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G = C24.18D4order 192 = 26·3

18th non-split extension by C24 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C24.18D4, C8.20D12, D12.20D4, Dic6.20D4, M4(2).9D6, (C2×C8).70D6, C8.C45S3, C8○D12.2C2, C4.135(S3×D4), C4.56(C2×D12), C12.136(C2×D4), C32(D4.5D4), C8.D6.2C2, (C2×Dic12)⋊21C2, C12.47D43C2, C6.49(C4⋊D4), C2.22(C12⋊D4), (C2×C24).102C22, (C2×C12).312C23, C4○D12.39C22, C22.6(Q83S3), (C2×Dic6).94C22, (C3×M4(2)).6C22, C4.Dic3.37C22, (C3×C8.C4)⋊6C2, (C2×C6).3(C4○D4), (C2×C4).113(C22×S3), SmallGroup(192,455)

Series: Derived Chief Lower central Upper central

C1C2×C12 — C24.18D4
C1C3C6C12C2×C12C4○D12C8○D12 — C24.18D4
C3C6C2×C12 — C24.18D4
C1C2C2×C4C8.C4

Generators and relations for C24.18D4
 G = < a,b,c | a24=1, b4=c2=a12, bab-1=a7, cac-1=a-1, cbc-1=b3 >

Subgroups: 288 in 100 conjugacy classes, 37 normal (27 characteristic)
C1, C2, C2 [×2], C3, C4 [×2], C4 [×3], C22, C22, S3, C6, C6, C8 [×2], C8 [×3], C2×C4, C2×C4 [×3], D4 [×2], Q8 [×5], Dic3 [×3], C12 [×2], D6, C2×C6, C2×C8, C2×C8, M4(2) [×2], M4(2) [×2], SD16 [×2], Q16 [×4], C2×Q8 [×2], C4○D4, C3⋊C8, C24 [×2], C24 [×2], Dic6, Dic6 [×4], C4×S3, D12, C2×Dic3 [×2], C3⋊D4, C2×C12, C4.10D4 [×2], C8.C4, C8○D4, C2×Q16, C8.C22 [×2], S3×C8, C8⋊S3, C24⋊C2 [×2], Dic12 [×4], C4.Dic3, C2×C24, C3×M4(2) [×2], C2×Dic6 [×2], C4○D12, D4.5D4, C12.47D4 [×2], C3×C8.C4, C8○D12, C2×Dic12, C8.D6 [×2], C24.18D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×4], C23, D6 [×3], C2×D4 [×2], C4○D4, D12 [×2], C22×S3, C4⋊D4, C2×D12, S3×D4, Q83S3, D4.5D4, C12⋊D4, C24.18D4

Character table of C24.18D4

 class 12A2B2C34A4B4C4D4E6A6B8A8B8C8D8E8F8G12A12B12C24A24B24C24D24E24F24G24H
 size 112122221224242422488121222444448888
ρ1111111111111111111111111111111    trivial
ρ2111-1111-11111111-1-1-1-11111111-1-1-1-1    linear of order 2
ρ311111111-1-111111-1-1111111111-1-1-1-1    linear of order 2
ρ4111-1111-1-1-11111111-1-111111111111    linear of order 2
ρ5111-1111-11-111-1-1-1-1111111-1-1-1-111-1-1    linear of order 2
ρ6111111111-111-1-1-11-1-1-1111-1-1-1-1-1-111    linear of order 2
ρ7111-1111-1-1111-1-1-11-111111-1-1-1-1-1-111    linear of order 2
ρ811111111-1111-1-1-1-11-1-1111-1-1-1-111-1-1    linear of order 2
ρ92220-122000-1-12222200-1-1-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ1022-2-222-22002-2000000022-200000000    orthogonal lifted from D4
ρ112220-122000-1-1-2-2-22-200-1-1-1111111-1-1    orthogonal lifted from D6
ρ1222-202-220002-222-20000-2-2222-2-20000    orthogonal lifted from D4
ρ132220-122000-1-1-2-2-2-2200-1-1-11111-1-111    orthogonal lifted from D6
ρ1422-2222-2-2002-2000000022-200000000    orthogonal lifted from D4
ρ152220-122000-1-1222-2-200-1-1-1-1-1-1-11111    orthogonal lifted from D6
ρ1622-202-220002-2-2-220000-2-22-2-2220000    orthogonal lifted from D4
ρ1722-20-1-22000-1122-2000011-1-1-1113-3-33    orthogonal lifted from D12
ρ1822-20-1-22000-1122-2000011-1-1-111-333-3    orthogonal lifted from D12
ρ1922-20-1-22000-11-2-22000011-111-1-13-33-3    orthogonal lifted from D12
ρ2022-20-1-22000-11-2-22000011-111-1-1-33-33    orthogonal lifted from D12
ρ2122202-2-20002200000-2i2i-2-2-200000000    complex lifted from C4○D4
ρ2222202-2-200022000002i-2i-2-2-200000000    complex lifted from C4○D4
ρ234440-2-4-4000-2-2000000022200000000    orthogonal lifted from Q83S3, Schur index 2
ρ2444-40-24-4000-220000000-2-2200000000    orthogonal lifted from S3×D4
ρ254-400400000-40-22220000000022-22000000    symplectic lifted from D4.5D4, Schur index 2
ρ264-400400000-4022-2200000000-2222000000    symplectic lifted from D4.5D4, Schur index 2
ρ274-400-20000020-222200000-23230-22-660000    symplectic faithful, Schur index 2
ρ284-400-2000002022-220000023-2302-2-660000    symplectic faithful, Schur index 2
ρ294-400-2000002022-2200000-232302-26-60000    symplectic faithful, Schur index 2
ρ304-400-20000020-22220000023-230-226-60000    symplectic faithful, Schur index 2

Smallest permutation representation of C24.18D4
On 96 points
Generators in S96
(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)
(1 38 76 62 13 26 88 50)(2 45 77 69 14 33 89 57)(3 28 78 52 15 40 90 64)(4 35 79 59 16 47 91 71)(5 42 80 66 17 30 92 54)(6 25 81 49 18 37 93 61)(7 32 82 56 19 44 94 68)(8 39 83 63 20 27 95 51)(9 46 84 70 21 34 96 58)(10 29 85 53 22 41 73 65)(11 36 86 60 23 48 74 72)(12 43 87 67 24 31 75 55)
(1 62 13 50)(2 61 14 49)(3 60 15 72)(4 59 16 71)(5 58 17 70)(6 57 18 69)(7 56 19 68)(8 55 20 67)(9 54 21 66)(10 53 22 65)(11 52 23 64)(12 51 24 63)(25 77 37 89)(26 76 38 88)(27 75 39 87)(28 74 40 86)(29 73 41 85)(30 96 42 84)(31 95 43 83)(32 94 44 82)(33 93 45 81)(34 92 46 80)(35 91 47 79)(36 90 48 78)

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

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), (1,38,76,62,13,26,88,50)(2,45,77,69,14,33,89,57)(3,28,78,52,15,40,90,64)(4,35,79,59,16,47,91,71)(5,42,80,66,17,30,92,54)(6,25,81,49,18,37,93,61)(7,32,82,56,19,44,94,68)(8,39,83,63,20,27,95,51)(9,46,84,70,21,34,96,58)(10,29,85,53,22,41,73,65)(11,36,86,60,23,48,74,72)(12,43,87,67,24,31,75,55), (1,62,13,50)(2,61,14,49)(3,60,15,72)(4,59,16,71)(5,58,17,70)(6,57,18,69)(7,56,19,68)(8,55,20,67)(9,54,21,66)(10,53,22,65)(11,52,23,64)(12,51,24,63)(25,77,37,89)(26,76,38,88)(27,75,39,87)(28,74,40,86)(29,73,41,85)(30,96,42,84)(31,95,43,83)(32,94,44,82)(33,93,45,81)(34,92,46,80)(35,91,47,79)(36,90,48,78) );

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)], [(1,38,76,62,13,26,88,50),(2,45,77,69,14,33,89,57),(3,28,78,52,15,40,90,64),(4,35,79,59,16,47,91,71),(5,42,80,66,17,30,92,54),(6,25,81,49,18,37,93,61),(7,32,82,56,19,44,94,68),(8,39,83,63,20,27,95,51),(9,46,84,70,21,34,96,58),(10,29,85,53,22,41,73,65),(11,36,86,60,23,48,74,72),(12,43,87,67,24,31,75,55)], [(1,62,13,50),(2,61,14,49),(3,60,15,72),(4,59,16,71),(5,58,17,70),(6,57,18,69),(7,56,19,68),(8,55,20,67),(9,54,21,66),(10,53,22,65),(11,52,23,64),(12,51,24,63),(25,77,37,89),(26,76,38,88),(27,75,39,87),(28,74,40,86),(29,73,41,85),(30,96,42,84),(31,95,43,83),(32,94,44,82),(33,93,45,81),(34,92,46,80),(35,91,47,79),(36,90,48,78)])

Matrix representation of C24.18D4 in GL4(𝔽73) generated by

016057
57161657
016016
57165716
,
14126514
6126596
65145961
5961247
,
12141465
2661659
14656159
6594712
G:=sub<GL(4,GF(73))| [0,57,0,57,16,16,16,16,0,16,0,57,57,57,16,16],[14,61,65,59,12,26,14,6,65,59,59,12,14,6,61,47],[12,26,14,6,14,61,65,59,14,6,61,47,65,59,59,12] >;

C24.18D4 in GAP, Magma, Sage, TeX

C_{24}._{18}D_4
% in TeX

G:=Group("C24.18D4");
// GroupNames label

G:=SmallGroup(192,455);
// by ID

G=gap.SmallGroup(192,455);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,254,219,58,1123,136,438,102,6278]);
// Polycyclic

G:=Group<a,b,c|a^24=1,b^4=c^2=a^12,b*a*b^-1=a^7,c*a*c^-1=a^-1,c*b*c^-1=b^3>;
// generators/relations

Export

Character table of C24.18D4 in TeX

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