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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, C8oD12:4C2, C4.66(S3xD4), (C2xC8).92D6, (C6xSD16):1C2, (C2xSD16):3S3, (C2xD4).78D6, (C2xQ8).83D6, C12.D4:8C2, C12.180(C2xD4), C3:5(D4.3D4), C8.32(C3:D4), C24.C4:10C2, Q8.11D6:4C2, C12.10D4:7C2, (C2xC24).48C22, D12:6C22.2C2, (C6xQ8).83C22, C2.21(D6:3D4), C6.118(C4:D4), (C2xC12).454C23, C4oD12.47C22, (C6xD4).103C22, C4.Dic3.20C22, C22.21(D4:2S3), C4.84(C2xC3:D4), (C2xC6).159(C4oD4), (C2xC4).127(C22xS3), SmallGroup(192,736)

Series: Derived Chief Lower central Upper central

C1C2xC12 — C24.44D4
C1C3C6C12C2xC12C4oD12C8oD12 — C24.44D4
C3C6C2xC12 — C24.44D4
C1C2C2xC4C2xSD16

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, C3, C4, C4, C22, C22, S3, C6, C6, C8, C8, C2xC4, C2xC4, D4, Q8, C23, Dic3, C12, C12, D6, C2xC6, C2xC6, C2xC8, C2xC8, M4(2), D8, SD16, Q16, C2xD4, C2xQ8, C4oD4, C3:C8, C24, Dic6, C4xS3, D12, C3:D4, C2xC12, C2xC12, C3xD4, C3xQ8, C22xC6, C4.D4, C4.10D4, C8.C4, C8oD4, C2xSD16, C8:C22, C8.C22, S3xC8, C8:S3, C4.Dic3, D4:S3, D4.S3, Q8:2S3, C3:Q16, C2xC24, C3xSD16, C4oD12, C6xD4, C6xQ8, D4.3D4, C24.C4, C12.D4, C12.10D4, C8oD12, D12:6C22, Q8.11D6, C6xSD16, C24.44D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, C4oD4, C3:D4, C22xS3, C4:D4, S3xD4, D4:2S3, C2xC3:D4, D4.3D4, D6:3D4, 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 C4oD4
ρ22222002-2-200222000002i-2i00-2-2000000    complex lifted from C4oD4
ρ2344-400-24-40022-20000000002-2000000    orthogonal lifted from S3xD4
ρ2444400-2-4-400-2-2-200000000022000000    symplectic lifted from D4:2S3, 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 31 19 37 13 43 7 25)(2 42 20 48 14 30 8 36)(3 29 21 35 15 41 9 47)(4 40 22 46 16 28 10 34)(5 27 23 33 17 39 11 45)(6 38 24 44 18 26 12 32)
(1 25)(2 42)(3 35)(4 28)(5 45)(6 38)(7 31)(8 48)(9 41)(10 34)(11 27)(12 44)(13 37)(14 30)(15 47)(16 40)(17 33)(18 26)(19 43)(20 36)(21 29)(22 46)(23 39)(24 32)

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

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

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

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

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