-1=a^-1*b^14>;
// generators/relations

Export

Subgroup lattice of (C4×C20)⋊C6 in TeX
Character table of (C4×C20)⋊C6 in TeX

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