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G = C24.6Q8  order 192 = 26·3

6th non-split extension by C24 of Q8 acting via Q8/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C24.6Q8, C8.2Dic6, C12.36SD16, C3⋊C16⋊2C4, C3⋊2(C8.Q8), C8.29(C4×S3), (C2×C8).42D6, C12.5(C4⋊C4), C24.33(C2×C4), (C2×C12).94D4, (C2×C6).3SD16, C6.3(C4.Q8), C8.C4.2S3, C8⋊Dic3.12C2, C12.C8.4C2, C4.5(Dic3⋊C4), C4.11(D4.S3), (C2×C24).153C22, C2.4(C12.Q8), C22.2(Q8⋊2S3), (C3×C8.C4).3C2, (C2×C4).17(C3⋊D4), SmallGroup(192,53)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C24 — C24.6Q8
Chief series
C1 — C3 — C6 — C12 — C2×C12 — C2×C24 — C12.C8 — C24.6Q8
Lower central
C3 — C6 — C12 — C24 — C24.6Q8
Upper central
C1 — C2 — C2×C4 — C2×C8 — C8.C4

Generators and relations for C24.6Q8
 G = < a,b,c | a24=1, b4=a12, c2=a21b2, bab-1=a7, cac-1=a5, cbc-1=a15b3 >

C1
C2
2C2
C3
C4
C22
C4
24C4
C6
2C6
C8
C8
C2×C4
4C8
12C2×C4
C2×C6
C12
C12
8Dic3
C2×C8
2M4(2)
3C16
3C16
6C4⋊C4
C24
C24
C2×C12
4C24
4C2×Dic3
C8.C4
3C4.Q8
3M5(2)
C2×C24
C3⋊C16
C3⋊C16
2C3×M4(2)
2C4⋊Dic3
3C8.Q8
C3×C8.C4
C12.C8
C8⋊Dic3
C24.6Q8

Character table of C24.6Q8

 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
ρ511-11-11i-i1-111-1-ii-1-111-1-111-11-1ii-i-i    linear of order 4
ρ611-11-11i-i1-111-1i-i-1-11-111-11-11-1-i-iii    linear of order 4
ρ711-11-11-ii1-111-1-ii-1-11-111-11-11-1ii-i-i    linear of order 4
ρ811-11-11-ii1-111-1i-i-1-111-1-111-11-1-i-iii    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
ρ11222-12200-1-1222-2-2-1-1-10000-1-1-1-11111    orthogonal lifted from D6
ρ1222-22-22002-2-2-2200-2-220000-22-220000    symplectic lifted from Q8, Schur index 2
ρ1322-2-1-2200-11-2-220011-100001-11-1-√3√3-√3√3    symplectic lifted from Dic6, Schur index 2
ρ1422-2-1-2200-11-2-220011-100001-11-1√3-√3√3-√3    symplectic lifted from Dic6, Schur index 2
ρ1522-2-1-2200-1122-2-2i2i11-10000-11-11-i-iii    complex lifted from C4×S3
ρ1622-2-1-2200-1122-22i-2i11-10000-11-11ii-i-i    complex lifted from C4×S3
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
ρ1922-222-2002-20000022-2√-2-√-2√-2-√-200000000    complex lifted from SD16
ρ2022-222-2002-20000022-2-√-2√-2-√-2√-200000000    complex lifted from SD16
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
ρ23444-2-4-400-2-200000222000000000000    orthogonal lifted from Q8⋊2S3
ρ2444-4-24-400-2200000-2-22000000000000    symplectic lifted from D4.S3, Schur index 2
ρ254-4040000-402√-2-2√-20000000000-2√-202√-200000    complex lifted from C8.Q8
ρ264-4040000-40-2√-22√-200000000002√-20-2√-200000    complex lifted from C8.Q8
ρ274-40-2000020-2√-22√-2000-2√32√300000-√-2√-6√-2-√-60000    complex faithful
ρ284-40-20000202√-2-2√-20002√3-2√300000√-2√-6-√-2-√-60000    complex faithful
ρ294-40-2000020-2√-22√-20002√3-2√300000-√-2-√-6√-2√-60000    complex faithful
ρ304-40-20000202√-2-2√-2000-2√32√300000√-2-√-6-√-2√-60000    complex faithful

Smallest permutation representation of C24.6Q8
►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 40 19 46 13 28 7 34)(2 47 20 29 14 35 8 41)(3 30 21 36 15 42 9 48)(4 37 22 43 16 25 10 31)(5 44 23 26 17 32 11 38)(6 27 24 33 18 39 12 45)

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

Matrix representation of C24.6Q8 ►in GL6(𝔽97)

010000
9610000
00575700
00405700
00004057
00004040
,
2200000
0220000
000010
000001
0009600
001000
,
0960000
9600000
00004040
00004057
001000
0009600

G:=sub<GL(6,GF(97))| [0,96,0,0,0,0,1,1,0,0,0,0,0,0,57,40,0,0,0,0,57,57,0,0,0,0,0,0,40,40,0,0,0,0,57,40],[22,0,0,0,0,0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,0,0,96,0,0,0,1,0,0,0,0,0,0,1,0,0],[0,96,0,0,0,0,96,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,96,0,0,40,40,0,0,0,0,40,57,0,0] >;
 

C24.6Q8 in GAP, Magma, Sage, TeX 

C_{24}._6Q_8
 
% in TeX
 
G:=Group("C24.6Q8");
 
// GroupNames label
 
G:=SmallGroup(192,53);
 
// by ID
 
G=gap.SmallGroup(192,53);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,365,36,758,184,346,80,851,102,6278]);
 
// Polycyclic
 
G:=Group<a,b,c|a^24=1,b^4=a^12,c^2=a^21*b^2,b*a*b^-1=a^7,c*a*c^-1=a^5,c*b*c^-1=a^15*b^3>;
 
// generators/relations
 

Export

Subgroup lattice of C24.6Q8 in TeX
Character table of C24.6Q8 in TeX

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