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## G = C12.10D4order 96 = 25·3

### 10th non-split extension by C12 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

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

Generators and relations for C12.10D4
G = < a,b,c | a12=1, b4=a6, c2=a9, bab-1=a-1, cac-1=a5, cbc-1=a9b3 >

Character table of C12.10D4

 class 1 2A 2B 3 4A 4B 4C 4D 6A 6B 6C 8A 8B 8C 8D 12A 12B 12C 12D 12E 12F size 1 1 2 2 2 2 4 4 2 2 2 12 12 12 12 4 4 4 4 4 4 ρ1 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 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 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 linear of order 2 ρ5 1 1 1 1 -1 -1 1 -1 1 1 1 -i i i -i 1 1 -1 -1 -1 -1 linear of order 4 ρ6 1 1 1 1 -1 -1 1 -1 1 1 1 i -i -i i 1 1 -1 -1 -1 -1 linear of order 4 ρ7 1 1 1 1 -1 -1 -1 1 1 1 1 i i -i -i -1 -1 -1 1 1 -1 linear of order 4 ρ8 1 1 1 1 -1 -1 -1 1 1 1 1 -i -i i i -1 -1 -1 1 1 -1 linear of order 4 ρ9 2 2 2 -1 2 2 -2 -2 -1 -1 -1 0 0 0 0 1 1 -1 1 1 -1 orthogonal lifted from D6 ρ10 2 2 2 -1 2 2 2 2 -1 -1 -1 0 0 0 0 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ11 2 2 -2 2 -2 2 0 0 -2 -2 2 0 0 0 0 0 0 -2 0 0 2 orthogonal lifted from D4 ρ12 2 2 -2 2 2 -2 0 0 -2 -2 2 0 0 0 0 0 0 2 0 0 -2 orthogonal lifted from D4 ρ13 2 2 2 -1 -2 -2 -2 2 -1 -1 -1 0 0 0 0 1 1 1 -1 -1 1 symplectic lifted from Dic3, Schur index 2 ρ14 2 2 2 -1 -2 -2 2 -2 -1 -1 -1 0 0 0 0 -1 -1 1 1 1 1 symplectic lifted from Dic3, Schur index 2 ρ15 2 2 -2 -1 -2 2 0 0 1 1 -1 0 0 0 0 √-3 -√-3 1 -√-3 √-3 -1 complex lifted from C3⋊D4 ρ16 2 2 -2 -1 2 -2 0 0 1 1 -1 0 0 0 0 -√-3 √-3 -1 -√-3 √-3 1 complex lifted from C3⋊D4 ρ17 2 2 -2 -1 2 -2 0 0 1 1 -1 0 0 0 0 √-3 -√-3 -1 √-3 -√-3 1 complex lifted from C3⋊D4 ρ18 2 2 -2 -1 -2 2 0 0 1 1 -1 0 0 0 0 -√-3 √-3 1 √-3 -√-3 -1 complex lifted from C3⋊D4 ρ19 4 -4 0 4 0 0 0 0 0 0 -4 0 0 0 0 0 0 0 0 0 0 symplectic lifted from C4.10D4, Schur index 2 ρ20 4 -4 0 -2 0 0 0 0 -2√-3 2√-3 2 0 0 0 0 0 0 0 0 0 0 complex faithful ρ21 4 -4 0 -2 0 0 0 0 2√-3 -2√-3 2 0 0 0 0 0 0 0 0 0 0 complex faithful

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

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

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

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

Matrix representation of C12.10D4 in GL4(𝔽7) generated by

 0 3 1 0 0 3 0 4 2 6 0 6 6 0 3 4
,
 5 4 4 1 5 6 0 6 2 3 5 1 1 5 1 5
,
 5 3 6 3 0 3 5 4 1 1 0 2 5 1 6 6
G:=sub<GL(4,GF(7))| [0,0,2,6,3,3,6,0,1,0,0,3,0,4,6,4],[5,5,2,1,4,6,3,5,4,0,5,1,1,6,1,5],[5,0,1,5,3,3,1,1,6,5,0,6,3,4,2,6] >;

C12.10D4 in GAP, Magma, Sage, TeX

C_{12}._{10}D_4
% in TeX

G:=Group("C12.10D4");
// GroupNames label

G:=SmallGroup(96,43);
// by ID

G=gap.SmallGroup(96,43);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,24,121,103,188,86,579,2309]);
// Polycyclic

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

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