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G = C24.8D4  order 192 = 26·3

8th non-split extension by C24 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C24.8D4, C12.49D8, C8.18D12, Dic12.4C4, C8.3(C4×S3), (C2×C8).44D6, C24.23(C2×C4), C4.5(D6⋊C4), (C2×C12).96D4, (C2×C6).5SD16, C4.22(D4⋊S3), C8.C4.3S3, C3⋊1(C8.17D4), C12.C8.3C2, C6.8(D4⋊C4), C12.5(C22⋊C4), (C2×C24).101C22, (C2×Dic12).12C2, C2.10(C6.D8), C22.4(Q8⋊2S3), (C3×C8.C4).2C2, (C2×C4).19(C3⋊D4), SmallGroup(192,55)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C24 — C24.8D4
Chief series
C1 — C3 — C6 — C12 — C2×C12 — C2×C24 — C2×Dic12 — C24.8D4
Lower central
C3 — C6 — C12 — C24 — C24.8D4
Upper central
C1 — C2 — C2×C4 — C2×C8 — C8.C4

Generators and relations for C24.8D4
 G = < a,b,c | a24=1, b4=a12, c2=a21, bab-1=a7, cac-1=a17, cbc-1=a21b3 >

C1
C2
2C2
C3
C4
C22
C4
12C4
12C4
C6
2C6
C8
C2×C4
C8
4C8
6Q8
6Q8
12C2×C4
12Q8
C12
C12
C2×C6
4Dic3
4Dic3
C2×C8
2M4(2)
3Q16
3Q16
6C16
6Q16
6C2×Q8
C2×C12
C24
C24
2Dic6
2Dic6
4Dic6
4C24
4C2×Dic3
C8.C4
3M5(2)
3C2×Q16
Dic12
Dic12
C2×C24
2Dic12
2C3⋊C16
2C3×M4(2)
2C2×Dic6
3C8.17D4
C12.C8
C3×C8.C4
C2×Dic12
C24.8D4

Character table of C24.8D4

 class 12A2B34A4B4C4D6A6B8A8B8C8D8E12A12B12C16A16B16C16D24A24B24C24D24E24F24G24H
 size 112222242424224882241212121244448888
ρ1111111111111111111111111111111    trivial
ρ21111111111111-1-1111-1-1-1-11111-1-1-1-1    linear of order 2
ρ3111111-1-11111111111-1-1-1-111111111    linear of order 2
ρ4111111-1-111111-1-111111111111-1-1-1-1    linear of order 2
ρ511-11-111-11-1-1-11-ii-1-11-ii-ii-11-11i-i-ii    linear of order 4
ρ611-11-11-111-1-1-11-ii-1-11i-ii-i-11-11i-i-ii    linear of order 4
ρ711-11-11-111-1-1-11i-i-1-11-ii-ii-11-11-iii-i    linear of order 4
ρ811-11-111-11-1-1-11i-i-1-11i-ii-i-11-11-iii-i    linear of order 4
ρ92222220022-2-2-2002220000-2-2-2-20000    orthogonal lifted from D4
ρ10222-12200-1-122222-1-1-10000-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ1122-22-22002-222-200-2-2200002-22-20000    orthogonal lifted from D4
ρ12222-12200-1-1222-2-2-1-1-10000-1-1-1-11111    orthogonal lifted from D6
ρ1322-2-1-2200-1122-20011-10000-11-11-√3-√3√3√3    orthogonal lifted from D12
ρ1422-2-1-2200-1122-20011-10000-11-11√3√3-√3-√3    orthogonal lifted from D12
ρ1522-222-2002-20000022-2-√2-√2√2√200000000    orthogonal lifted from D8
ρ1622-222-2002-20000022-2√2√2-√2-√200000000    orthogonal lifted from D8
ρ17222-12200-1-1-2-2-200-1-1-100001111-√-3√-3-√-3√-3    complex lifted from C3⋊D4
ρ18222-12200-1-1-2-2-200-1-1-100001111√-3-√-3√-3-√-3    complex lifted from C3⋊D4
ρ1922-2-1-2200-11-2-222i-2i11-100001-11-1i-i-ii    complex lifted from C4×S3
ρ2022-2-1-2200-11-2-22-2i2i11-100001-11-1-iii-i    complex lifted from C4×S3
ρ212222-2-2002200000-2-2-2-√-2√-2√-2-√-200000000    complex lifted from SD16
ρ222222-2-2002200000-2-2-2√-2-√-2-√-2√-200000000    complex lifted from SD16
ρ2344-4-24-400-2200000-2-22000000000000    orthogonal lifted from D4⋊S3, Schur index 2
ρ24444-2-4-400-2-200000222000000000000    orthogonal lifted from Q8⋊2S3
ρ254-4040000-40-2√22√200000000002√20-2√200000    symplectic lifted from C8.17D4, Schur index 2
ρ264-4040000-402√2-2√20000000000-2√202√200000    symplectic lifted from C8.17D4, Schur index 2
ρ274-40-20000202√2-2√20002√3-2√300000√2-√6-√2√60000    symplectic faithful, Schur index 2
ρ284-40-2000020-2√22√2000-2√32√300000-√2-√6√2√60000    symplectic faithful, Schur index 2
ρ294-40-20000202√2-2√2000-2√32√300000√2√6-√2-√60000    symplectic faithful, Schur index 2
ρ304-40-2000020-2√22√20002√3-2√300000-√2√6√2-√60000    symplectic faithful, Schur index 2

Smallest permutation representation of C24.8D4
►On 96 points
Generators in S96
(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)(49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96)

(1 40 7 34 13 28 19 46)(2 47 8 41 14 35 20 29)(3 30 9 48 15 42 21 36)(4 37 10 31 16 25 22 43)(5 44 11 38 17 32 23 26)(6 27 12 45 18 39 24 33)(49 92 55 86 61 80 67 74)(50 75 56 93 62 87 68 81)(51 82 57 76 63 94 69 88)(52 89 58 83 64 77 70 95)(53 96 59 90 65 84 71 78)(54 79 60 73 66 91 72 85)

(1 83 22 80 19 77 16 74 13 95 10 92 7 89 4 86)(2 76 23 73 20 94 17 91 14 88 11 85 8 82 5 79)(3 93 24 90 21 87 18 84 15 81 12 78 9 75 6 96)(25 52 46 49 43 70 40 67 37 64 34 61 31 58 28 55)(26 69 47 66 44 63 41 60 38 57 35 54 32 51 29 72)(27 62 48 59 45 56 42 53 39 50 36 71 33 68 30 65)
 
G:=sub<Sym(96)| (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)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96), (1,40,7,34,13,28,19,46)(2,47,8,41,14,35,20,29)(3,30,9,48,15,42,21,36)(4,37,10,31,16,25,22,43)(5,44,11,38,17,32,23,26)(6,27,12,45,18,39,24,33)(49,92,55,86,61,80,67,74)(50,75,56,93,62,87,68,81)(51,82,57,76,63,94,69,88)(52,89,58,83,64,77,70,95)(53,96,59,90,65,84,71,78)(54,79,60,73,66,91,72,85), (1,83,22,80,19,77,16,74,13,95,10,92,7,89,4,86)(2,76,23,73,20,94,17,91,14,88,11,85,8,82,5,79)(3,93,24,90,21,87,18,84,15,81,12,78,9,75,6,96)(25,52,46,49,43,70,40,67,37,64,34,61,31,58,28,55)(26,69,47,66,44,63,41,60,38,57,35,54,32,51,29,72)(27,62,48,59,45,56,42,53,39,50,36,71,33,68,30,65)>;
 
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)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96), (1,40,7,34,13,28,19,46)(2,47,8,41,14,35,20,29)(3,30,9,48,15,42,21,36)(4,37,10,31,16,25,22,43)(5,44,11,38,17,32,23,26)(6,27,12,45,18,39,24,33)(49,92,55,86,61,80,67,74)(50,75,56,93,62,87,68,81)(51,82,57,76,63,94,69,88)(52,89,58,83,64,77,70,95)(53,96,59,90,65,84,71,78)(54,79,60,73,66,91,72,85), (1,83,22,80,19,77,16,74,13,95,10,92,7,89,4,86)(2,76,23,73,20,94,17,91,14,88,11,85,8,82,5,79)(3,93,24,90,21,87,18,84,15,81,12,78,9,75,6,96)(25,52,46,49,43,70,40,67,37,64,34,61,31,58,28,55)(26,69,47,66,44,63,41,60,38,57,35,54,32,51,29,72)(27,62,48,59,45,56,42,53,39,50,36,71,33,68,30,65) );
 
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),(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96)], [(1,40,7,34,13,28,19,46),(2,47,8,41,14,35,20,29),(3,30,9,48,15,42,21,36),(4,37,10,31,16,25,22,43),(5,44,11,38,17,32,23,26),(6,27,12,45,18,39,24,33),(49,92,55,86,61,80,67,74),(50,75,56,93,62,87,68,81),(51,82,57,76,63,94,69,88),(52,89,58,83,64,77,70,95),(53,96,59,90,65,84,71,78),(54,79,60,73,66,91,72,85)], [(1,83,22,80,19,77,16,74,13,95,10,92,7,89,4,86),(2,76,23,73,20,94,17,91,14,88,11,85,8,82,5,79),(3,93,24,90,21,87,18,84,15,81,12,78,9,75,6,96),(25,52,46,49,43,70,40,67,37,64,34,61,31,58,28,55),(26,69,47,66,44,63,41,60,38,57,35,54,32,51,29,72),(27,62,48,59,45,56,42,53,39,50,36,71,33,68,30,65)]])
 

Matrix representation of C24.8D4 ►in GL6(𝔽97)

110000
9600000
00148300
007000
0000014
00009014
,
7500000
0750000
000010
000001
0096200
0096100
,
88810000
9090000
00005477
00004443
00342000
00446300

G:=sub<GL(6,GF(97))| [1,96,0,0,0,0,1,0,0,0,0,0,0,0,14,7,0,0,0,0,83,0,0,0,0,0,0,0,0,90,0,0,0,0,14,14],[75,0,0,0,0,0,0,75,0,0,0,0,0,0,0,0,96,96,0,0,0,0,2,1,0,0,1,0,0,0,0,0,0,1,0,0],[88,90,0,0,0,0,81,9,0,0,0,0,0,0,0,0,34,44,0,0,0,0,20,63,0,0,54,44,0,0,0,0,77,43,0,0] >;
 

C24.8D4 in GAP, Magma, Sage, TeX 

C_{24}._8D_4
 
% in TeX
 
G:=Group("C24.8D4");
 
// GroupNames label
 
G:=SmallGroup(192,55);
 
// by ID
 
G=gap.SmallGroup(192,55);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,141,36,758,184,675,794,192,1684,851,102,6278]);
 
// Polycyclic
 
G:=Group<a,b,c|a^24=1,b^4=a^12,c^2=a^21,b*a*b^-1=a^7,c*a*c^-1=a^17,c*b*c^-1=a^21*b^3>;
 
// generators/relations
 

Export

Subgroup lattice of C24.8D4 in TeX
Character table of C24.8D4 in TeX

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