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G = C4⋊Dic3⋊C4order 192 = 26·3

2nd semidirect product of C4⋊Dic3 and C4 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — C4⋊Dic3⋊C4
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — C22×C12 — C12.48D4 — C4⋊Dic3⋊C4
 Lower central C3 — C2×C6 — C2×C12 — C4⋊Dic3⋊C4
 Upper central C1 — C22 — C22×C4 — C2.C42

Generators and relations for C4⋊Dic3⋊C4
G = < a,b,c,d | a4=b6=d4=1, c2=b3, ab=ba, cac-1=a-1, dad-1=ab3, cbc-1=b-1, bd=db, dcd-1=a-1b3c >

Subgroups: 224 in 80 conjugacy classes, 29 normal (all characteristic)
C1, C2, C2, C3, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, Q8, C23, Dic3, C12, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×Q8, C3⋊C8, Dic6, C2×Dic3, C2×C12, C2×C12, C22×C6, C2.C42, C22⋊C8, C22⋊Q8, C2×C3⋊C8, Dic3⋊C4, C4⋊Dic3, C6.D4, C2×Dic6, C22×C12, C22×C12, C23.31D4, C12.55D4, C3×C2.C42, C12.48D4, C4⋊Dic3⋊C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D6, C22⋊C4, SD16, Q16, C4×S3, D12, C3⋊D4, C23⋊C4, Q8⋊C4, C4≀C2, D6⋊C4, D4.S3, C3⋊Q16, C23.31D4, C424S3, C23.6D6, C6.SD16, C4⋊Dic3⋊C4

Smallest permutation representation of C4⋊Dic3⋊C4
On 48 points
Generators in S48
(1 22 10 13)(2 23 11 14)(3 24 12 15)(4 19 7 16)(5 20 8 17)(6 21 9 18)(25 37 34 46)(26 38 35 47)(27 39 36 48)(28 40 31 43)(29 41 32 44)(30 42 33 45)
(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 26 4 29)(2 25 5 28)(3 30 6 27)(7 32 10 35)(8 31 11 34)(9 36 12 33)(13 38 16 41)(14 37 17 40)(15 42 18 39)(19 44 22 47)(20 43 23 46)(21 48 24 45)
(13 16)(14 17)(15 18)(19 22)(20 23)(21 24)(25 46 31 40)(26 47 32 41)(27 48 33 42)(28 43 34 37)(29 44 35 38)(30 45 36 39)

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

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

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

39 conjugacy classes

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

39 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + - + + - - image C1 C2 C2 C2 C4 C4 S3 D4 D4 D6 SD16 Q16 C4×S3 D12 C3⋊D4 C4≀C2 C42⋊4S3 C23⋊C4 D4.S3 C3⋊Q16 C23.6D6 kernel C4⋊Dic3⋊C4 C12.55D4 C3×C2.C42 C12.48D4 C4⋊Dic3 C2×Dic6 C2.C42 C2×C12 C22×C6 C22×C4 C2×C6 C2×C6 C2×C4 C2×C4 C23 C6 C2 C6 C22 C22 C2 # reps 1 1 1 1 2 2 1 1 1 1 2 2 2 2 2 4 8 1 1 1 2

Matrix representation of C4⋊Dic3⋊C4 in GL4(𝔽73) generated by

 46 0 0 0 64 27 0 0 0 0 27 0 0 0 63 46
,
 8 0 0 0 15 64 0 0 0 0 72 0 0 0 0 72
,
 12 1 0 0 3 61 0 0 0 0 22 2 0 0 13 51
,
 1 0 0 0 29 46 0 0 0 0 1 20 0 0 0 72
G:=sub<GL(4,GF(73))| [46,64,0,0,0,27,0,0,0,0,27,63,0,0,0,46],[8,15,0,0,0,64,0,0,0,0,72,0,0,0,0,72],[12,3,0,0,1,61,0,0,0,0,22,13,0,0,2,51],[1,29,0,0,0,46,0,0,0,0,1,0,0,0,20,72] >;

C4⋊Dic3⋊C4 in GAP, Magma, Sage, TeX

C_4\rtimes {\rm Dic}_3\rtimes C_4
% in TeX

G:=Group("C4:Dic3:C4");
// GroupNames label

G:=SmallGroup(192,11);
// by ID

G=gap.SmallGroup(192,11);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,141,36,422,1571,570,192,6278]);
// Polycyclic

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

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