Copied to
clipboard

G = C8.24D12  order 192 = 26·3

10th non-split extension by C8 of D12 acting via D12/D6=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C8.24D12, C24.42D4, D12.22D4, Dic6.22D4, M4(2).11D6, C8○D12⋊9C2, (C2×C8).72D6, C8.C4⋊7S3, C8⋊D6.2C2, C4.58(C2×D12), C4.137(S3×D4), C8.D6⋊10C2, C12.138(C2×D4), C3⋊3(D4.3D4), M4(2)⋊S3⋊4C2, C12.47D4⋊4C2, C6.51(C4⋊D4), C2.24(C12⋊D4), (C2×C12).314C23, (C2×C24).155C22, C4○D12.41C22, (C2×D12).89C22, C22.8(Q8⋊3S3), (C2×Dic6).95C22, (C3×M4(2)).8C22, C4.Dic3.39C22, (C2×C24⋊C2)⋊26C2, (C3×C8.C4)⋊8C2, (C2×C6).5(C4○D4), (C2×C4).115(C22×S3), SmallGroup(192,457)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C12 — C8.24D12
Chief series
C1 — C3 — C6 — C12 — C2×C12 — C4○D12 — C8○D12 — C8.24D12
Lower central
C3 — C6 — C2×C12 — C8.24D12
Upper central
C1 — C2 — C2×C4 — C8.C4

Generators and relations for C8.24D12
 G = < a,b,c | a8=1, b12=c2=a4, bab-1=cac-1=a3, cbc-1=b11 >

Subgroups: 352 in 104 conjugacy classes, 37 normal (all characteristic)
C1, C2, C2 [×3], C3, C4 [×2], C4 [×2], C22, C22 [×3], S3 [×2], C6, C6, C8 [×2], C8 [×3], C2×C4, C2×C4 [×2], D4 [×4], Q8 [×3], C23, Dic3 [×2], C12 [×2], D6 [×3], C2×C6, C2×C8, C2×C8, M4(2) [×2], M4(2) [×2], D8, SD16 [×4], Q16, C2×D4, C2×Q8, C4○D4, C3⋊C8, C24 [×2], C24 [×2], Dic6, Dic6 [×2], C4×S3, D12, D12 [×2], C2×Dic3, C3⋊D4, C2×C12, C22×S3, C4.D4, C4.10D4, C8.C4, C8○D4, C2×SD16, C8⋊C22, C8.C22, S3×C8, C8⋊S3, C24⋊C2 [×4], D24, Dic12, C4.Dic3, C2×C24, C3×M4(2) [×2], C2×Dic6, C2×D12, C4○D12, D4.3D4, M4(2)⋊S3, C12.47D4, C3×C8.C4, C8○D12, C2×C24⋊C2, C8⋊D6, C8.D6, C8.24D12
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, Q8⋊3S3, D4.3D4, C12⋊D4, C8.24D12

Character table of C8.24D12

 class 12A2B2C2D34A4B4C4D6A6B8A8B8C8D8E8F8G12A12B12C24A24B24C24D24E24F24G24H
 size 112122422212242422488121222444448888
ρ1111111111111111111111111111111    trivial
ρ2111-1-1111-1-11111111-1-111111111111    linear of order 2
ρ31111-11111111-1-1-1-11-1-1111-1-1-1-111-1-1    linear of order 2
ρ4111-11111-1-111-1-1-1-1111111-1-1-1-111-1-1    linear of order 2
ρ5111-1-1111-1111-1-1-11-111111-1-1-1-1-1-111    linear of order 2
ρ6111111111-111-1-1-11-1-1-1111-1-1-1-1-1-111    linear of order 2
ρ71111-11111-111111-1-1111111111-1-1-1-1    linear of order 2
ρ8111-11111-1111111-1-1-1-11111111-1-1-1-1    linear of order 2
ρ922200-12200-1-1-2-2-2-2200-1-1-11111-1-111    orthogonal lifted from D6
ρ1022-2002-22002-222-20000-2-2222-2-20000    orthogonal lifted from D4
ρ1122-2002-22002-2-2-220000-2-22-2-2220000    orthogonal lifted from D4
ρ1222-2-2022-2202-2000000022-200000000    orthogonal lifted from D4
ρ1322200-12200-1-1222-2-200-1-1-1-1-1-1-11111    orthogonal lifted from D6
ρ1422-22022-2-202-2000000022-200000000    orthogonal lifted from D4
ρ1522200-12200-1-12222200-1-1-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ1622200-12200-1-1-2-2-22-200-1-1-1111111-1-1    orthogonal lifted from D6
ρ1722-200-1-2200-1122-2000011-1-1-111√3-√3-√3√3    orthogonal lifted from D12
ρ1822-200-1-2200-1122-2000011-1-1-111-√3√3√3-√3    orthogonal lifted from D12
ρ1922-200-1-2200-11-2-22000011-111-1-1√3-√3√3-√3    orthogonal lifted from D12
ρ2022-200-1-2200-11-2-22000011-111-1-1-√3√3-√3√3    orthogonal lifted from D12
ρ21222002-2-20022000002i-2i-2-2-200000000    complex lifted from C4○D4
ρ22222002-2-2002200000-2i2i-2-2-200000000    complex lifted from C4○D4
ρ2344-400-24-400-220000000-2-2200000000    orthogonal lifted from S3×D4
ρ2444400-2-4-400-2-2000000022200000000    orthogonal lifted from Q8⋊3S3, Schur index 2
ρ254-400040000-402√-2-2√-200000000-2√-22√-2000000    complex lifted from D4.3D4
ρ264-400040000-40-2√-22√-2000000002√-2-2√-2000000    complex lifted from D4.3D4
ρ274-4000-2000020-2√-22√-2000002√3-2√30-√-2√-2√-6-√-60000    complex faithful
ρ284-4000-20000202√-2-2√-200000-2√32√30√-2-√-2√-6-√-60000    complex faithful
ρ294-4000-20000202√-2-2√-2000002√3-2√30√-2-√-2-√-6√-60000    complex faithful
ρ304-4000-2000020-2√-22√-200000-2√32√30-√-2√-2-√-6√-60000    complex faithful

Smallest permutation representation of C8.24D12
►On 48 points
Generators in S48
(1 40 7 46 13 28 19 34)(2 47 20 41 14 35 8 29)(3 42 9 48 15 30 21 36)(4 25 22 43 16 37 10 31)(5 44 11 26 17 32 23 38)(6 27 24 45 18 39 12 33)

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

Matrix representation of C8.24D12 ►in GL4(𝔽73) generated by

4862530
11375353
004811
006237
,
14661110
77111
014667
5906659
,
25116761
36486167
002562
003748
G:=sub<GL(4,GF(73))| [48,11,0,0,62,37,0,0,53,53,48,62,0,53,11,37],[14,7,0,59,66,7,14,0,11,1,66,66,10,11,7,59],[25,36,0,0,11,48,0,0,67,61,25,37,61,67,62,48] >;
 

C8.24D12 in GAP, Magma, Sage, TeX 

C_8._{24}D_{12}
 
% in TeX
 
G:=Group("C8.24D12");
 
// GroupNames label
 
G:=SmallGroup(192,457);
 
// by ID
 
G=gap.SmallGroup(192,457);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,254,555,58,1123,136,438,102,6278]);
 
// Polycyclic
 
G:=Group<a,b,c|a^8=1,b^12=c^2=a^4,b*a*b^-1=c*a*c^-1=a^3,c*b*c^-1=b^11>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁