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G = (C6×D4)⋊6C4order 192 = 26·3

2nd semidirect product of C6×D4 and C4 acting via C4/C2=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C6×D4)⋊6C4, (C2×D4)⋊4Dic3, (C2×D4).196D6, (C2×C12).189D4, C12.203(C2×D4), D4.5(C2×Dic3), (C22×D4).3S3, D4⋊Dic337C2, C12.80(C22×C4), C4⋊Dic368C22, (C22×C4).163D6, (C22×C6).195D4, C6.101(C8⋊C22), C12.31(C22⋊C4), (C2×C12).470C23, C2.5(D126C22), C4.8(C6.D4), (C6×D4).238C22, C23.91(C3⋊D4), C34(C23.37D4), C4.10(C22×Dic3), C23.26D618C2, (C22×C12).195C22, C22.20(C6.D4), (D4×C2×C6).2C2, (C2×C3⋊C8)⋊9C22, C4.89(C2×C3⋊D4), (C3×D4).22(C2×C4), (C2×C6).552(C2×D4), C6.72(C2×C22⋊C4), (C2×C12).116(C2×C4), (C2×C4.Dic3)⋊17C2, (C2×C4).23(C2×Dic3), C2.8(C2×C6.D4), C22.90(C2×C3⋊D4), (C2×C4).196(C3⋊D4), (C2×C4).557(C22×S3), (C2×C6).111(C22⋊C4), SmallGroup(192,774)

Series: Derived Chief Lower central Upper central

C1C12 — (C6×D4)⋊6C4
C1C3C6C2×C6C2×C12C4⋊Dic3C23.26D6 — (C6×D4)⋊6C4
C3C6C12 — (C6×D4)⋊6C4
C1C22C22×C4C22×D4

Generators and relations for (C6×D4)⋊6C4
 G = < a,b,c,d | a6=b4=c2=d4=1, ab=ba, ac=ca, dad-1=a-1b2, cbc=dbd-1=b-1, dcd-1=b-1c >

Subgroups: 456 in 190 conjugacy classes, 71 normal (23 characteristic)
C1, C2, C2 [×2], C2 [×6], C3, C4 [×2], C4 [×2], C4 [×2], C22, C22 [×2], C22 [×18], C6, C6 [×2], C6 [×6], C8 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×2], D4 [×4], D4 [×6], C23, C23 [×10], Dic3 [×2], C12 [×2], C12 [×2], C2×C6, C2×C6 [×2], C2×C6 [×18], C42, C22⋊C4, C4⋊C4 [×2], C2×C8 [×2], M4(2) [×2], C22×C4, C2×D4 [×6], C2×D4 [×3], C24, C3⋊C8 [×2], C2×Dic3 [×2], C2×C12 [×2], C2×C12 [×4], C3×D4 [×4], C3×D4 [×6], C22×C6, C22×C6 [×10], D4⋊C4 [×4], C42⋊C2, C2×M4(2), C22×D4, C2×C3⋊C8 [×2], C4.Dic3 [×2], C4×Dic3, C4⋊Dic3 [×2], C6.D4, C22×C12, C6×D4 [×6], C6×D4 [×3], C23×C6, C23.37D4, D4⋊Dic3 [×4], C2×C4.Dic3, C23.26D6, D4×C2×C6, (C6×D4)⋊6C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×4], C23, Dic3 [×4], D6 [×3], C22⋊C4 [×4], C22×C4, C2×D4 [×2], C2×Dic3 [×6], C3⋊D4 [×4], C22×S3, C2×C22⋊C4, C8⋊C22 [×2], C6.D4 [×4], C22×Dic3, C2×C3⋊D4 [×2], C23.37D4, D126C22 [×2], C2×C6.D4, (C6×D4)⋊6C4

Smallest permutation representation of (C6×D4)⋊6C4
On 48 points
Generators in S48
(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 7 13 11)(2 8 14 12)(3 9 15 10)(4 16 24 20)(5 17 22 21)(6 18 23 19)(25 31 28 34)(26 32 29 35)(27 33 30 36)(37 48 40 45)(38 43 41 46)(39 44 42 47)
(1 23)(2 24)(3 22)(4 14)(5 15)(6 13)(7 18)(8 16)(9 17)(10 21)(11 19)(12 20)(25 42)(26 37)(27 38)(28 39)(29 40)(30 41)(31 44)(32 45)(33 46)(34 47)(35 48)(36 43)
(1 28 6 47)(2 30 4 43)(3 26 5 45)(7 31 18 42)(8 33 16 38)(9 35 17 40)(10 32 21 37)(11 34 19 39)(12 36 20 41)(13 25 23 44)(14 27 24 46)(15 29 22 48)

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

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

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

42 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I 3 4A4B4C4D4E4F4G4H6A···6G6H···6O8A8B8C8D12A12B12C12D
order12222222223444444446···66···6888812121212
size111122444422222121212122···24···4121212124444

42 irreducible representations

dim1111112222222244
type+++++++++-++
imageC1C2C2C2C2C4S3D4D4D6Dic3D6C3⋊D4C3⋊D4C8⋊C22D126C22
kernel(C6×D4)⋊6C4D4⋊Dic3C2×C4.Dic3C23.26D6D4×C2×C6C6×D4C22×D4C2×C12C22×C6C22×C4C2×D4C2×D4C2×C4C23C6C2
# reps1411181311426224

Matrix representation of (C6×D4)⋊6C4 in GL6(𝔽73)

6500000
090000
0072000
0007200
000010
000001
,
100000
010000
0012900
00107200
00007244
0000631
,
7200000
0720000
0012900
0007200
000010
00001072
,
0280000
1300000
000010
000001
001000
000100

G:=sub<GL(6,GF(73))| [65,0,0,0,0,0,0,9,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,10,0,0,0,0,29,72,0,0,0,0,0,0,72,63,0,0,0,0,44,1],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,29,72,0,0,0,0,0,0,1,10,0,0,0,0,0,72],[0,13,0,0,0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0] >;

(C6×D4)⋊6C4 in GAP, Magma, Sage, TeX

(C_6\times D_4)\rtimes_6C_4
% in TeX

G:=Group("(C6xD4):6C4");
// GroupNames label

G:=SmallGroup(192,774);
// by ID

G=gap.SmallGroup(192,774);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,232,422,387,1684,438,102,6278]);
// Polycyclic

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

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