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G = C3xC4:1D4order 96 = 25·3

Direct product of C3 and C4:1D4

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

Aliases: C3xC4:1D4, C12:6D4, C42:9C6, C4:1(C3xD4), (C2xD4):3C6, C2.9(C6xD4), (C4xC12):13C2, (C6xD4):12C2, C6.72(C2xD4), C23.4(C2xC6), (C2xC6).82C23, (C22xC6).4C22, (C2xC12).125C22, C22.17(C22xC6), (C2xC4).23(C2xC6), SmallGroup(96,174)

Series: Derived Chief Lower central Upper central

C1C22 — C3xC4:1D4
C1C2C22C2xC6C22xC6C6xD4 — C3xC4:1D4
C1C22 — C3xC4:1D4
C1C2xC6 — C3xC4:1D4

Generators and relations for C3xC4:1D4
 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, C2, C3, C4, C22, C22, C6, C6, C2xC4, D4, C23, C12, C2xC6, C2xC6, C42, C2xD4, C2xC12, C3xD4, C22xC6, C4:1D4, C4xC12, C6xD4, C3xC4:1D4
Quotients: C1, C2, C3, C22, C6, D4, C23, C2xC6, C2xD4, C3xD4, C22xC6, C4:1D4, C6xD4, C3xC4:1D4

Smallest permutation representation of C3xC4:1D4
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)]])

C3xC4:1D4 is a maximal subgroup of
C12.9D8  C42:5Dic3  C12.16D8  C42.72D6  C12:2D8  C12:D8  C42.74D6  Dic6:9D4  C12:4SD16  C42:8D6  C42.166D6  C42:28D6  C42.238D6  D12:11D4  Dic6:11D4  C42.168D6  C42:30D6  C3xD42  C42:2C18

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++++
imageC1C2C2C3C6C6D4C3xD4
kernelC3xC4:1D4C4xC12C6xD4C4:1D4C42C2xD4C12C4
# reps1162212612

Matrix representation of C3xC4:1D4 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] >;

C3xC4:1D4 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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