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## G = (C6×D4)⋊6C4order 192 = 26·3

### 2nd 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)⋊6C4
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — C4⋊Dic3 — C23.26D6 — (C6×D4)⋊6C4
 Lower central C3 — C6 — C12 — (C6×D4)⋊6C4
 Upper central C1 — C22 — C22×C4 — C22×D4

Generators and relations for (C6×D4)⋊6C4
G = < a,b,c,d | a6=b4=c2=d4=1, ab=ba, ac=ca, dad-1=a-1b2, cbc=dbd-1=b-1, dcd-1=b-1c >

Subgroups: 456 in 190 conjugacy classes, 71 normal (23 characteristic)
C1, C2, C2, C2, C3, C4, C4, C4, C22, C22, C22, C6, C6, C6, C8, C2×C4, C2×C4, C2×C4, D4, D4, C23, C23, Dic3, C12, C12, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C2×D4, C2×D4, C24, C3⋊C8, C2×Dic3, C2×C12, C2×C12, C3×D4, C3×D4, C22×C6, C22×C6, D4⋊C4, C42⋊C2, C2×M4(2), C22×D4, C2×C3⋊C8, C4.Dic3, C4×Dic3, C4⋊Dic3, C6.D4, C22×C12, C6×D4, C6×D4, C23×C6, C23.37D4, D4⋊Dic3, C2×C4.Dic3, C23.26D6, D4×C2×C6, (C6×D4)⋊6C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, Dic3, D6, C22⋊C4, C22×C4, C2×D4, C2×Dic3, C3⋊D4, C22×S3, C2×C22⋊C4, C8⋊C22, C6.D4, C22×Dic3, C2×C3⋊D4, C23.37D4, D126C22, C2×C6.D4, (C6×D4)⋊6C4

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

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

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

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

42 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 3 4A 4B 4C 4D 4E 4F 4G 4H 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 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 2 2 12 12 12 12 2 ··· 2 4 ··· 4 12 12 12 12 4 4 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 C4 S3 D4 D4 D6 Dic3 D6 C3⋊D4 C3⋊D4 C8⋊C22 D12⋊6C22 kernel (C6×D4)⋊6C4 D4⋊Dic3 C2×C4.Dic3 C23.26D6 D4×C2×C6 C6×D4 C22×D4 C2×C12 C22×C6 C22×C4 C2×D4 C2×D4 C2×C4 C23 C6 C2 # reps 1 4 1 1 1 8 1 3 1 1 4 2 6 2 2 4

Matrix representation of (C6×D4)⋊6C4 in GL6(𝔽73)

 65 0 0 0 0 0 0 9 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 29 0 0 0 0 10 72 0 0 0 0 0 0 72 44 0 0 0 0 63 1
,
 72 0 0 0 0 0 0 72 0 0 0 0 0 0 1 29 0 0 0 0 0 72 0 0 0 0 0 0 1 0 0 0 0 0 10 72
,
 0 28 0 0 0 0 13 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0

G:=sub<GL(6,GF(73))| [65,0,0,0,0,0,0,9,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,10,0,0,0,0,29,72,0,0,0,0,0,0,72,63,0,0,0,0,44,1],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,29,72,0,0,0,0,0,0,1,10,0,0,0,0,0,72],[0,13,0,0,0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0] >;

(C6×D4)⋊6C4 in GAP, Magma, Sage, TeX

(C_6\times D_4)\rtimes_6C_4
% in TeX

G:=Group("(C6xD4):6C4");
// GroupNames label

G:=SmallGroup(192,774);
// by ID

G=gap.SmallGroup(192,774);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,232,422,387,1684,438,102,6278]);
// 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=d*b*d^-1=b^-1,d*c*d^-1=b^-1*c>;
// generators/relations

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