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G = C30.D4order 240 = 24·3·5

4th non-split extension by C30 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C30.4D4, C20.3D6, C151SD16, Dic62D5, D20.2S3, C12.3D10, C60.23C22, C32(Q8⋊D5), C153C87C2, C52(D4.S3), C4.16(S3×D5), (C5×Dic6)⋊4C2, (C3×D20).2C2, C6.8(C5⋊D4), C10.8(C3⋊D4), C2.5(C15⋊D4), SmallGroup(240,16)

Series: Derived Chief Lower central Upper central

C1C60 — C30.D4
C1C5C15C30C60C3×D20 — C30.D4
C15C30C60 — C30.D4
C1C2C4

Generators and relations for C30.D4
 G = < a,b,c | a20=b6=1, c2=a15, bab-1=a-1, cac-1=a9, cbc-1=a5b-1 >

20C2
6C4
10C22
20C6
4D5
3Q8
5D4
15C8
2Dic3
10C2×C6
2D10
6C20
4C3×D5
15SD16
5C3×D4
5C3⋊C8
3C52C8
3C5×Q8
2C5×Dic3
2C6×D5
5D4.S3
3Q8⋊D5

Character table of C30.D4

 class 12A2B34A4B5A5B6A6B6C8A8B10A10B1215A15B20A20B20C20D20E20F30A30B60A60B60C60D
 size 1120221222220203030224444412121212444444
ρ1111111111111111111111111111111    trivial
ρ211111-111111-1-11111111-1-1-1-1111111    linear of order 2
ρ311-111-1111-1-1111111111-1-1-1-1111111    linear of order 2
ρ411-1111111-1-1-1-111111111111111111    linear of order 2
ρ52202-20222000022-222-2-2000022-2-2-2-2    orthogonal lifted from D4
ρ622-2-12022-1110022-1-1-1220000-1-1-1-1-1-1    orthogonal lifted from D6
ρ7222-12022-1-1-10022-1-1-1220000-1-1-1-1-1-1    orthogonal lifted from S3
ρ8220222-1+5/2-1-5/220000-1+5/2-1-5/22-1+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
ρ9220222-1-5/2-1+5/220000-1-5/2-1+5/22-1-5/2-1+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
ρ1022022-2-1+5/2-1-5/220000-1+5/2-1-5/22-1+5/2-1-5/2-1+5/2-1-5/21-5/21-5/21+5/21+5/2-1-5/2-1+5/2-1+5/2-1-5/2-1-5/2-1+5/2    orthogonal lifted from D10
ρ1122022-2-1-5/2-1+5/220000-1-5/2-1+5/22-1-5/2-1+5/2-1-5/2-1+5/21+5/21+5/21-5/21-5/2-1+5/2-1-5/2-1-5/2-1+5/2-1+5/2-1-5/2    orthogonal lifted from D10
ρ12220-1-2022-1-3--300221-1-1-2-20000-1-11111    complex lifted from C3⋊D4
ρ13220-1-2022-1--3-300221-1-1-2-20000-1-11111    complex lifted from C3⋊D4
ρ142202-20-1-5/2-1+5/220000-1-5/2-1+5/2-2-1-5/2-1+5/21+5/21-5/2ζ53525352ζ545545-1+5/2-1-5/21+5/21-5/21-5/21+5/2    complex lifted from C5⋊D4
ρ152202-20-1+5/2-1-5/220000-1+5/2-1-5/2-2-1+5/2-1-5/21-5/21+5/2ζ5455455352ζ5352-1-5/2-1+5/21-5/21+5/21+5/21-5/2    complex lifted from C5⋊D4
ρ162202-20-1-5/2-1+5/220000-1-5/2-1+5/2-2-1-5/2-1+5/21+5/21-5/25352ζ5352545ζ545-1+5/2-1-5/21+5/21-5/21-5/21+5/2    complex lifted from C5⋊D4
ρ172202-20-1+5/2-1-5/220000-1+5/2-1-5/2-2-1+5/2-1-5/21-5/21+5/2545ζ545ζ53525352-1-5/2-1+5/21-5/21+5/21+5/21-5/2    complex lifted from C5⋊D4
ρ182-2020022-200-2--2-2-2022000000-2-20000    complex lifted from SD16
ρ192-2020022-200--2-2-2-2022000000-2-20000    complex lifted from SD16
ρ20440-240-1-5-1+5-20000-1-5-1+5-21+5/21-5/2-1-5-1+500001-5/21+5/21+5/21-5/21-5/21+5/2    orthogonal lifted from S3×D5
ρ21440-240-1+5-1-5-20000-1+5-1-5-21-5/21+5/2-1+5-1-500001+5/21-5/21-5/21+5/21+5/21-5/2    orthogonal lifted from S3×D5
ρ224-40400-1-5-1+5-400001+51-50-1-5-1+50000001-51+50000    orthogonal lifted from Q8⋊D5, Schur index 2
ρ234-40400-1+5-1-5-400001-51+50-1+5-1-50000001+51-50000    orthogonal lifted from Q8⋊D5, Schur index 2
ρ244-40-2004420000-4-40-2-2000000220000    symplectic lifted from D4.S3, Schur index 2
ρ25440-2-40-1+5-1-5-20000-1+5-1-521-5/21+5/21-51+500001+5/21-5/2-1+5/2-1-5/2-1-5/2-1+5/2    symplectic lifted from C15⋊D4, Schur index 2
ρ26440-2-40-1-5-1+5-20000-1-5-1+521+5/21-5/21+51-500001-5/21+5/2-1-5/2-1+5/2-1+5/2-1-5/2    symplectic lifted from C15⋊D4, Schur index 2
ρ274-40-200-1-5-1+5200001+51-501+5/21-5/2000000-1+5/2-1-5/2-2ζ4ζ3ζ53+2ζ4ζ3ζ524ζ534ζ524ζ3ζ54-2ζ4ζ3ζ54ζ544ζ543ζ3ζ54-2ζ43ζ3ζ543ζ5443ζ5-2ζ43ζ3ζ53+2ζ43ζ3ζ5243ζ5343ζ52    complex faithful
ρ284-40-200-1-5-1+5200001+51-501+5/21-5/2000000-1+5/2-1-5/2-2ζ43ζ3ζ53+2ζ43ζ3ζ5243ζ5343ζ5243ζ3ζ54-2ζ43ζ3ζ543ζ5443ζ54ζ3ζ54-2ζ4ζ3ζ54ζ544ζ5-2ζ4ζ3ζ53+2ζ4ζ3ζ524ζ534ζ52    complex faithful
ρ294-40-200-1+5-1-5200001-51+501-5/21+5/2000000-1-5/2-1+5/24ζ3ζ54-2ζ4ζ3ζ54ζ544ζ5-2ζ43ζ3ζ53+2ζ43ζ3ζ5243ζ5343ζ52-2ζ4ζ3ζ53+2ζ4ζ3ζ524ζ534ζ5243ζ3ζ54-2ζ43ζ3ζ543ζ5443ζ5    complex faithful
ρ304-40-200-1+5-1-5200001-51+501-5/21+5/2000000-1-5/2-1+5/243ζ3ζ54-2ζ43ζ3ζ543ζ5443ζ5-2ζ4ζ3ζ53+2ζ4ζ3ζ524ζ534ζ52-2ζ43ζ3ζ53+2ζ43ζ3ζ5243ζ5343ζ524ζ3ζ54-2ζ4ζ3ζ54ζ544ζ5    complex faithful

Smallest permutation representation of C30.D4
On 120 points
Generators in S120
(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)(81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100)(101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120)
(1 79 34)(2 78 35 20 80 33)(3 77 36 19 61 32)(4 76 37 18 62 31)(5 75 38 17 63 30)(6 74 39 16 64 29)(7 73 40 15 65 28)(8 72 21 14 66 27)(9 71 22 13 67 26)(10 70 23 12 68 25)(11 69 24)(41 119 100 60 120 99)(42 118 81 59 101 98)(43 117 82 58 102 97)(44 116 83 57 103 96)(45 115 84 56 104 95)(46 114 85 55 105 94)(47 113 86 54 106 93)(48 112 87 53 107 92)(49 111 88 52 108 91)(50 110 89 51 109 90)
(1 43 16 58 11 53 6 48)(2 52 17 47 12 42 7 57)(3 41 18 56 13 51 8 46)(4 50 19 45 14 60 9 55)(5 59 20 54 15 49 10 44)(21 105 36 120 31 115 26 110)(22 114 37 109 32 104 27 119)(23 103 38 118 33 113 28 108)(24 112 39 107 34 102 29 117)(25 101 40 116 35 111 30 106)(61 100 76 95 71 90 66 85)(62 89 77 84 72 99 67 94)(63 98 78 93 73 88 68 83)(64 87 79 82 74 97 69 92)(65 96 80 91 75 86 70 81)

G:=sub<Sym(120)| (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)(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100)(101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120), (1,79,34)(2,78,35,20,80,33)(3,77,36,19,61,32)(4,76,37,18,62,31)(5,75,38,17,63,30)(6,74,39,16,64,29)(7,73,40,15,65,28)(8,72,21,14,66,27)(9,71,22,13,67,26)(10,70,23,12,68,25)(11,69,24)(41,119,100,60,120,99)(42,118,81,59,101,98)(43,117,82,58,102,97)(44,116,83,57,103,96)(45,115,84,56,104,95)(46,114,85,55,105,94)(47,113,86,54,106,93)(48,112,87,53,107,92)(49,111,88,52,108,91)(50,110,89,51,109,90), (1,43,16,58,11,53,6,48)(2,52,17,47,12,42,7,57)(3,41,18,56,13,51,8,46)(4,50,19,45,14,60,9,55)(5,59,20,54,15,49,10,44)(21,105,36,120,31,115,26,110)(22,114,37,109,32,104,27,119)(23,103,38,118,33,113,28,108)(24,112,39,107,34,102,29,117)(25,101,40,116,35,111,30,106)(61,100,76,95,71,90,66,85)(62,89,77,84,72,99,67,94)(63,98,78,93,73,88,68,83)(64,87,79,82,74,97,69,92)(65,96,80,91,75,86,70,81)>;

G:=Group( (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)(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100)(101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120), (1,79,34)(2,78,35,20,80,33)(3,77,36,19,61,32)(4,76,37,18,62,31)(5,75,38,17,63,30)(6,74,39,16,64,29)(7,73,40,15,65,28)(8,72,21,14,66,27)(9,71,22,13,67,26)(10,70,23,12,68,25)(11,69,24)(41,119,100,60,120,99)(42,118,81,59,101,98)(43,117,82,58,102,97)(44,116,83,57,103,96)(45,115,84,56,104,95)(46,114,85,55,105,94)(47,113,86,54,106,93)(48,112,87,53,107,92)(49,111,88,52,108,91)(50,110,89,51,109,90), (1,43,16,58,11,53,6,48)(2,52,17,47,12,42,7,57)(3,41,18,56,13,51,8,46)(4,50,19,45,14,60,9,55)(5,59,20,54,15,49,10,44)(21,105,36,120,31,115,26,110)(22,114,37,109,32,104,27,119)(23,103,38,118,33,113,28,108)(24,112,39,107,34,102,29,117)(25,101,40,116,35,111,30,106)(61,100,76,95,71,90,66,85)(62,89,77,84,72,99,67,94)(63,98,78,93,73,88,68,83)(64,87,79,82,74,97,69,92)(65,96,80,91,75,86,70,81) );

G=PermutationGroup([(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),(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100),(101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120)], [(1,79,34),(2,78,35,20,80,33),(3,77,36,19,61,32),(4,76,37,18,62,31),(5,75,38,17,63,30),(6,74,39,16,64,29),(7,73,40,15,65,28),(8,72,21,14,66,27),(9,71,22,13,67,26),(10,70,23,12,68,25),(11,69,24),(41,119,100,60,120,99),(42,118,81,59,101,98),(43,117,82,58,102,97),(44,116,83,57,103,96),(45,115,84,56,104,95),(46,114,85,55,105,94),(47,113,86,54,106,93),(48,112,87,53,107,92),(49,111,88,52,108,91),(50,110,89,51,109,90)], [(1,43,16,58,11,53,6,48),(2,52,17,47,12,42,7,57),(3,41,18,56,13,51,8,46),(4,50,19,45,14,60,9,55),(5,59,20,54,15,49,10,44),(21,105,36,120,31,115,26,110),(22,114,37,109,32,104,27,119),(23,103,38,118,33,113,28,108),(24,112,39,107,34,102,29,117),(25,101,40,116,35,111,30,106),(61,100,76,95,71,90,66,85),(62,89,77,84,72,99,67,94),(63,98,78,93,73,88,68,83),(64,87,79,82,74,97,69,92),(65,96,80,91,75,86,70,81)])

C30.D4 is a maximal subgroup of
C4014D6  C408D6  D405S3  D30.3D4  D20.34D6  C60.36D4  D20.37D6  D60.C22  D5×D4.S3  D20.9D6  D20.24D6  S3×Q8⋊D5  D20.13D6  D20.D6  D20.28D6
C30.D4 is a maximal quotient of
C30.D8  Dic6⋊Dic5  C30.SD16

Matrix representation of C30.D4 in GL6(𝔽241)

24000000
02400000
00024000
00118900
0000230192
00003211
,
1500000
20160000
001895200
002405200
000010
000093240
,
77400000
1531640000
001895200
002405200
00005133
0000126228

G:=sub<GL(6,GF(241))| [240,0,0,0,0,0,0,240,0,0,0,0,0,0,0,1,0,0,0,0,240,189,0,0,0,0,0,0,230,32,0,0,0,0,192,11],[15,20,0,0,0,0,0,16,0,0,0,0,0,0,189,240,0,0,0,0,52,52,0,0,0,0,0,0,1,93,0,0,0,0,0,240],[77,153,0,0,0,0,40,164,0,0,0,0,0,0,189,240,0,0,0,0,52,52,0,0,0,0,0,0,51,126,0,0,0,0,33,228] >;

C30.D4 in GAP, Magma, Sage, TeX

C_{30}.D_4
% in TeX

G:=Group("C30.D4");
// GroupNames label

G:=SmallGroup(240,16);
// by ID

G=gap.SmallGroup(240,16);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-5,73,55,218,116,50,490,6917]);
// Polycyclic

G:=Group<a,b,c|a^20=b^6=1,c^2=a^15,b*a*b^-1=a^-1,c*a*c^-1=a^9,c*b*c^-1=a^5*b^-1>;
// generators/relations

Export

Subgroup lattice of C30.D4 in TeX
Character table of C30.D4 in TeX

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