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G = C3×C41D4order 96 = 25·3

Direct product of C3 and C41D4

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

Aliases: C3×C41D4, C126D4, C429C6, C41(C3×D4), (C2×D4)⋊3C6, C2.9(C6×D4), (C4×C12)⋊13C2, (C6×D4)⋊12C2, C6.72(C2×D4), C23.4(C2×C6), (C2×C6).82C23, (C22×C6).4C22, (C2×C12).125C22, C22.17(C22×C6), (C2×C4).23(C2×C6), SmallGroup(96,174)

Series: Derived Chief Lower central Upper central

C1C22 — C3×C41D4
C1C2C22C2×C6C22×C6C6×D4 — C3×C41D4
C1C22 — C3×C41D4
C1C2×C6 — C3×C41D4

Generators and relations for C3×C41D4
 G = < a,b,c,d | a3=b4=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >

Subgroups: 180 in 108 conjugacy classes, 52 normal (8 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×6], C22, C22 [×12], C6 [×3], C6 [×4], C2×C4 [×3], D4 [×12], C23 [×4], C12 [×6], C2×C6, C2×C6 [×12], C42, C2×D4 [×6], C2×C12 [×3], C3×D4 [×12], C22×C6 [×4], C41D4, C4×C12, C6×D4 [×6], C3×C41D4
Quotients: C1, C2 [×7], C3, C22 [×7], C6 [×7], D4 [×6], C23, C2×C6 [×7], C2×D4 [×3], C3×D4 [×6], C22×C6, C41D4, C6×D4 [×3], C3×C41D4

Smallest permutation representation of C3×C41D4
On 48 points
Generators in S48
(1 26 14)(2 27 15)(3 28 16)(4 25 13)(5 34 42)(6 35 43)(7 36 44)(8 33 41)(9 37 46)(10 38 47)(11 39 48)(12 40 45)(17 22 30)(18 23 31)(19 24 32)(20 21 29)
(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 31 10 35)(2 32 11 36)(3 29 12 33)(4 30 9 34)(5 13 22 46)(6 14 23 47)(7 15 24 48)(8 16 21 45)(17 37 42 25)(18 38 43 26)(19 39 44 27)(20 40 41 28)
(1 30)(2 29)(3 32)(4 31)(5 47)(6 46)(7 45)(8 48)(9 35)(10 34)(11 33)(12 36)(13 23)(14 22)(15 21)(16 24)(17 26)(18 25)(19 28)(20 27)(37 43)(38 42)(39 41)(40 44)

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

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

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

C3×C41D4 is a maximal subgroup of
C12.9D8  C425Dic3  C12.16D8  C42.72D6  C122D8  C12⋊D8  C42.74D6  Dic69D4  C124SD16  C428D6  C42.166D6  C4228D6  C42.238D6  D1211D4  Dic611D4  C42.168D6  C4230D6  C3×D42  C422C18

42 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B4A···4F6A···6F6G···6N12A···12L
order12222222334···46···66···612···12
size11114444112···21···14···42···2

42 irreducible representations

dim11111122
type++++
imageC1C2C2C3C6C6D4C3×D4
kernelC3×C41D4C4×C12C6×D4C41D4C42C2×D4C12C4
# reps1162212612

Matrix representation of C3×C41D4 in GL4(𝔽13) generated by

9000
0900
0010
0001
,
1000
0100
00012
0010
,
11200
21200
00120
00012
,
1000
21200
0001
0010
G:=sub<GL(4,GF(13))| [9,0,0,0,0,9,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,12,0],[1,2,0,0,12,12,0,0,0,0,12,0,0,0,0,12],[1,2,0,0,0,12,0,0,0,0,0,1,0,0,1,0] >;

C3×C41D4 in GAP, Magma, Sage, TeX

C_3\times C_4\rtimes_1D_4
% in TeX

G:=Group("C3xC4:1D4");
// GroupNames label

G:=SmallGroup(96,174);
// by ID

G=gap.SmallGroup(96,174);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-2,-2,313,151,938,230]);
// 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,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

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