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G = C42.196D6order 192 = 26·3

16th non-split extension by C42 of D6 acting via D6/S3=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.196D6, M4(2).23D6, C4≀C27S3, D4⋊S34C4, C32(C8○D8), C3⋊C8.37D4, D4.S34C4, D4.4(C4×S3), C3⋊Q164C4, C6.39(C4×D4), Q8.9(C4×S3), D12.C49C2, Q82S34C4, C4○D4.37D6, D12.7(C2×C4), C4.203(S3×D4), C424S37C2, C12.362(C2×D4), D4.Dic32C2, Dic6.7(C2×C4), C12.20(C22×C4), (C4×C12).51C22, Q8.13D6.2C2, C12.53D410C2, (C2×C12).264C23, C4○D12.13C22, C4.Dic3.9C22, C22.9(D42S3), C2.23(Dic34D4), (C3×M4(2)).25C22, (C4×C3⋊C8)⋊3C2, C3⋊C8.8(C2×C4), C4.20(S3×C2×C4), (C3×C4≀C2)⋊12C2, (C3×D4).7(C2×C4), (C3×Q8).7(C2×C4), (C2×C6).35(C4○D4), (C2×C3⋊C8).222C22, (C3×C4○D4).5C22, (C2×C4).370(C22×S3), SmallGroup(192,383)

Series: Derived Chief Lower central Upper central

C1C12 — C42.196D6
C1C3C6C12C2×C12C4○D12Q8.13D6 — C42.196D6
C3C6C12 — C42.196D6
C1C4C2×C4C4≀C2

Generators and relations for C42.196D6
 G = < a,b,c,d | a4=b4=c6=1, d2=cbc-1=b-1, ab=ba, cac-1=ab-1, ad=da, bd=db, dcd-1=b-1c-1 >

Subgroups: 240 in 106 conjugacy classes, 45 normal (all characteristic)
C1, C2, C2 [×3], C3, C4 [×2], C4 [×4], C22, C22 [×2], S3, C6, C6 [×2], C8 [×6], C2×C4, C2×C4 [×3], D4, D4 [×3], Q8, Q8, Dic3, C12 [×2], C12 [×3], D6, C2×C6, C2×C6, C42, C2×C8 [×4], M4(2), M4(2) [×3], D8, SD16 [×2], Q16, C4○D4, C4○D4, C3⋊C8 [×4], C3⋊C8, C24, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C2×C12 [×2], C3×D4, C3×D4, C3×Q8, C4×C8, C4≀C2, C4≀C2, C8.C4, C8○D4 [×2], C4○D8, S3×C8, C8⋊S3, C2×C3⋊C8 [×2], C2×C3⋊C8, C4.Dic3, C4.Dic3, D4⋊S3, D4.S3, Q82S3, C3⋊Q16, C4×C12, C3×M4(2), C4○D12, C3×C4○D4, C8○D8, C4×C3⋊C8, C424S3, C12.53D4, C3×C4≀C2, D12.C4, D4.Dic3, Q8.13D6, C42.196D6
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×2], C23, D6 [×3], C22×C4, C2×D4, C4○D4, C4×S3 [×2], C22×S3, C4×D4, S3×C2×C4, S3×D4, D42S3, C8○D8, Dic34D4, C42.196D6

Smallest permutation representation of C42.196D6
On 48 points
Generators in S48
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 32)(33 35 37 39)(34 36 38 40)(41 45)(42 46)(43 47)(44 48)
(1 7 5 3)(2 8 6 4)(9 15 13 11)(10 16 14 12)(17 23 21 19)(18 24 22 20)(25 31 29 27)(26 32 30 28)(33 39 37 35)(34 40 38 36)(41 47 45 43)(42 48 46 44)
(1 20 11 33 46 26)(2 25 47 40 12 19)(3 18 13 39 48 32)(4 31 41 38 14 17)(5 24 15 37 42 30)(6 29 43 36 16 23)(7 22 9 35 44 28)(8 27 45 34 10 21)
(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)

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

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

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

42 conjugacy classes

class 1 2A2B2C2D 3 4A4B4C···4G4H4I6A6B6C8A8B8C8D8E8F8G···8L8M8N12A12B12C···12G12H24A24B
order122223444···4446668888888···888121212···12122424
size1124122112···24122483333446···61212224···4888

42 irreducible representations

dim111111111111222222222444
type++++++++++++++-
imageC1C2C2C2C2C2C2C2C4C4C4C4S3D4D6D6D6C4○D4C4×S3C4×S3C8○D8S3×D4D42S3C42.196D6
kernelC42.196D6C4×C3⋊C8C424S3C12.53D4C3×C4≀C2D12.C4D4.Dic3Q8.13D6D4⋊S3D4.S3Q82S3C3⋊Q16C4≀C2C3⋊C8C42M4(2)C4○D4C2×C6D4Q8C3C4C22C1
# reps111111112222121112228114

Matrix representation of C42.196D6 in GL4(𝔽5) generated by

0002
0031
2130
4003
,
4003
0424
3410
1001
,
0100
1302
0001
0412
,
3400
3203
4004
0010
G:=sub<GL(4,GF(5))| [0,0,2,4,0,0,1,0,0,3,3,0,2,1,0,3],[4,0,3,1,0,4,4,0,0,2,1,0,3,4,0,1],[0,1,0,0,1,3,0,4,0,0,0,1,0,2,1,2],[3,3,4,0,4,2,0,0,0,0,0,1,0,3,4,0] >;

C42.196D6 in GAP, Magma, Sage, TeX

C_4^2._{196}D_6
% in TeX

G:=Group("C4^2.196D6");
// GroupNames label

G:=SmallGroup(192,383);
// by ID

G=gap.SmallGroup(192,383);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,555,58,136,1684,851,438,102,6278]);
// Polycyclic

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

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