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## G = D4.12D12order 192 = 26·3

### 2nd non-split extension by D4 of D12 acting through Inn(D4)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — D4.12D12
 Chief series C1 — C3 — C6 — C12 — D12 — C2×D12 — D4○D12 — D4.12D12
 Lower central C3 — C6 — C12 — D4.12D12
 Upper central C1 — C2 — C4○D4 — C8○D4

Generators and relations for D4.12D12
G = < a,b,c,d | a4=b2=d2=1, c12=a2, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=a2c11 >

Subgroups: 880 in 268 conjugacy classes, 107 normal (16 characteristic)
C1, C2, C2 [×9], C3, C4, C4 [×3], C4 [×2], C22 [×3], C22 [×12], S3 [×6], C6, C6 [×3], C8, C8 [×3], C2×C4 [×3], C2×C4 [×6], D4 [×3], D4 [×18], Q8, Q8 [×2], C23 [×6], Dic3 [×2], C12, C12 [×3], D6 [×12], C2×C6 [×3], C2×C8 [×3], M4(2) [×3], D8 [×9], SD16 [×6], Q16, C2×D4 [×12], C4○D4, C4○D4 [×8], C24, C24 [×3], Dic6 [×2], C4×S3 [×6], D12 [×6], D12 [×6], C3⋊D4 [×6], C2×C12 [×3], C3×D4 [×3], C3×Q8, C22×S3 [×6], C8○D4, C2×D8 [×3], C4○D8 [×3], C8⋊C22 [×6], 2+ 1+4 [×2], C24⋊C2 [×6], D24 [×9], Dic12, C2×C24 [×3], C3×M4(2) [×3], C2×D12 [×6], C4○D12 [×6], S3×D4 [×6], Q83S3 [×2], C3×C4○D4, D4○D8, C2×D24 [×3], C4○D24 [×3], C8⋊D6 [×6], C3×C8○D4, D4○D12 [×2], D4.12D12
Quotients: C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C24, D12 [×4], C22×S3 [×7], C22×D4, C2×D12 [×6], S3×C23, D4○D8, C22×D12, D4.12D12

Smallest permutation representation of D4.12D12
On 48 points
Generators in S48
(1 34 13 46)(2 35 14 47)(3 36 15 48)(4 37 16 25)(5 38 17 26)(6 39 18 27)(7 40 19 28)(8 41 20 29)(9 42 21 30)(10 43 22 31)(11 44 23 32)(12 45 24 33)
(25 37)(26 38)(27 39)(28 40)(29 41)(30 42)(31 43)(32 44)(33 45)(34 46)(35 47)(36 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 24)(2 23)(3 22)(4 21)(5 20)(6 19)(7 18)(8 17)(9 16)(10 15)(11 14)(12 13)(25 42)(26 41)(27 40)(28 39)(29 38)(30 37)(31 36)(32 35)(33 34)(43 48)(44 47)(45 46)

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

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

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

42 conjugacy classes

 class 1 2A 2B 2C 2D 2E ··· 2J 3 4A 4B 4C 4D 4E 4F 6A 6B 6C 6D 8A 8B 8C 8D 8E 12A 12B 12C 12D 12E 24A 24B 24C 24D 24E ··· 24J order 1 2 2 2 2 2 ··· 2 3 4 4 4 4 4 4 6 6 6 6 8 8 8 8 8 12 12 12 12 12 24 24 24 24 24 ··· 24 size 1 1 2 2 2 12 ··· 12 2 2 2 2 2 12 12 2 4 4 4 2 2 4 4 4 2 2 4 4 4 2 2 2 2 4 ··· 4

42 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 S3 D4 D4 D6 D6 D6 D12 D12 D4○D8 D4.12D12 kernel D4.12D12 C2×D24 C4○D24 C8⋊D6 C3×C8○D4 D4○D12 C8○D4 C3×D4 C3×Q8 C2×C8 M4(2) C4○D4 D4 Q8 C3 C1 # reps 1 3 3 6 1 2 1 3 1 3 3 1 6 2 2 4

Matrix representation of D4.12D12 in GL4(𝔽73) generated by

 0 0 72 0 0 0 0 72 1 0 0 0 0 1 0 0
,
 1 0 0 0 0 1 0 0 0 0 72 0 0 0 0 72
,
 68 55 0 0 18 50 0 0 0 0 68 55 0 0 18 50
,
 50 18 0 0 68 23 0 0 0 0 50 18 0 0 68 23
G:=sub<GL(4,GF(73))| [0,0,1,0,0,0,0,1,72,0,0,0,0,72,0,0],[1,0,0,0,0,1,0,0,0,0,72,0,0,0,0,72],[68,18,0,0,55,50,0,0,0,0,68,18,0,0,55,50],[50,68,0,0,18,23,0,0,0,0,50,68,0,0,18,23] >;

D4.12D12 in GAP, Magma, Sage, TeX

D_4._{12}D_{12}
% in TeX

G:=Group("D4.12D12");
// GroupNames label

G:=SmallGroup(192,1311);
// by ID

G=gap.SmallGroup(192,1311);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,387,675,192,1684,102,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^2=d^2=1,c^12=a^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=a^2*c^11>;
// generators/relations

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