Copied to
clipboard

## G = C3×C4⋊D4order 96 = 25·3

### Direct product of C3 and C4⋊D4

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

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

Generators and relations for C3×C4⋊D4
G = < a,b,c,d | a3=b4=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 148 in 94 conjugacy classes, 48 normal (24 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×2], C4 [×3], C22, C22 [×2], C22 [×8], C6 [×3], C6 [×4], C2×C4 [×2], C2×C4 [×2], C2×C4 [×2], D4 [×6], C23, C23 [×2], C12 [×2], C12 [×3], C2×C6, C2×C6 [×2], C2×C6 [×8], C22⋊C4 [×2], C4⋊C4, C22×C4, C2×D4, C2×D4 [×2], C2×C12 [×2], C2×C12 [×2], C2×C12 [×2], C3×D4 [×6], C22×C6, C22×C6 [×2], C4⋊D4, C3×C22⋊C4 [×2], C3×C4⋊C4, C22×C12, C6×D4, C6×D4 [×2], C3×C4⋊D4
Quotients: C1, C2 [×7], C3, C22 [×7], C6 [×7], D4 [×4], C23, C2×C6 [×7], C2×D4 [×2], C4○D4, C3×D4 [×4], C22×C6, C4⋊D4, C6×D4 [×2], C3×C4○D4, C3×C4⋊D4

Smallest permutation representation of C3×C4⋊D4
On 48 points
Generators in S48
(1 11 31)(2 12 32)(3 9 29)(4 10 30)(5 13 33)(6 14 34)(7 15 35)(8 16 36)(17 37 41)(18 38 42)(19 39 43)(20 40 44)(21 28 45)(22 25 46)(23 26 47)(24 27 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 35 39 23)(2 34 40 22)(3 33 37 21)(4 36 38 24)(5 41 28 9)(6 44 25 12)(7 43 26 11)(8 42 27 10)(13 17 45 29)(14 20 46 32)(15 19 47 31)(16 18 48 30)
(2 4)(5 28)(6 27)(7 26)(8 25)(10 12)(13 45)(14 48)(15 47)(16 46)(18 20)(21 33)(22 36)(23 35)(24 34)(30 32)(38 40)(42 44)

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

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

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

42 conjugacy classes

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

42 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + + + image C1 C2 C2 C2 C2 C3 C6 C6 C6 C6 D4 D4 C4○D4 C3×D4 C3×D4 C3×C4○D4 kernel C3×C4⋊D4 C3×C22⋊C4 C3×C4⋊C4 C22×C12 C6×D4 C4⋊D4 C22⋊C4 C4⋊C4 C22×C4 C2×D4 C12 C2×C6 C6 C4 C22 C2 # reps 1 2 1 1 3 2 4 2 2 6 2 2 2 4 4 4

Matrix representation of C3×C4⋊D4 in GL4(𝔽13) generated by

 3 0 0 0 0 3 0 0 0 0 1 0 0 0 0 1
,
 5 0 0 0 4 8 0 0 0 0 12 0 0 0 0 12
,
 12 9 0 0 0 1 0 0 0 0 0 12 0 0 1 0
,
 1 4 0 0 0 12 0 0 0 0 1 0 0 0 0 12
G:=sub<GL(4,GF(13))| [3,0,0,0,0,3,0,0,0,0,1,0,0,0,0,1],[5,4,0,0,0,8,0,0,0,0,12,0,0,0,0,12],[12,0,0,0,9,1,0,0,0,0,0,1,0,0,12,0],[1,0,0,0,4,12,0,0,0,0,1,0,0,0,0,12] >;

C3×C4⋊D4 in GAP, Magma, Sage, TeX

C_3\times C_4\rtimes D_4
% in TeX

G:=Group("C3xC4:D4");
// GroupNames label

G:=SmallGroup(96,168);
// by ID

G=gap.SmallGroup(96,168);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-2,-2,313,151,938]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^4=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

׿
×
𝔽