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G = C3×C4⋊D4order 96 = 25·3

Direct product of C3 and C4⋊D4

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

Aliases: C3×C4⋊D4, C129D4, C4⋊C42C6, C42(C3×D4), (C2×C6)⋊4D4, (C2×D4)⋊2C6, C2.5(C6×D4), (C6×D4)⋊11C2, C22⋊C43C6, (C22×C4)⋊6C6, C6.68(C2×D4), C222(C3×D4), (C22×C12)⋊11C2, C6.41(C4○D4), (C2×C6).76C23, C23.11(C2×C6), (C2×C12).123C22, C22.11(C22×C6), (C22×C6).27C22, (C3×C4⋊C4)⋊11C2, (C2×C4).3(C2×C6), C2.4(C3×C4○D4), (C3×C22⋊C4)⋊11C2, SmallGroup(96,168)

Series: Derived Chief Lower central Upper central

C1C22 — C3×C4⋊D4
C1C2C22C2×C6C22×C6C6×D4 — C3×C4⋊D4
C1C22 — C3×C4⋊D4
C1C2×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)])

C3×C4⋊D4 is a maximal subgroup of
(C6×D4)⋊C4  (C2×C6).D8  C4⋊D4.S3  C6.Q16⋊C2  D1216D4  D1217D4  C3⋊C822D4  C4⋊D4⋊S3  Dic617D4  C3⋊C823D4  C3⋊C85D4  C12⋊(C4○D4)  C6.322+ 1+4  Dic619D4  Dic620D4  C4⋊C4.178D6  C6.342+ 1+4  C6.702- 1+4  C6.712- 1+4  C6.372+ 1+4  C4⋊C421D6  C6.382+ 1+4  C6.722- 1+4  D1219D4  C6.402+ 1+4  C6.732- 1+4  D1220D4  C6.422+ 1+4  C6.432+ 1+4  C6.442+ 1+4  C6.452+ 1+4  C6.462+ 1+4  C6.1152+ 1+4  C6.472+ 1+4  C6.482+ 1+4  C6.492+ 1+4  C3×D42

42 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B4A4B4C4D4E4F6A···6F6G6H6I6J6K6L6M6N12A···12H12I12J12K12L
order12222222334444446···66666666612···1212121212
size11112244112222441···1222244442···24444

42 irreducible representations

dim1111111111222222
type+++++++
imageC1C2C2C2C2C3C6C6C6C6D4D4C4○D4C3×D4C3×D4C3×C4○D4
kernelC3×C4⋊D4C3×C22⋊C4C3×C4⋊C4C22×C12C6×D4C4⋊D4C22⋊C4C4⋊C4C22×C4C2×D4C12C2×C6C6C4C22C2
# reps1211324226222444

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

3000
0300
0010
0001
,
5000
4800
00120
00012
,
12900
0100
00012
0010
,
1400
01200
0010
00012
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

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