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G = (C3×D4).31D4order 192 = 26·3

1st non-split extension by C3×D4 of D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C3×D4).31D4, (C2×C6)⋊11SD16, C6.72C22≀C2, (C2×D4).199D6, C12.206(C2×D4), (C2×C12).301D4, C35(C22⋊SD16), C6.64(C2×SD16), (C22×D4).5S3, D4⋊Dic339C2, D4.13(C3⋊D4), C4⋊Dic322C22, C224(D4.S3), (C22×C6).197D4, (C22×C4).166D6, C12.55D416C2, C12.48D426C2, C6.103(C8⋊C22), (C2×C12).473C23, C2.5(C244S3), (C2×Dic6)⋊15C22, (C6×D4).241C22, C23.93(C3⋊D4), C2.23(D126C22), (C22×C12).198C22, (D4×C2×C6).4C2, (C2×C3⋊C8)⋊11C22, C4.59(C2×C3⋊D4), (C2×D4.S3)⋊23C2, (C2×C6).554(C2×D4), C2.17(C2×D4.S3), (C2×C4).84(C3⋊D4), (C2×C4).559(C22×S3), C22.217(C2×C3⋊D4), SmallGroup(192,777)

Series: Derived Chief Lower central Upper central

C1C2×C12 — (C3×D4).31D4
C1C3C6C2×C6C2×C12C2×Dic6C2×D4.S3 — (C3×D4).31D4
C3C6C2×C12 — (C3×D4).31D4
C1C22C22×C4C22×D4

Generators and relations for (C3×D4).31D4
 G = < a,b,c,d,e | a3=b4=c2=1, d4=e2=b2, ab=ba, ac=ca, dad-1=eae-1=a-1, cbc=ebe-1=b-1, bd=db, dcd-1=bc, ece-1=b-1c, ede-1=d3 >

Subgroups: 488 in 188 conjugacy classes, 51 normal (25 characteristic)
C1, C2 [×3], C2 [×6], C3, C4 [×2], C4 [×3], C22, C22 [×2], C22 [×18], C6 [×3], C6 [×6], C8 [×2], C2×C4 [×2], C2×C4 [×4], D4 [×4], D4 [×6], Q8 [×2], C23, C23 [×10], Dic3 [×2], C12 [×2], C12, C2×C6, C2×C6 [×2], C2×C6 [×18], C22⋊C4, C4⋊C4 [×2], C2×C8 [×2], SD16 [×4], C22×C4, C2×D4 [×2], C2×D4 [×5], C2×Q8, C24, C3⋊C8 [×2], Dic6 [×2], C2×Dic3 [×2], C2×C12 [×2], C2×C12 [×2], C3×D4 [×4], C3×D4 [×6], C22×C6, C22×C6 [×10], C22⋊C8, D4⋊C4 [×2], C22⋊Q8, C2×SD16 [×2], C22×D4, C2×C3⋊C8 [×2], Dic3⋊C4, C4⋊Dic3, D4.S3 [×4], C6.D4, C2×Dic6, C22×C12, C6×D4 [×2], C6×D4 [×5], C23×C6, C22⋊SD16, C12.55D4, D4⋊Dic3 [×2], C12.48D4, C2×D4.S3 [×2], D4×C2×C6, (C3×D4).31D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×6], C23, D6 [×3], SD16 [×2], C2×D4 [×3], C3⋊D4 [×6], C22×S3, C22≀C2, C2×SD16, C8⋊C22, D4.S3 [×2], C2×C3⋊D4 [×3], C22⋊SD16, D126C22, C2×D4.S3, C244S3, (C3×D4).31D4

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

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

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

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

39 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I 3 4A4B4C4D4E6A···6G6H···6O8A8B8C8D12A12B12C12D
order12222222223444446···66···6888812121212
size1111224444222424242···24···4121212124444

39 irreducible representations

dim1111112222222222444
type+++++++++++++-
imageC1C2C2C2C2C2S3D4D4D4D6D6SD16C3⋊D4C3⋊D4C3⋊D4C8⋊C22D4.S3D126C22
kernel(C3×D4).31D4C12.55D4D4⋊Dic3C12.48D4C2×D4.S3D4×C2×C6C22×D4C2×C12C3×D4C22×C6C22×C4C2×D4C2×C6C2×C4D4C23C6C22C2
# reps1121211141124282122

Matrix representation of (C3×D4).31D4 in GL6(𝔽73)

6400000
080000
001000
000100
000010
000001
,
7200000
0720000
001000
000100
000001
0000720
,
100000
0720000
0072000
0007200
0000720
000001
,
0720000
100000
0007200
001000
00006767
0000667
,
010000
100000
000100
001000
0000676
000066

G:=sub<GL(6,GF(73))| [64,0,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,1,0],[1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,1],[0,1,0,0,0,0,72,0,0,0,0,0,0,0,0,1,0,0,0,0,72,0,0,0,0,0,0,0,67,6,0,0,0,0,67,67],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,67,6,0,0,0,0,6,6] >;

(C3×D4).31D4 in GAP, Magma, Sage, TeX

(C_3\times D_4)._{31}D_4
% in TeX

G:=Group("(C3xD4).31D4");
// GroupNames label

G:=SmallGroup(192,777);
// by ID

G=gap.SmallGroup(192,777);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,253,254,1684,851,102,6278]);
// Polycyclic

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

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