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## G = C4×C3⋊D4order 96 = 25·3

### Direct product of C4 and C3⋊D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C4×C3⋊D4
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — C2×C3⋊D4 — C4×C3⋊D4
 Lower central C3 — C6 — C4×C3⋊D4
 Upper central C1 — C2×C4 — C22×C4

Generators and relations for C4×C3⋊D4
G = < a,b,c,d | a4=b3=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 194 in 94 conjugacy classes, 45 normal (29 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×2], C4 [×5], C22, C22 [×2], C22 [×6], S3 [×2], C6 [×3], C6 [×2], C2×C4 [×2], C2×C4 [×7], D4 [×4], C23, C23, Dic3 [×2], Dic3 [×2], C12 [×2], C12, D6 [×2], D6 [×2], C2×C6, C2×C6 [×2], C2×C6 [×2], C42, C22⋊C4 [×2], C4⋊C4, C22×C4, C22×C4, C2×D4, C4×S3 [×2], C2×Dic3 [×3], C3⋊D4 [×4], C2×C12 [×2], C2×C12 [×2], C22×S3, C22×C6, C4×D4, C4×Dic3, Dic3⋊C4, D6⋊C4, C6.D4, S3×C2×C4, C2×C3⋊D4, C22×C12, C4×C3⋊D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×2], C23, D6 [×3], C22×C4, C2×D4, C4○D4, C4×S3 [×2], C3⋊D4 [×2], C22×S3, C4×D4, S3×C2×C4, C4○D12, C2×C3⋊D4, C4×C3⋊D4

Smallest permutation representation of C4×C3⋊D4
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 37 23)(2 38 24)(3 39 21)(4 40 22)(5 36 46)(6 33 47)(7 34 48)(8 35 45)(9 29 27)(10 30 28)(11 31 25)(12 32 26)(13 43 17)(14 44 18)(15 41 19)(16 42 20)
(1 47 27 14)(2 48 28 15)(3 45 25 16)(4 46 26 13)(5 32 43 22)(6 29 44 23)(7 30 41 24)(8 31 42 21)(9 18 37 33)(10 19 38 34)(11 20 39 35)(12 17 40 36)
(1 3)(2 4)(5 19)(6 20)(7 17)(8 18)(9 31)(10 32)(11 29)(12 30)(13 48)(14 45)(15 46)(16 47)(21 37)(22 38)(23 39)(24 40)(25 27)(26 28)(33 42)(34 43)(35 44)(36 41)

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

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

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

36 conjugacy classes

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

36 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C4 S3 D4 D6 D6 C4○D4 C3⋊D4 C4×S3 C4○D12 kernel C4×C3⋊D4 C4×Dic3 Dic3⋊C4 D6⋊C4 C6.D4 S3×C2×C4 C2×C3⋊D4 C22×C12 C3⋊D4 C22×C4 C12 C2×C4 C23 C6 C4 C22 C2 # reps 1 1 1 1 1 1 1 1 8 1 2 2 1 2 4 4 4

Matrix representation of C4×C3⋊D4 in GL3(𝔽13) generated by

 8 0 0 0 12 0 0 0 12
,
 1 0 0 0 12 12 0 1 0
,
 12 0 0 0 11 9 0 11 2
,
 1 0 0 0 1 0 0 12 12
G:=sub<GL(3,GF(13))| [8,0,0,0,12,0,0,0,12],[1,0,0,0,12,1,0,12,0],[12,0,0,0,11,11,0,9,2],[1,0,0,0,1,12,0,0,12] >;

C4×C3⋊D4 in GAP, Magma, Sage, TeX

C_4\times C_3\rtimes D_4
% in TeX

G:=Group("C4xC3:D4");
// GroupNames label

G:=SmallGroup(96,135);
// by ID

G=gap.SmallGroup(96,135);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,217,50,2309]);
// Polycyclic

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

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