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G = C24.23D4order 192 = 26·3

23rd non-split extension by C24 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C24.23D4, D12.25D4, Dic6.25D4, (C6×D8)⋊2C2, C8○D121C2, (C2×D8)⋊10S3, (C2×C8).87D6, C4.61(S3×D4), (C2×D4).67D6, C24.C43C2, D126C224C2, C12.169(C2×D4), C12.D47C2, C34(D4.4D4), C8.27(C3⋊D4), (C2×C24).32C22, (C6×D4).86C22, C2.18(D63D4), C6.111(C4⋊D4), (C2×C12).437C23, C4○D12.46C22, C4.Dic3.19C22, C22.20(D42S3), C4.81(C2×C3⋊D4), (C2×C6).158(C4○D4), (C2×C4).126(C22×S3), SmallGroup(192,719)

Series: Derived Chief Lower central Upper central

C1C2×C12 — C24.23D4
C1C3C6C12C2×C12C4○D12C8○D12 — C24.23D4
C3C6C2×C12 — C24.23D4
C1C2C2×C4C2×D8

Generators and relations for C24.23D4
 G = < a,b,c | a24=c2=1, b4=a12, bab-1=a-1, cac=a17, cbc=a12b3 >

Subgroups: 312 in 108 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, C23, Dic3, C12, D6, C2×C6, C2×C6, C2×C8, C2×C8, M4(2), D8, SD16, C2×D4, C4○D4, C3⋊C8, C24, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C3×D4, C22×C6, C4.D4, C8.C4, C8○D4, C2×D8, C8⋊C22, S3×C8, C8⋊S3, C4.Dic3, C4.Dic3, D4⋊S3, D4.S3, C2×C24, C3×D8, C4○D12, C6×D4, D4.4D4, C24.C4, C12.D4, C8○D12, D126C22, C6×D8, C24.23D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C3⋊D4, C22×S3, C4⋊D4, S3×D4, D42S3, C2×C3⋊D4, D4.4D4, D63D4, C24.23D4

Character table of C24.23D4

 class 12A2B2C2D2E34A4B4C6A6B6C6D6E6F6G8A8B8C8D8E8F8G12A12B24A24B24C24D
 size 112881222212222888822412122424444444
ρ1111111111111111111111111111111    trivial
ρ2111-1-111111111-1-1-1-111111-1-1111111    linear of order 2
ρ311111-1111-11111111111-1-1-1-1111111    linear of order 2
ρ4111-1-1-1111-1111-1-1-1-1111-1-111111111    linear of order 2
ρ51111-1111111111-11-1-1-1-1-1-1-1111-1-1-1-1    linear of order 2
ρ6111-1111111111-11-11-1-1-1-1-11-111-1-1-1-1    linear of order 2
ρ71111-1-1111-11111-11-1-1-1-1111-111-1-1-1-1    linear of order 2
ρ8111-11-1111-1111-11-11-1-1-111-1111-1-1-1-1    linear of order 2
ρ9222-220-1220-1-1-11-11-1-2-2-20000-1-11111    orthogonal lifted from D6
ρ10222220-1220-1-1-1-1-1-1-12220000-1-1-1-1-1-1    orthogonal lifted from S3
ρ1122-200022-20-2-220000-2-2200002-2-2-222    orthogonal lifted from D4
ρ1222-20022-22-2-2-2200000000000-220000    orthogonal lifted from D4
ρ1322-200022-20-2-22000022-200002-222-2-2    orthogonal lifted from D4
ρ14222-2-20-1220-1-1-111112220000-1-1-1-1-1-1    orthogonal lifted from D6
ρ1522-200-22-222-2-2200000000000-220000    orthogonal lifted from D4
ρ162222-20-1220-1-1-1-11-11-2-2-20000-1-11111    orthogonal lifted from D6
ρ1722-2000-12-2011-1-3--3--3-3-2-220000-1111-1-1    complex lifted from C3⋊D4
ρ1822-2000-12-2011-1-3-3--3--322-20000-11-1-111    complex lifted from C3⋊D4
ρ1922-2000-12-2011-1--3--3-3-322-20000-11-1-111    complex lifted from C3⋊D4
ρ2022-2000-12-2011-1--3-3-3--3-2-220000-1111-1-1    complex lifted from C3⋊D4
ρ212220002-2-202220000000-2i2i00-2-20000    complex lifted from C4○D4
ρ222220002-2-2022200000002i-2i00-2-20000    complex lifted from C4○D4
ρ2344-4000-2-44022-2000000000002-20000    orthogonal lifted from S3×D4
ρ244-40000400000-40000-2222000000022-2200    orthogonal lifted from D4.4D4
ρ254-40000400000-4000022-220000000-222200    orthogonal lifted from D4.4D4
ρ26444000-2-4-40-2-2-200000000000220000    symplectic lifted from D42S3, Schur index 2
ρ274-40000-20002-3-2-32000022-2200000002-2-6--6    complex faithful
ρ284-40000-2000-2-32-32000022-2200000002-2--6-6    complex faithful
ρ294-40000-20002-3-2-320000-22220000000-22--6-6    complex faithful
ρ304-40000-2000-2-32-320000-22220000000-22-6--6    complex faithful

Smallest permutation representation of C24.23D4
On 48 points
Generators in S48
(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)
(1 48 19 30 13 36 7 42)(2 47 20 29 14 35 8 41)(3 46 21 28 15 34 9 40)(4 45 22 27 16 33 10 39)(5 44 23 26 17 32 11 38)(6 43 24 25 18 31 12 37)
(1 33)(2 26)(3 43)(4 36)(5 29)(6 46)(7 39)(8 32)(9 25)(10 42)(11 35)(12 28)(13 45)(14 38)(15 31)(16 48)(17 41)(18 34)(19 27)(20 44)(21 37)(22 30)(23 47)(24 40)

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

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), (1,48,19,30,13,36,7,42)(2,47,20,29,14,35,8,41)(3,46,21,28,15,34,9,40)(4,45,22,27,16,33,10,39)(5,44,23,26,17,32,11,38)(6,43,24,25,18,31,12,37), (1,33)(2,26)(3,43)(4,36)(5,29)(6,46)(7,39)(8,32)(9,25)(10,42)(11,35)(12,28)(13,45)(14,38)(15,31)(16,48)(17,41)(18,34)(19,27)(20,44)(21,37)(22,30)(23,47)(24,40) );

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)], [(1,48,19,30,13,36,7,42),(2,47,20,29,14,35,8,41),(3,46,21,28,15,34,9,40),(4,45,22,27,16,33,10,39),(5,44,23,26,17,32,11,38),(6,43,24,25,18,31,12,37)], [(1,33),(2,26),(3,43),(4,36),(5,29),(6,46),(7,39),(8,32),(9,25),(10,42),(11,35),(12,28),(13,45),(14,38),(15,31),(16,48),(17,41),(18,34),(19,27),(20,44),(21,37),(22,30),(23,47),(24,40)]])

Matrix representation of C24.23D4 in GL4(𝔽7) generated by

2121
6131
6433
2204
,
1361
3015
1350
3521
,
5100
4200
6351
5142
G:=sub<GL(4,GF(7))| [2,6,6,2,1,1,4,2,2,3,3,0,1,1,3,4],[1,3,1,3,3,0,3,5,6,1,5,2,1,5,0,1],[5,4,6,5,1,2,3,1,0,0,5,4,0,0,1,2] >;

C24.23D4 in GAP, Magma, Sage, TeX

C_{24}._{23}D_4
% in TeX

G:=Group("C24.23D4");
// GroupNames label

G:=SmallGroup(192,719);
// by ID

G=gap.SmallGroup(192,719);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,254,219,1123,297,136,438,102,6278]);
// 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=a^12*b^3>;
// generators/relations

Export

Character table of C24.23D4 in TeX

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