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## G = (C2×C12).D4order 192 = 26·3

### 16th non-split extension by C2×C12 of D4 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — (C2×C12).D4
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — C6×Q8 — Dic3⋊Q8 — (C2×C12).D4
 Lower central C3 — C6 — C2×C6 — C2×C12 — (C2×C12).D4
 Upper central C1 — C2 — C22 — C2×Q8 — C4.10D4

Generators and relations for (C2×C12).D4
G = < a,b,c,d | a2=b12=1, c4=d2=b6, ab=ba, cac-1=ab6, ad=da, cbc-1=dbd-1=ab5, dcd-1=b9c3 >

Subgroups: 176 in 60 conjugacy classes, 21 normal (all characteristic)
C1, C2, C2, C3, C4, C22, C6, C6, C8, C2×C4, C2×C4, Q8, Dic3, C12, C2×C6, C42, C4⋊C4, M4(2), C2×Q8, C2×Q8, C3⋊C8, C24, Dic6, C2×Dic3, C2×C12, C3×Q8, C4.10D4, C4.10D4, C4⋊Q8, C4.Dic3, C4×Dic3, Dic3⋊C4, C3×M4(2), C2×Dic6, C6×Q8, C42.3C4, C12.10D4, C3×C4.10D4, Dic3⋊Q8, (C2×C12).D4
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, (C2×C12).D4

Character table of (C2×C12).D4

 class 1 2A 2B 3 4A 4B 4C 4D 4E 4F 6A 6B 8A 8B 8C 8D 12A 12B 12C 12D 24A 24B 24C 24D size 1 1 2 2 4 4 4 12 12 24 2 4 8 8 24 24 4 4 8 8 8 8 8 8 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ3 1 1 1 1 1 1 1 -1 -1 -1 1 1 -1 -1 1 1 1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ4 1 1 1 1 1 1 1 -1 -1 -1 1 1 1 1 -1 -1 1 1 1 1 1 1 1 1 linear of order 2 ρ5 1 1 1 1 -1 1 -1 -1 -1 1 1 1 i -i i -i -1 -1 -1 1 i i -i -i linear of order 4 ρ6 1 1 1 1 -1 1 -1 1 1 -1 1 1 i -i -i i -1 -1 -1 1 i i -i -i linear of order 4 ρ7 1 1 1 1 -1 1 -1 1 1 -1 1 1 -i i i -i -1 -1 -1 1 -i -i i i linear of order 4 ρ8 1 1 1 1 -1 1 -1 -1 -1 1 1 1 -i i -i i -1 -1 -1 1 -i -i i i linear of order 4 ρ9 2 2 2 -1 2 2 2 0 0 0 -1 -1 -2 -2 0 0 -1 -1 -1 -1 1 1 1 1 orthogonal lifted from D6 ρ10 2 2 2 2 -2 -2 2 0 0 0 2 2 0 0 0 0 -2 -2 2 -2 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 2 2 2 -2 -2 0 0 0 2 2 0 0 0 0 2 2 -2 -2 0 0 0 0 orthogonal lifted from D4 ρ12 2 2 2 -1 2 2 2 0 0 0 -1 -1 2 2 0 0 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ13 2 2 2 -1 -2 -2 2 0 0 0 -1 -1 0 0 0 0 1 1 -1 1 √3 -√3 √3 -√3 orthogonal lifted from D12 ρ14 2 2 2 -1 -2 -2 2 0 0 0 -1 -1 0 0 0 0 1 1 -1 1 -√3 √3 -√3 √3 orthogonal lifted from D12 ρ15 2 2 2 -1 -2 2 -2 0 0 0 -1 -1 -2i 2i 0 0 1 1 1 -1 i i -i -i complex lifted from C4×S3 ρ16 2 2 2 -1 -2 2 -2 0 0 0 -1 -1 2i -2i 0 0 1 1 1 -1 -i -i i i complex lifted from C4×S3 ρ17 2 2 2 -1 2 -2 -2 0 0 0 -1 -1 0 0 0 0 -1 -1 1 1 -√-3 √-3 √-3 -√-3 complex lifted from C3⋊D4 ρ18 2 2 2 -1 2 -2 -2 0 0 0 -1 -1 0 0 0 0 -1 -1 1 1 √-3 -√-3 -√-3 √-3 complex lifted from C3⋊D4 ρ19 4 4 -4 4 0 0 0 0 0 0 4 -4 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C23⋊C4 ρ20 4 -4 0 4 0 0 0 2 -2 0 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from C42.3C4, Schur index 2 ρ21 4 -4 0 4 0 0 0 -2 2 0 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from C42.3C4, Schur index 2 ρ22 4 4 -4 -2 0 0 0 0 0 0 -2 2 0 0 0 0 2√-3 -2√-3 0 0 0 0 0 0 complex lifted from C23.6D6 ρ23 4 4 -4 -2 0 0 0 0 0 0 -2 2 0 0 0 0 -2√-3 2√-3 0 0 0 0 0 0 complex lifted from C23.6D6 ρ24 8 -8 0 -4 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2

Smallest permutation representation of (C2×C12).D4
On 48 points
Generators in S48
(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)
(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 38 26 7 13 44 32)(2 24 45 25 8 18 39 31)(3 17 40 36 9 23 46 30)(4 22 47 35 10 16 41 29)(5 15 42 34 11 21 48 28)(6 20 37 33 12 14 43 27)
(1 10 7 4)(2 3 8 9)(5 6 11 12)(13 26 19 32)(14 25 20 31)(15 36 21 30)(16 35 22 29)(17 34 23 28)(18 33 24 27)(37 42 43 48)(38 47 44 41)(39 40 45 46)

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

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

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

Matrix representation of (C2×C12).D4 in GL6(𝔽73)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 0 0 0 58 58 0 72
,
 1 72 0 0 0 0 1 0 0 0 0 0 0 0 50 57 0 0 0 0 24 23 0 0 0 0 21 21 50 32 0 0 19 47 61 23
,
 59 7 0 0 0 0 66 14 0 0 0 0 0 0 69 69 37 53 0 0 44 44 9 1 0 0 72 0 0 0 0 0 65 29 20 33
,
 0 72 0 0 0 0 72 0 0 0 0 0 0 0 50 57 0 0 0 0 24 23 0 0 0 0 69 69 37 53 0 0 7 35 32 36

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,58,0,0,0,1,0,58,0,0,0,0,72,0,0,0,0,0,0,72],[1,1,0,0,0,0,72,0,0,0,0,0,0,0,50,24,21,19,0,0,57,23,21,47,0,0,0,0,50,61,0,0,0,0,32,23],[59,66,0,0,0,0,7,14,0,0,0,0,0,0,69,44,72,65,0,0,69,44,0,29,0,0,37,9,0,20,0,0,53,1,0,33],[0,72,0,0,0,0,72,0,0,0,0,0,0,0,50,24,69,7,0,0,57,23,69,35,0,0,0,0,37,32,0,0,0,0,53,36] >;

(C2×C12).D4 in GAP, Magma, Sage, TeX

(C_2\times C_{12}).D_4
% in TeX

G:=Group("(C2xC12).D4");
// GroupNames label

G:=SmallGroup(192,37);
// by ID

G=gap.SmallGroup(192,37);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,141,36,422,184,1123,794,297,136,851,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^12=1,c^4=d^2=b^6,a*b=b*a,c*a*c^-1=a*b^6,a*d=d*a,c*b*c^-1=d*b*d^-1=a*b^5,d*c*d^-1=b^9*c^3>;
// generators/relations

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