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G = C12.23D4order 96 = 25·3

23rd non-split extension by C12 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C12.23D4, (C2×Q8)⋊6S3, (C6×Q8)⋊4C2, D6⋊C416C2, C6.58(C2×D4), (C2×C4).57D6, (C4×Dic3)⋊7C2, (C2×D12).9C2, C34(C4.4D4), C6.37(C4○D4), C4.11(C3⋊D4), (C2×C6).59C23, (C2×C12).40C22, C2.9(Q83S3), C22.65(C22×S3), (C22×S3).13C22, (C2×Dic3).41C22, C2.22(C2×C3⋊D4), SmallGroup(96,154)

Series: Derived Chief Lower central Upper central

C1C2×C6 — C12.23D4
C1C3C6C2×C6C22×S3C2×D12 — C12.23D4
C3C2×C6 — C12.23D4
C1C22C2×Q8

Generators and relations for C12.23D4
 G = < a,b,c | a12=b4=c2=1, bab-1=a5, cac=a-1, cbc=a6b-1 >

Subgroups: 194 in 76 conjugacy classes, 33 normal (13 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×2], C4 [×4], C22, C22 [×6], S3 [×2], C6, C6 [×2], C2×C4, C2×C4 [×2], C2×C4 [×2], D4 [×2], Q8 [×2], C23 [×2], Dic3 [×2], C12 [×2], C12 [×2], D6 [×6], C2×C6, C42, C22⋊C4 [×4], C2×D4, C2×Q8, D12 [×2], C2×Dic3 [×2], C2×C12, C2×C12 [×2], C3×Q8 [×2], C22×S3 [×2], C4.4D4, C4×Dic3, D6⋊C4 [×4], C2×D12, C6×Q8, C12.23D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], C2×D4, C4○D4 [×2], C3⋊D4 [×2], C22×S3, C4.4D4, Q83S3 [×2], C2×C3⋊D4, C12.23D4

Character table of C12.23D4

 class 12A2B2C2D2E34A4B4C4D4E4F4G4H6A6B6C12A12B12C12D12E12F
 size 11111212222446666222444444
ρ1111111111111111111111111    trivial
ρ2111111111-1-1-1-1-1-11111-1-11-1-1    linear of order 2
ρ31111-111-1-1-11-11-11111-11-1-1-11    linear of order 2
ρ41111-111-1-11-11-11-1111-1-11-11-1    linear of order 2
ρ51111-1-111111-1-1-1-1111111111    linear of order 2
ρ61111-1-1111-1-111111111-1-11-1-1    linear of order 2
ρ711111-11-1-1-111-11-1111-11-1-1-11    linear of order 2
ρ811111-11-1-11-1-11-11111-1-11-11-1    linear of order 2
ρ9222200-1-2-22-20000-1-1-111-11-11    orthogonal lifted from D6
ρ1022-2-20022-2000000-22-2200-200    orthogonal lifted from D4
ρ11222200-122-2-20000-1-1-1-111-111    orthogonal lifted from D6
ρ1222-2-2002-22000000-22-2-200200    orthogonal lifted from D4
ρ13222200-122220000-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ14222200-1-2-2-220000-1-1-11-1111-1    orthogonal lifted from D6
ρ1522-2-200-1-220000001-111--3-3-1--3-3    complex lifted from C3⋊D4
ρ1622-2-200-12-20000001-11-1--3--31-3-3    complex lifted from C3⋊D4
ρ1722-2-200-1-220000001-111-3--3-1-3--3    complex lifted from C3⋊D4
ρ1822-2-200-12-20000001-11-1-3-31--3--3    complex lifted from C3⋊D4
ρ192-22-2002000002i0-2i2-2-2000000    complex lifted from C4○D4
ρ202-22-200200000-2i02i2-2-2000000    complex lifted from C4○D4
ρ212-2-220020000-2i02i0-2-22000000    complex lifted from C4○D4
ρ222-2-2200200002i0-2i0-2-22000000    complex lifted from C4○D4
ρ234-4-4400-20000000022-2000000    orthogonal lifted from Q83S3, Schur index 2
ρ244-44-400-200000000-222000000    orthogonal lifted from Q83S3, Schur index 2

Smallest permutation representation of C12.23D4
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 19 26 47)(2 24 27 40)(3 17 28 45)(4 22 29 38)(5 15 30 43)(6 20 31 48)(7 13 32 41)(8 18 33 46)(9 23 34 39)(10 16 35 44)(11 21 36 37)(12 14 25 42)
(2 12)(3 11)(4 10)(5 9)(6 8)(13 47)(14 46)(15 45)(16 44)(17 43)(18 42)(19 41)(20 40)(21 39)(22 38)(23 37)(24 48)(25 27)(28 36)(29 35)(30 34)(31 33)

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

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

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

C12.23D4 is a maximal subgroup of
(C2×C4).D12  D12.5D4  (C2×C8).D6  Dic6.11D4  Q8⋊C4⋊S3  D12.12D4  (C3×D4).D4  C24.43D4  C249D4  (C2×Q16)⋊S3  C24.37D4  C24.28D4  D12.39D4  2- 1+44S3  C42.122D6  C42.131D6  C42.133D6  C42.136D6  C4⋊C4.187D6  D1221D4  Dic622D4  C6.532+ 1+4  C6.222- 1+4  C6.242- 1+4  C6.562+ 1+4  C6.592+ 1+4  C42.138D6  S3×C4.4D4  C4220D6  C4222D6  C42.143D6  C4224D6  C42.171D6  C42.240D6  C42.177D6  C42.178D6  C42.179D6  C42.180D6  C6.442- 1+4  C6.452- 1+4  C6.1452+ 1+4  C6.1462+ 1+4  (C2×C12)⋊17D4  C36.23D4  C62.33C23  C12.28D12  C62.77C23  C62.262C23  (C2×C20).D6  C60.89D4  C60.47D4  C60.23D4
C12.23D4 is a maximal quotient of
(C4×Dic3)⋊9C4  (C2×C12).55D4  (C2×D12)⋊10C4  D6⋊C47C4  (C2×C4)⋊3D12  (C2×C12).289D4  C42.70D6  C42.216D6  C42.71D6  C12.D8  C42.82D6  C12.Q16  (C6×Q8)⋊7C4  (C22×Q8)⋊9S3  C36.23D4  C62.33C23  C12.28D12  C62.77C23  C62.262C23  (C2×C20).D6  C60.89D4  C60.47D4  C60.23D4

Matrix representation of C12.23D4 in GL4(𝔽13) generated by

0100
121200
0053
0008
,
11900
11200
00122
0001
,
12000
1100
0010
00112
G:=sub<GL(4,GF(13))| [0,12,0,0,1,12,0,0,0,0,5,0,0,0,3,8],[11,11,0,0,9,2,0,0,0,0,12,0,0,0,2,1],[12,1,0,0,0,1,0,0,0,0,1,1,0,0,0,12] >;

C12.23D4 in GAP, Magma, Sage, TeX

C_{12}._{23}D_4
% in TeX

G:=Group("C12.23D4");
// GroupNames label

G:=SmallGroup(96,154);
// by ID

G=gap.SmallGroup(96,154);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,217,103,218,188,86,2309]);
// Polycyclic

G:=Group<a,b,c|a^12=b^4=c^2=1,b*a*b^-1=a^5,c*a*c=a^-1,c*b*c=a^6*b^-1>;
// generators/relations

Export

Character table of C12.23D4 in TeX

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