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G = C24.29D4order 192 = 26·3

29th non-split extension by C24 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C24.29D4, D12.27D4, Dic6.27D4, (C6×Q16)⋊2C2, C4.69(S3×D4), (C2×C8).97D6, (C2×Q16)⋊10S3, C8○D12.1C2, (C2×Q8).92D6, C24.C44C2, C12.190(C2×D4), C34(D4.5D4), C8.29(C3⋊D4), C12.10D48C2, (C2×C24).34C22, (C6×Q8).95C22, C2.24(D63D4), C6.123(C4⋊D4), (C2×C12).466C23, Q8.11D6.2C2, C4○D12.48C22, C4.Dic3.21C22, C22.22(D42S3), C4.87(C2×C3⋊D4), (C2×C6).160(C4○D4), (C2×C4).128(C22×S3), SmallGroup(192,751)

Series: Derived Chief Lower central Upper central

C1C2×C12 — C24.29D4
C1C3C6C12C2×C12C4○D12C8○D12 — C24.29D4
C3C6C2×C12 — C24.29D4
C1C2C2×C4C2×Q16

Generators and relations for C24.29D4
 G = < a,b,c | a24=c2=1, b4=a12, bab-1=a-1, cac=a17, cbc=b3 >

Subgroups: 248 in 100 conjugacy classes, 37 normal (27 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C8, C2×C4, C2×C4, D4, Q8, Dic3, C12, C12, D6, C2×C6, C2×C8, C2×C8, M4(2), SD16, Q16, C2×Q8, C4○D4, C3⋊C8, C24, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C2×C12, C3×Q8, C4.10D4, C8.C4, C8○D4, C2×Q16, C8.C22, S3×C8, C8⋊S3, C4.Dic3, C4.Dic3, Q82S3, C3⋊Q16, C2×C24, C3×Q16, C4○D12, C6×Q8, D4.5D4, C24.C4, C12.10D4, C8○D12, Q8.11D6, C6×Q16, C24.29D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C3⋊D4, C22×S3, C4⋊D4, S3×D4, D42S3, C2×C3⋊D4, D4.5D4, D63D4, C24.29D4

Character table of C24.29D4

 class 12A2B2C34A4B4C4D4E6A6B6C8A8B8C8D8E8F8G12A12B12C12D12E12F24A24B24C24D
 size 112122228812222224121224244488884444
ρ1111111111111111111111111111111    trivial
ρ211111111-11111-1-1-1-1-1-1111-111-1-1-1-1-1    linear of order 2
ρ3111-1111-11-1111-1-1-111-11111-1-11-1-1-1-1    linear of order 2
ρ4111-1111-1-1-1111111-1-11111-1-1-1-11111    linear of order 2
ρ5111-11111-1-1111-1-1-1111-111-111-1-1-1-1-1    linear of order 2
ρ6111-111111-1111111-1-1-1-11111111111    linear of order 2
ρ71111111-1-1111111111-1-111-1-1-1-11111    linear of order 2
ρ81111111-111111-1-1-1-1-11-1111-1-11-1-1-1-1    linear of order 2
ρ92220-1222-20-1-1-1-2-2-20000-1-11-1-111111    orthogonal lifted from D6
ρ102220-122-220-1-1-1-2-2-20000-1-1-111-11111    orthogonal lifted from D6
ρ1122-2222-200-22-2-200000002-200000000    orthogonal lifted from D4
ρ1222-202-220002-2-2-2-220000-2200002-2-22    orthogonal lifted from D4
ρ132220-122-2-20-1-1-12220000-1-11111-1-1-1-1    orthogonal lifted from D6
ρ1422-2-222-20022-2-200000002-200000000    orthogonal lifted from D4
ρ152220-122220-1-1-12220000-1-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ1622-202-220002-2-222-20000-220000-222-2    orthogonal lifted from D4
ρ1722-20-1-22000-111-2-2200001-1--3--3-3-3-111-1    complex lifted from C3⋊D4
ρ1822-20-1-22000-111-2-2200001-1-3-3--3--3-111-1    complex lifted from C3⋊D4
ρ1922-20-1-22000-11122-200001-1-3--3-3--31-1-11    complex lifted from C3⋊D4
ρ2022-20-1-22000-11122-200001-1--3-3--3-31-1-11    complex lifted from C3⋊D4
ρ2122202-2-20002220002i-2i00-2-200000000    complex lifted from C4○D4
ρ2222202-2-2000222000-2i2i00-2-200000000    complex lifted from C4○D4
ρ2344-40-24-4000-2220000000-2200000000    orthogonal lifted from S3×D4
ρ244440-2-4-4000-2-2-200000002200000000    symplectic lifted from D42S3, Schur index 2
ρ254-400400000-40022-22000000000000-22220    symplectic lifted from D4.5D4, Schur index 2
ρ264-400400000-400-222200000000000022-220    symplectic lifted from D4.5D4, Schur index 2
ρ274-400-20000022-3-2-322-2200000000000-62-2--6    complex faithful
ρ284-400-2000002-2-32-3-222200000000000-6-22--6    complex faithful
ρ294-400-2000002-2-32-322-2200000000000--62-2-6    complex faithful
ρ304-400-20000022-3-2-3-222200000000000--6-22-6    complex faithful

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

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

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

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

Matrix representation of C24.29D4 in GL4(𝔽7) generated by

2121
6131
6433
2204
,
6411
1066
2134
2535
,
6121
4145
6354
5122
G:=sub<GL(4,GF(7))| [2,6,6,2,1,1,4,2,2,3,3,0,1,1,3,4],[6,1,2,2,4,0,1,5,1,6,3,3,1,6,4,5],[6,4,6,5,1,1,3,1,2,4,5,2,1,5,4,2] >;

C24.29D4 in GAP, Magma, Sage, TeX

C_{24}._{29}D_4
% in TeX

G:=Group("C24.29D4");
// GroupNames label

G:=SmallGroup(192,751);
// by ID

G=gap.SmallGroup(192,751);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,254,219,184,1123,297,136,438,102,6278]);
// Polycyclic

G:=Group<a,b,c|a^24=c^2=1,b^4=a^12,b*a*b^-1=a^-1,c*a*c=a^17,c*b*c=b^3>;
// generators/relations

Export

Character table of C24.29D4 in TeX

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