Copied to
clipboard

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 [×4], C3, C4 [×2], C4, C22, C22 [×5], S3, C6, C6 [×3], C8 [×2], C8 [×3], C2×C4, C2×C4, D4 [×6], Q8, C23 [×2], Dic3, C12 [×2], D6, C2×C6, C2×C6 [×4], C2×C8, C2×C8, M4(2) [×4], D8 [×4], SD16 [×2], C2×D4 [×2], C4○D4, C3⋊C8 [×3], C24 [×2], Dic6, C4×S3, D12, C3⋊D4, C2×C12, C3×D4 [×4], C22×C6 [×2], C4.D4 [×2], C8.C4, C8○D4, C2×D8, C8⋊C22 [×2], S3×C8, C8⋊S3, C4.Dic3, C4.Dic3 [×2], D4⋊S3 [×2], D4.S3 [×2], C2×C24, C3×D8 [×2], C4○D12, C6×D4 [×2], D4.4D4, C24.C4, C12.D4 [×2], C8○D12, D126C22 [×2], C6×D8, C24.23D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×4], C23, D6 [×3], C2×D4 [×2], C4○D4, C3⋊D4 [×2], 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 26 19 32 13 38 7 44)(2 25 20 31 14 37 8 43)(3 48 21 30 15 36 9 42)(4 47 22 29 16 35 10 41)(5 46 23 28 17 34 11 40)(6 45 24 27 18 33 12 39)
(1 35)(2 28)(3 45)(4 38)(5 31)(6 48)(7 41)(8 34)(9 27)(10 44)(11 37)(12 30)(13 47)(14 40)(15 33)(16 26)(17 43)(18 36)(19 29)(20 46)(21 39)(22 32)(23 25)(24 42)

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

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

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

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

׿
×
𝔽