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G = C3×C4⋊Dic3order 144 = 24·32

Direct product of C3 and C4⋊Dic3

direct product, metacyclic, supersoluble, monomial

Aliases: C3×C4⋊Dic3, C121C12, C123Dic3, C6.22D12, C6.9Dic6, C62.18C22, C4⋊(C3×Dic3), (C3×C12)⋊4C4, C6.4(C3×D4), (C3×C6).5Q8, C6.2(C3×Q8), C327(C4⋊C4), (C6×C12).8C2, C6.8(C2×C12), (C2×C12).4C6, C2.1(C3×D12), (C3×C6).20D4, (C2×C6).43D6, (C2×C12).19S3, C22.5(S3×C6), C2.2(C3×Dic6), C2.4(C6×Dic3), (C6×Dic3).2C2, (C2×Dic3).2C6, C6.20(C2×Dic3), C32(C3×C4⋊C4), (C2×C6).8(C2×C6), (C2×C4).3(C3×S3), (C3×C6).29(C2×C4), SmallGroup(144,78)

Series: Derived Chief Lower central Upper central

C1C6 — C3×C4⋊Dic3
C1C3C6C2×C6C62C6×Dic3 — C3×C4⋊Dic3
C3C6 — C3×C4⋊Dic3
C1C2×C6C2×C12

Generators and relations for C3×C4⋊Dic3
 G = < a,b,c,d | a3=b4=c6=1, d2=c3, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >

Subgroups: 104 in 60 conjugacy classes, 38 normal (26 characteristic)
C1, C2 [×3], C3 [×2], C3, C4 [×2], C4 [×2], C22, C6 [×6], C6 [×3], C2×C4, C2×C4 [×2], C32, Dic3 [×2], C12 [×4], C12 [×4], C2×C6 [×2], C2×C6, C4⋊C4, C3×C6 [×3], C2×Dic3 [×2], C2×C12 [×2], C2×C12 [×3], C3×Dic3 [×2], C3×C12 [×2], C62, C4⋊Dic3, C3×C4⋊C4, C6×Dic3 [×2], C6×C12, C3×C4⋊Dic3
Quotients: C1, C2 [×3], C3, C4 [×2], C22, S3, C6 [×3], C2×C4, D4, Q8, Dic3 [×2], C12 [×2], D6, C2×C6, C4⋊C4, C3×S3, Dic6, D12, C2×Dic3, C2×C12, C3×D4, C3×Q8, C3×Dic3 [×2], S3×C6, C4⋊Dic3, C3×C4⋊C4, C3×Dic6, C3×D12, C6×Dic3, C3×C4⋊Dic3

Smallest permutation representation of C3×C4⋊Dic3
On 48 points
Generators in S48
(1 5 3)(2 6 4)(7 9 11)(8 10 12)(13 17 15)(14 18 16)(19 23 21)(20 24 22)(25 29 27)(26 30 28)(31 33 35)(32 34 36)(37 39 41)(38 40 42)(43 45 47)(44 46 48)
(1 30 17 22)(2 25 18 23)(3 26 13 24)(4 27 14 19)(5 28 15 20)(6 29 16 21)(7 32 44 40)(8 33 45 41)(9 34 46 42)(10 35 47 37)(11 36 48 38)(12 31 43 39)
(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 4 32)(2 34 5 31)(3 33 6 36)(7 30 10 27)(8 29 11 26)(9 28 12 25)(13 41 16 38)(14 40 17 37)(15 39 18 42)(19 44 22 47)(20 43 23 46)(21 48 24 45)

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

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

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

C3×C4⋊Dic3 is a maximal subgroup of
D123Dic3  C6.17D24  Dic6⋊Dic3  C12.73D12  C12.Dic6  C12.6Dic6  C6.18D24  C12.8Dic6  C62.11C23  C62.13C23  Dic36Dic6  C62.16C23  C62.17C23  C62.18C23  C62.19C23  C62.24C23  D67Dic6  C62.28C23  C62.39C23  C12.30D12  C62.42C23  D6.D12  Dic35D12  C62.65C23  D12⋊Dic3  D64Dic6  C62.70C23  C127D12  C122D12  C123Dic6  C12⋊Dic6  C12×Dic6  C12×D12  C3×S3×C4⋊C4  C3×D4×Dic3  C3×Q8×Dic3  C62.20D6  C36⋊C12  C62.30D6
C3×C4⋊Dic3 is a maximal quotient of
C62.20D6  C36⋊C12

54 conjugacy classes

class 1 2A2B2C3A3B3C3D3E4A4B4C4D4E4F6A···6F6G···6O12A···12P12Q···12X
order1222333334444446···66···612···1212···12
size1111112222266661···12···22···26···6

54 irreducible representations

dim1111111122222222222222
type+++++--+-+
imageC1C2C2C3C4C6C6C12S3D4Q8Dic3D6C3×S3Dic6D12C3×D4C3×Q8C3×Dic3S3×C6C3×Dic6C3×D12
kernelC3×C4⋊Dic3C6×Dic3C6×C12C4⋊Dic3C3×C12C2×Dic3C2×C12C12C2×C12C3×C6C3×C6C12C2×C6C2×C4C6C6C6C6C4C22C2C2
# reps1212442811121222224244

Matrix representation of C3×C4⋊Dic3 in GL4(𝔽13) generated by

3000
0300
0030
0003
,
1000
0100
0050
0008
,
10000
0400
00100
0004
,
0100
12000
0001
00120
G:=sub<GL(4,GF(13))| [3,0,0,0,0,3,0,0,0,0,3,0,0,0,0,3],[1,0,0,0,0,1,0,0,0,0,5,0,0,0,0,8],[10,0,0,0,0,4,0,0,0,0,10,0,0,0,0,4],[0,12,0,0,1,0,0,0,0,0,0,12,0,0,1,0] >;

C3×C4⋊Dic3 in GAP, Magma, Sage, TeX

C_3\times C_4\rtimes {\rm Dic}_3
% in TeX

G:=Group("C3xC4:Dic3");
// GroupNames label

G:=SmallGroup(144,78);
// by ID

G=gap.SmallGroup(144,78);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-3,72,313,151,3461]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^4=c^6=1,d^2=c^3,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations

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