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## G = (C3×D4).31D4order 192 = 26·3

### 1st non-split extension by C3×D4 of D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — (C3×D4).31D4
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — C2×Dic6 — C2×D4.S3 — (C3×D4).31D4
 Lower central C3 — C6 — C2×C12 — (C3×D4).31D4
 Upper central C1 — C22 — C22×C4 — C22×D4

Generators and relations for (C3×D4).31D4
G = < a,b,c,d,e | a3=b4=c2=1, d4=e2=b2, ab=ba, ac=ca, dad-1=eae-1=a-1, cbc=ebe-1=b-1, bd=db, dcd-1=bc, ece-1=b-1c, ede-1=d3 >

Subgroups: 488 in 188 conjugacy classes, 51 normal (25 characteristic)
C1, C2 [×3], C2 [×6], C3, C4 [×2], C4 [×3], C22, C22 [×2], C22 [×18], C6 [×3], C6 [×6], C8 [×2], C2×C4 [×2], C2×C4 [×4], D4 [×4], D4 [×6], Q8 [×2], C23, C23 [×10], Dic3 [×2], C12 [×2], C12, C2×C6, C2×C6 [×2], C2×C6 [×18], C22⋊C4, C4⋊C4 [×2], C2×C8 [×2], SD16 [×4], C22×C4, C2×D4 [×2], C2×D4 [×5], C2×Q8, C24, C3⋊C8 [×2], Dic6 [×2], C2×Dic3 [×2], C2×C12 [×2], C2×C12 [×2], C3×D4 [×4], C3×D4 [×6], C22×C6, C22×C6 [×10], C22⋊C8, D4⋊C4 [×2], C22⋊Q8, C2×SD16 [×2], C22×D4, C2×C3⋊C8 [×2], Dic3⋊C4, C4⋊Dic3, D4.S3 [×4], C6.D4, C2×Dic6, C22×C12, C6×D4 [×2], C6×D4 [×5], C23×C6, C22⋊SD16, C12.55D4, D4⋊Dic3 [×2], C12.48D4, C2×D4.S3 [×2], D4×C2×C6, (C3×D4).31D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×6], C23, D6 [×3], SD16 [×2], C2×D4 [×3], C3⋊D4 [×6], C22×S3, C22≀C2, C2×SD16, C8⋊C22, D4.S3 [×2], C2×C3⋊D4 [×3], C22⋊SD16, D126C22, C2×D4.S3, C244S3, (C3×D4).31D4

Smallest permutation representation of (C3×D4).31D4
On 48 points
Generators in S48
(1 39 45)(2 46 40)(3 33 47)(4 48 34)(5 35 41)(6 42 36)(7 37 43)(8 44 38)(9 19 29)(10 30 20)(11 21 31)(12 32 22)(13 23 25)(14 26 24)(15 17 27)(16 28 18)
(1 15 5 11)(2 16 6 12)(3 9 7 13)(4 10 8 14)(17 35 21 39)(18 36 22 40)(19 37 23 33)(20 38 24 34)(25 47 29 43)(26 48 30 44)(27 41 31 45)(28 42 32 46)
(1 13)(2 8)(3 11)(4 6)(5 9)(7 15)(10 16)(12 14)(17 37)(18 20)(19 35)(21 33)(22 24)(23 39)(25 45)(26 32)(27 43)(28 30)(29 41)(31 47)(34 36)(38 40)(42 48)(44 46)
(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 2 5 6)(3 8 7 4)(9 10 13 14)(11 16 15 12)(17 22 21 18)(19 20 23 24)(25 26 29 30)(27 32 31 28)(33 38 37 34)(35 36 39 40)(41 42 45 46)(43 48 47 44)

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

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

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

39 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 3 4A 4B 4C 4D 4E 6A ··· 6G 6H ··· 6O 8A 8B 8C 8D 12A 12B 12C 12D order 1 2 2 2 2 2 2 2 2 2 3 4 4 4 4 4 6 ··· 6 6 ··· 6 8 8 8 8 12 12 12 12 size 1 1 1 1 2 2 4 4 4 4 2 2 2 4 24 24 2 ··· 2 4 ··· 4 12 12 12 12 4 4 4 4

39 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 S3 D4 D4 D4 D6 D6 SD16 C3⋊D4 C3⋊D4 C3⋊D4 C8⋊C22 D4.S3 D12⋊6C22 kernel (C3×D4).31D4 C12.55D4 D4⋊Dic3 C12.48D4 C2×D4.S3 D4×C2×C6 C22×D4 C2×C12 C3×D4 C22×C6 C22×C4 C2×D4 C2×C6 C2×C4 D4 C23 C6 C22 C2 # reps 1 1 2 1 2 1 1 1 4 1 1 2 4 2 8 2 1 2 2

Matrix representation of (C3×D4).31D4 in GL6(𝔽73)

 64 0 0 0 0 0 0 8 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 1
,
 72 0 0 0 0 0 0 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 72 0
,
 1 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 1
,
 0 72 0 0 0 0 1 0 0 0 0 0 0 0 0 72 0 0 0 0 1 0 0 0 0 0 0 0 67 67 0 0 0 0 6 67
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 67 6 0 0 0 0 6 6

G:=sub<GL(6,GF(73))| [64,0,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,1,0],[1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,1],[0,1,0,0,0,0,72,0,0,0,0,0,0,0,0,1,0,0,0,0,72,0,0,0,0,0,0,0,67,6,0,0,0,0,67,67],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,67,6,0,0,0,0,6,6] >;

(C3×D4).31D4 in GAP, Magma, Sage, TeX

(C_3\times D_4)._{31}D_4
% in TeX

G:=Group("(C3xD4).31D4");
// GroupNames label

G:=SmallGroup(192,777);
// by ID

G=gap.SmallGroup(192,777);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,253,254,1684,851,102,6278]);
// Polycyclic

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

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