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## G = (C6×D4)⋊9C4order 192 = 26·3

### 5th semidirect product of C6×D4 and C4 acting via C4/C2=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — (C6×D4)⋊9C4
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C4.Dic3 — C2×C4.Dic3 — (C6×D4)⋊9C4
 Lower central C3 — C6 — C12 — (C6×D4)⋊9C4
 Upper central C1 — C4 — C22×C4 — C2×C4○D4

Generators and relations for (C6×D4)⋊9C4
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=bc >

Subgroups: 328 in 154 conjugacy classes, 67 normal (43 characteristic)
C1, C2, C2 [×5], C3, C4 [×4], C4 [×4], C22 [×3], C22 [×5], C6, C6 [×5], C8 [×2], C2×C4 [×6], C2×C4 [×7], D4 [×2], D4 [×5], Q8 [×2], Q8, C23, C23, Dic3 [×2], C12 [×4], C12 [×2], C2×C6 [×3], C2×C6 [×5], C42 [×2], C22⋊C4, C4⋊C4, C2×C8, M4(2) [×3], C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4 [×4], C4○D4 [×2], C3⋊C8 [×2], C2×Dic3 [×2], C2×C12 [×6], C2×C12 [×5], C3×D4 [×2], C3×D4 [×5], C3×Q8 [×2], C3×Q8, C22×C6, C22×C6, C4≀C2 [×4], C42⋊C2, C2×M4(2), C2×C4○D4, C2×C3⋊C8, C4.Dic3 [×2], C4.Dic3, C4×Dic3 [×2], C4⋊Dic3, C6.D4, C22×C12, C22×C12, C6×D4, C6×D4, C6×Q8, C3×C4○D4 [×4], C3×C4○D4 [×2], C42⋊C22, Q83Dic3 [×4], C2×C4.Dic3, C23.26D6, C6×C4○D4, (C6×D4)⋊9C4
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], C42⋊C22, C2×C6.D4, (C6×D4)⋊9C4

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

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

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

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

42 conjugacy classes

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

42 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + - - - + image C1 C2 C2 C2 C2 C4 C4 C4 S3 D4 D4 D6 Dic3 Dic3 Dic3 D6 C3⋊D4 C3⋊D4 C42⋊C22 (C6×D4)⋊9C4 kernel (C6×D4)⋊9C4 Q8⋊3Dic3 C2×C4.Dic3 C23.26D6 C6×C4○D4 C6×D4 C6×Q8 C3×C4○D4 C2×C4○D4 C2×C12 C22×C6 C22×C4 C2×D4 C2×Q8 C4○D4 C4○D4 C2×C4 C23 C3 C1 # reps 1 4 1 1 1 2 2 4 1 3 1 1 1 1 2 2 6 2 2 4

Matrix representation of (C6×D4)⋊9C4 in GL4(𝔽73) generated by

 60 30 0 0 43 30 0 0 0 0 60 30 0 0 43 30
,
 27 0 46 0 0 27 0 46 0 0 46 0 0 0 0 46
,
 7 14 66 59 59 66 14 7 14 28 66 59 45 59 14 7
,
 0 72 0 60 72 0 60 0 0 0 0 46 0 0 46 0
G:=sub<GL(4,GF(73))| [60,43,0,0,30,30,0,0,0,0,60,43,0,0,30,30],[27,0,0,0,0,27,0,0,46,0,46,0,0,46,0,46],[7,59,14,45,14,66,28,59,66,14,66,14,59,7,59,7],[0,72,0,0,72,0,0,0,0,60,0,46,60,0,46,0] >;

(C6×D4)⋊9C4 in GAP, Magma, Sage, TeX

(C_6\times D_4)\rtimes_9C_4
% in TeX

G:=Group("(C6xD4):9C4");
// GroupNames label

G:=SmallGroup(192,795);
// by ID

G=gap.SmallGroup(192,795);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,422,387,136,1684,438,102,6278]);
// 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=b*c>;
// generators/relations

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