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## G = (C2×D12)⋊13C4order 192 = 26·3

### 9th semidirect product of C2×D12 and C4 acting via C4/C2=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — (C2×D12)⋊13C4
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — C2×C3⋊D4 — C2×C4○D12 — (C2×D12)⋊13C4
 Lower central C3 — C6 — C2×C6 — (C2×D12)⋊13C4
 Upper central C1 — C4 — C22×C4 — C42⋊C2

Generators and relations for (C2×D12)⋊13C4
G = < a,b,c,d | a2=b12=c2=d4=1, ab=ba, dcd-1=ac=ca, dad-1=ab6, cbc=b-1, bd=db >

Subgroups: 456 in 158 conjugacy classes, 59 normal (41 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, C12, C12, D6, C2×C6, C2×C6, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, Dic6, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×C6, C23⋊C4, C42⋊C2, C42⋊C2, C2×C4○D4, C4×Dic3, C4⋊Dic3, C6.D4, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, C2×D12, C4○D12, C2×C3⋊D4, C22×C12, C23.C23, C23.6D6, C23.26D6, C3×C42⋊C2, C2×C4○D12, (C2×D12)⋊13C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22⋊C4, C22×C4, C2×D4, C4×S3, D12, C3⋊D4, C22×S3, C2×C22⋊C4, D6⋊C4, S3×C2×C4, C2×D12, C2×C3⋊D4, C23.C23, C2×D6⋊C4, (C2×D12)⋊13C4

Smallest permutation representation of (C2×D12)⋊13C4
On 48 points
Generators in S48
(1 43)(2 44)(3 45)(4 46)(5 47)(6 48)(7 37)(8 38)(9 39)(10 40)(11 41)(12 42)(13 29)(14 30)(15 31)(16 32)(17 33)(18 34)(19 35)(20 36)(21 25)(22 26)(23 27)(24 28)
(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 21)(2 20)(3 19)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)(10 24)(11 23)(12 22)(25 43)(26 42)(27 41)(28 40)(29 39)(30 38)(31 37)(32 48)(33 47)(34 46)(35 45)(36 44)
(1 19 37 35)(2 20 38 36)(3 21 39 25)(4 22 40 26)(5 23 41 27)(6 24 42 28)(7 13 43 29)(8 14 44 30)(9 15 45 31)(10 16 46 32)(11 17 47 33)(12 18 48 34)

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

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

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

42 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J ··· 4O 6A 6B 6C 6D 6E 12A 12B 12C 12D 12E ··· 12N 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 12 12 12 12 12 ··· 12 size 1 1 2 2 2 12 12 2 1 1 2 2 2 4 4 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 1 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + image C1 C2 C2 C2 C2 C4 C4 C4 S3 D4 D6 D6 C4×S3 D12 C3⋊D4 C23.C23 (C2×D12)⋊13C4 kernel (C2×D12)⋊13C4 C23.6D6 C23.26D6 C3×C42⋊C2 C2×C4○D12 C2×Dic6 S3×C2×C4 C2×D12 C42⋊C2 C2×C12 C22⋊C4 C22×C4 C2×C4 C2×C4 C2×C4 C3 C1 # reps 1 4 1 1 1 2 4 2 1 4 2 1 4 4 4 2 4

Matrix representation of (C2×D12)⋊13C4 in GL4(𝔽13) generated by

 2 9 8 3 4 11 5 8 0 0 2 4 0 0 9 11
,
 10 10 0 0 3 7 0 0 0 0 7 10 0 0 3 10
,
 0 5 6 6 5 0 6 0 0 9 8 8 9 4 0 5
,
 8 0 4 1 0 8 3 4 4 0 5 0 9 4 0 5
G:=sub<GL(4,GF(13))| [2,4,0,0,9,11,0,0,8,5,2,9,3,8,4,11],[10,3,0,0,10,7,0,0,0,0,7,3,0,0,10,10],[0,5,0,9,5,0,9,4,6,6,8,0,6,0,8,5],[8,0,4,9,0,8,0,4,4,3,5,0,1,4,0,5] >;

(C2×D12)⋊13C4 in GAP, Magma, Sage, TeX

(C_2\times D_{12})\rtimes_{13}C_4
% in TeX

G:=Group("(C2xD12):13C4");
// GroupNames label

G:=SmallGroup(192,565);
// by ID

G=gap.SmallGroup(192,565);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,422,58,1123,438,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^12=c^2=d^4=1,a*b=b*a,d*c*d^-1=a*c=c*a,d*a*d^-1=a*b^6,c*b*c=b^-1,b*d=d*b>;
// generators/relations

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