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G = C23.D12  order 192 = 26·3

1st non-split extension by C23 of D12 acting via D12/C3=D4

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C23.1D12, (C2×D4).3D6, (C4×Dic3)⋊1C4, (C2×Dic6)⋊3C4, C23⋊C4.2S3, C3⋊1(C42⋊3C4), C6.9(C23⋊C4), (C6×D4).3C22, (C22×C6).10D4, C23.3(C3⋊D4), C22.10(D6⋊C4), C23.12D6.1C2, C23.7D6.1C2, C2.10(C23.6D6), (C2×C4).1(C4×S3), (C2×C12).1(C2×C4), (C3×C23⋊C4).2C2, (C2×C6).3(C22⋊C4), SmallGroup(192,32)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C12 — C23.D12
Chief series
C1 — C3 — C6 — C2×C6 — C22×C6 — C6×D4 — C23.12D6 — C23.D12
Lower central
C3 — C6 — C2×C6 — C2×C12 — C23.D12
Upper central
C1 — C2 — C22 — C2×D4 — C23⋊C4

Generators and relations for C23.D12
 G = < a,b,c,d,e | a2=b2=c2=d12=1, e2=ca=ac, dad-1=ab=ba, ae=ea, dbd-1=ebe-1=bc=cb, cd=dc, ce=ec, ede-1=ad-1 >

Subgroups: 256 in 70 conjugacy classes, 21 normal (all characteristic)
C1, C2, C2, C3, C4, C22, C22, C6, C6, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, C2×C6, C2×C6, C42, C22⋊C4, C2×D4, C2×Q8, Dic6, C2×Dic3, C2×C12, C2×C12, C3×D4, C22×C6, C23⋊C4, C23⋊C4, C4.4D4, C4×Dic3, C6.D4, C3×C22⋊C4, C2×Dic6, C6×D4, C42⋊3C4, C23.7D6, C3×C23⋊C4, C23.12D6, C23.D12
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D6, C22⋊C4, C4×S3, D12, C3⋊D4, C23⋊C4, D6⋊C4, C42⋊3C4, C23.6D6, C23.D12

Character table of C23.D12

 class 12A2B2C2D34A4B4C4D4E4F4G4H6A6B6C6D6E12A12B12C12D12E
 size 11244248812122424242444888888
ρ1111111111111111111111111    trivial
ρ21111111-1-1-1-111-111111-1-1-1-11    linear of order 2
ρ3111111111-1-1-1-1-11111111111    linear of order 2
ρ41111111-1-111-1-1111111-1-1-1-11    linear of order 2
ρ5111-1-111i-i11-ii-11-1-11-1-iii-i1    linear of order 4
ρ6111-1-111-ii-1-1-ii11-1-11-1i-i-ii1    linear of order 4
ρ7111-1-111i-i-1-1i-i11-1-11-1-iii-i1    linear of order 4
ρ8111-1-111-ii11i-i-11-1-11-1i-i-ii1    linear of order 4
ρ922222-12-2-200000-1-1-1-1-11111-1    orthogonal lifted from D6
ρ1022222-122200000-1-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ112222-22-200000002222-20000-2    orthogonal lifted from D4
ρ12222-222-200000002-2-2220000-2    orthogonal lifted from D4
ρ13222-22-1-20000000-111-1-1√3-√3√3-√31    orthogonal lifted from D12
ρ14222-22-1-20000000-111-1-1-√3√3-√3√31    orthogonal lifted from D12
ρ15222-2-2-122i-2i00000-111-11i-i-ii-1    complex lifted from C4×S3
ρ16222-2-2-12-2i2i00000-111-11-iii-i-1    complex lifted from C4×S3
ρ172222-2-1-20000000-1-1-1-11√-3√-3-√-3-√-31    complex lifted from C3⋊D4
ρ182222-2-1-20000000-1-1-1-11-√-3-√-3√-3√-31    complex lifted from C3⋊D4
ρ1944-400400000000400-4000000    orthogonal lifted from C23⋊C4
ρ2044-400-200000000-2-2√-32√-32000000    complex lifted from C23.6D6
ρ2144-400-200000000-22√-3-2√-32000000    complex lifted from C23.6D6
ρ224-40004000-2i2i000-4000000000    complex lifted from C42⋊3C4
ρ234-400040002i-2i000-4000000000    complex lifted from C42⋊3C4
ρ248-8000-4000000004000000000    symplectic faithful, Schur index 2

Smallest permutation representation of C23.D12
►On 48 points
Generators in S48
(1 13)(3 30)(4 46)(5 17)(7 34)(8 38)(9 21)(11 26)(12 42)(15 45)(16 31)(19 37)(20 35)(23 41)(24 27)(28 43)(32 47)(36 39)

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

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

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

Matrix representation of C23.D12 ►in GL8(𝔽13)

120000000
012000000
120100000
012010000
00000100
00001000
000000012
000000120
,
120000000
012000000
001200000
000120000
00000100
00001000
00000001
00000010
,
10000000
01000000
00100000
00010000
000012000
000001200
000000120
000000012
,
012020000
1121120000
25010000
871210000
00009444
00009499
00009949
00004449
,
1111000000
92000000
561120000
1180120000
000000012
00000010
000012000
00000100

G:=sub<GL(8,GF(13))| [12,0,12,0,0,0,0,0,0,12,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12],[0,1,2,8,0,0,0,0,12,12,5,7,0,0,0,0,0,11,0,12,0,0,0,0,2,2,1,1,0,0,0,0,0,0,0,0,9,9,9,4,0,0,0,0,4,4,9,4,0,0,0,0,4,9,4,4,0,0,0,0,4,9,9,9],[11,9,5,11,0,0,0,0,11,2,6,8,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0] >;
 

C23.D12 in GAP, Magma, Sage, TeX 

C_2^3.D_{12}
 
% in TeX
 
G:=Group("C2^3.D12");
 
// GroupNames label
 
G:=SmallGroup(192,32);
 
// by ID
 
G=gap.SmallGroup(192,32);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,141,36,422,1123,794,297,136,851,6278]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^12=1,e^2=c*a=a*c,d*a*d^-1=a*b=b*a,a*e=e*a,d*b*d^-1=e*b*e^-1=b*c=c*b,c*d=d*c,c*e=e*c,e*d*e^-1=a*d^-1>;
 
// generators/relations
 

Export

Character table of C23.D12 in TeX

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