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## G = (C6×D4)⋊10C4order 192 = 26·3

### 6th semidirect product of C6×D4 and C4 acting via C4/C2=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — (C6×D4)⋊10C4
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — C6.D4 — C23.26D6 — (C6×D4)⋊10C4
 Lower central C3 — C6 — C2×C6 — (C6×D4)⋊10C4
 Upper central C1 — C4 — C22×C4 — C2×C4○D4

Generators and relations for (C6×D4)⋊10C4
G = < a,b,c,d | a6=b4=c2=d4=1, ab=ba, ac=ca, dad-1=a-1b2, cbc=b-1, bd=db, dcd-1=a3b2c >

Subgroups: 360 in 158 conjugacy classes, 67 normal (25 characteristic)
C1, C2, C2 [×5], C3, C4 [×2], C4 [×2], C4 [×6], C22, C22 [×2], C22 [×5], C6, C6 [×5], C2×C4 [×2], C2×C4 [×6], C2×C4 [×8], D4 [×6], Q8 [×2], C23, C23 [×2], Dic3 [×4], C12 [×2], C12 [×2], C12 [×2], C2×C6, C2×C6 [×2], C2×C6 [×5], C42 [×2], C22⋊C4 [×4], C4⋊C4 [×2], C22×C4, C22×C4 [×2], C2×D4, C2×D4 [×2], C2×Q8, C4○D4 [×4], C2×Dic3 [×4], C2×C12 [×2], C2×C12 [×6], C2×C12 [×4], C3×D4 [×6], C3×Q8 [×2], C22×C6, C22×C6 [×2], C23⋊C4 [×4], C42⋊C2 [×2], C2×C4○D4, C4×Dic3 [×2], C4⋊Dic3 [×2], C6.D4 [×4], C22×C12, C22×C12 [×2], C6×D4, C6×D4 [×2], C6×Q8, C3×C4○D4 [×4], C23.C23, C23.7D6 [×4], C23.26D6 [×2], C6×C4○D4, (C6×D4)⋊10C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×4], C23, Dic3 [×4], D6 [×3], C22⋊C4 [×4], C22×C4, C2×D4 [×2], C2×Dic3 [×6], C3⋊D4 [×4], C22×S3, C2×C22⋊C4, C6.D4 [×4], C22×Dic3, C2×C3⋊D4 [×2], C23.C23, C2×C6.D4, (C6×D4)⋊10C4

Smallest permutation representation of (C6×D4)⋊10C4
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 38 16 34)(2 39 17 35)(3 40 18 36)(4 41 13 31)(5 42 14 32)(6 37 15 33)(7 27 47 23)(8 28 48 24)(9 29 43 19)(10 30 44 20)(11 25 45 21)(12 26 46 22)
(1 43)(2 44)(3 45)(4 46)(5 47)(6 48)(7 14)(8 15)(9 16)(10 17)(11 18)(12 13)(19 34)(20 35)(21 36)(22 31)(23 32)(24 33)(25 40)(26 41)(27 42)(28 37)(29 38)(30 39)
(1 38 16 34)(2 33 17 37)(3 42 18 32)(4 31 13 41)(5 40 14 36)(6 35 15 39)(7 24)(8 27)(9 22)(10 25)(11 20)(12 29)(19 46)(21 44)(23 48)(26 43)(28 47)(30 45)

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

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

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

42 conjugacy classes

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

42 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + - + - + - image C1 C2 C2 C2 C4 C4 C4 S3 D4 Dic3 D6 Dic3 D6 Dic3 C3⋊D4 C23.C23 (C6×D4)⋊10C4 kernel (C6×D4)⋊10C4 C23.7D6 C23.26D6 C6×C4○D4 C22×C12 C6×D4 C6×Q8 C2×C4○D4 C2×C12 C22×C4 C22×C4 C2×D4 C2×D4 C2×Q8 C2×C4 C3 C1 # reps 1 4 2 1 4 2 2 1 4 2 1 1 2 1 8 2 4

Matrix representation of (C6×D4)⋊10C4 in GL6(𝔽13)

 1 1 0 0 0 0 12 0 0 0 0 0 0 0 0 12 12 0 0 0 12 0 12 0 0 0 0 0 1 0 0 0 0 0 2 12
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 8 0 0 8 0 0 0 8 0 8 0 0 0 0 5 0 0 0 0 0 0 5
,
 2 4 0 0 0 0 9 11 0 0 0 0 0 0 8 0 0 8 0 0 8 0 5 8 0 0 5 8 0 0 0 0 10 0 0 5
,
 5 0 0 0 0 0 8 8 0 0 0 0 0 0 8 0 8 0 0 0 0 5 8 5 0 0 0 0 5 8 0 0 0 0 10 8

G:=sub<GL(6,GF(13))| [1,12,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,0,12,12,1,2,0,0,0,0,0,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,5,0,0,0,8,8,0,5],[2,9,0,0,0,0,4,11,0,0,0,0,0,0,8,8,5,10,0,0,0,0,8,0,0,0,0,5,0,0,0,0,8,8,0,5],[5,8,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,5,0,0,0,0,8,8,5,10,0,0,0,5,8,8] >;

(C6×D4)⋊10C4 in GAP, Magma, Sage, TeX

(C_6\times D_4)\rtimes_{10}C_4
% in TeX

G:=Group("(C6xD4):10C4");
// GroupNames label

G:=SmallGroup(192,799);
// by ID

G=gap.SmallGroup(192,799);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,232,422,297,1684,6278]);
// Polycyclic

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

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