Copied to
clipboard

G = C24.44D4order 192 = 26·3

44th non-split extension by C24 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C24.44D4, D12.26D4, Dic6.26D4, C8○D124C2, C4.66(S3×D4), (C2×C8).92D6, (C6×SD16)⋊1C2, (C2×SD16)⋊3S3, (C2×D4).78D6, (C2×Q8).83D6, C12.D48C2, C12.180(C2×D4), C35(D4.3D4), C8.32(C3⋊D4), C24.C410C2, Q8.11D64C2, C12.10D47C2, (C2×C24).48C22, D126C22.2C2, (C6×Q8).83C22, C2.21(D63D4), C6.118(C4⋊D4), (C2×C12).454C23, C4○D12.47C22, (C6×D4).103C22, C4.Dic3.20C22, C22.21(D42S3), C4.84(C2×C3⋊D4), (C2×C6).159(C4○D4), (C2×C4).127(C22×S3), SmallGroup(192,736)

Series: Derived Chief Lower central Upper central

C1C2×C12 — C24.44D4
C1C3C6C12C2×C12C4○D12C8○D12 — C24.44D4
C3C6C2×C12 — C24.44D4
C1C2C2×C4C2×SD16

Generators and relations for C24.44D4
 G = < a,b,c | a24=c2=1, b4=a12, bab-1=a11, cac=a17, cbc=a12b3 >

Subgroups: 280 in 104 conjugacy classes, 37 normal (all characteristic)
C1, C2, C2 [×3], C3, C4 [×2], C4 [×2], C22, C22 [×3], S3, C6, C6 [×2], C8 [×2], C8 [×3], C2×C4, C2×C4 [×2], D4 [×4], Q8 [×3], C23, Dic3, C12 [×2], C12, D6, C2×C6, C2×C6 [×2], C2×C8, C2×C8, M4(2) [×4], D8, SD16 [×4], Q16, C2×D4, C2×Q8, C4○D4, C3⋊C8 [×3], C24 [×2], Dic6, C4×S3, D12, C3⋊D4, C2×C12, C2×C12, C3×D4 [×2], C3×Q8 [×2], C22×C6, C4.D4, C4.10D4, C8.C4, C8○D4, C2×SD16, C8⋊C22, C8.C22, S3×C8, C8⋊S3, C4.Dic3 [×3], D4⋊S3, D4.S3, Q82S3, C3⋊Q16, C2×C24, C3×SD16 [×2], C4○D12, C6×D4, C6×Q8, D4.3D4, C24.C4, C12.D4, C12.10D4, C8○D12, D126C22, Q8.11D6, C6×SD16, C24.44D4
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, D4.3D4, D63D4, C24.44D4

Character table of C24.44D4

 class 12A2B2C2D34A4B4C4D6A6B6C6D6E8A8B8C8D8E8F8G12A12B12C12D24A24B24C24D
 size 112812222812222882241212242444884444
ρ1111111111111111111111111111111    trivial
ρ2111-11111-11111-1-111111-1-111-1-11111    linear of order 2
ρ311111111-1111111-1-1-1-1-1-1111-1-1-1-1-1-1    linear of order 2
ρ4111-1111111111-1-1-1-1-1-1-11-11111-1-1-1-1    linear of order 2
ρ5111-1-11111-1111-1-1-1-1-111-111111-1-1-1-1    linear of order 2
ρ61111-1111-1-111111-1-1-1111-111-1-1-1-1-1-1    linear of order 2
ρ7111-1-1111-1-1111-1-1111-1-11111-1-11111    linear of order 2
ρ81111-11111-111111111-1-1-1-111111111    linear of order 2
ρ922220-12220-1-1-1-1-12220000-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ10222-20-122-20-1-1-1112220000-1-111-1-1-1-1    orthogonal lifted from D6
ρ1122-2002-2200-2-2200-2-2200002-200-222-2    orthogonal lifted from D4
ρ1222-20222-20-2-2-22000000000-22000000    orthogonal lifted from D4
ρ1322-20-222-202-2-22000000000-22000000    orthogonal lifted from D4
ρ1422220-122-20-1-1-1-1-1-2-2-20000-1-1111111    orthogonal lifted from D6
ρ15222-20-12220-1-1-111-2-2-20000-1-1-1-11111    orthogonal lifted from D6
ρ1622-2002-2200-2-220022-200002-2002-2-22    orthogonal lifted from D4
ρ1722-200-1-220011-1--3-322-20000-11--3-3-111-1    complex lifted from C3⋊D4
ρ1822-200-1-220011-1-3--3-2-220000-11--3-31-1-11    complex lifted from C3⋊D4
ρ1922-200-1-220011-1-3--322-20000-11-3--3-111-1    complex lifted from C3⋊D4
ρ2022-200-1-220011-1--3-3-2-220000-11-3--31-1-11    complex lifted from C3⋊D4
ρ21222002-2-20022200000-2i2i00-2-2000000    complex lifted from C4○D4
ρ22222002-2-200222000002i-2i00-2-2000000    complex lifted from C4○D4
ρ2344-400-24-40022-20000000002-2000000    orthogonal lifted from S3×D4
ρ2444400-2-4-400-2-2-200000000022000000    symplectic lifted from D42S3, Schur index 2
ρ254-40004000000-400-2-22-20000000002-200-2-2    complex lifted from D4.3D4
ρ264-40004000000-4002-2-2-2000000000-2-2002-2    complex lifted from D4.3D4
ρ274-4000-200002-3-2-3200-2-22-2000000000--2-66-2    complex faithful
ρ284-4000-20000-2-32-3200-2-22-2000000000--26-6-2    complex faithful
ρ294-4000-200002-3-2-32002-2-2-2000000000-26-6--2    complex faithful
ρ304-4000-20000-2-32-32002-2-2-2000000000-2-66--2    complex faithful

Smallest permutation representation of C24.44D4
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 41 19 47 13 29 7 35)(2 28 20 34 14 40 8 46)(3 39 21 45 15 27 9 33)(4 26 22 32 16 38 10 44)(5 37 23 43 17 25 11 31)(6 48 24 30 18 36 12 42)
(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), (1,41,19,47,13,29,7,35)(2,28,20,34,14,40,8,46)(3,39,21,45,15,27,9,33)(4,26,22,32,16,38,10,44)(5,37,23,43,17,25,11,31)(6,48,24,30,18,36,12,42), (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), (1,41,19,47,13,29,7,35)(2,28,20,34,14,40,8,46)(3,39,21,45,15,27,9,33)(4,26,22,32,16,38,10,44)(5,37,23,43,17,25,11,31)(6,48,24,30,18,36,12,42), (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)], [(1,41,19,47,13,29,7,35),(2,28,20,34,14,40,8,46),(3,39,21,45,15,27,9,33),(4,26,22,32,16,38,10,44),(5,37,23,43,17,25,11,31),(6,48,24,30,18,36,12,42)], [(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 C24.44D4 in GL4(𝔽73) generated by

541900
545400
002548
002525
,
0010
00072
0100
1000
,
0010
0001
1000
0100
G:=sub<GL(4,GF(73))| [54,54,0,0,19,54,0,0,0,0,25,25,0,0,48,25],[0,0,0,1,0,0,1,0,1,0,0,0,0,72,0,0],[0,0,1,0,0,0,0,1,1,0,0,0,0,1,0,0] >;

C24.44D4 in GAP, Magma, Sage, TeX

C_{24}._{44}D_4
% in TeX

G:=Group("C24.44D4");
// GroupNames label

G:=SmallGroup(192,736);
// by ID

G=gap.SmallGroup(192,736);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,254,555,1123,297,136,438,102,6278]);
// Polycyclic

G:=Group<a,b,c|a^24=c^2=1,b^4=a^12,b*a*b^-1=a^11,c*a*c=a^17,c*b*c=a^12*b^3>;
// generators/relations

Export

Character table of C24.44D4 in TeX

׿
×
𝔽