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G = C3×M4(2).C4order 192 = 26·3

Direct product of C3 and M4(2).C4

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Aliases: C3×M4(2).C4, M4(2).1C12, C8.4(C2×C12), C4.78(C6×D4), C8.C43C6, C24.39(C2×C4), C12.65(C4⋊C4), (C2×C12).44Q8, (C2×C12).523D4, C12.483(C2×D4), (C22×C6).5Q8, C23.5(C3×Q8), C22.2(C6×Q8), C4.29(C22×C12), (C3×M4(2)).2C4, (C2×M4(2)).2C6, (C2×C24).198C22, C12.187(C22×C4), (C2×C12).903C23, (C6×M4(2)).34C2, M4(2).10(C2×C6), (C22×C12).415C22, (C3×M4(2)).44C22, C2.16(C6×C4⋊C4), C4.16(C3×C4⋊C4), C6.72(C2×C4⋊C4), (C2×C4).7(C3×Q8), (C2×C8).17(C2×C6), (C2×C6).15(C2×Q8), (C2×C6).28(C4⋊C4), (C2×C4).26(C2×C12), (C2×C4).126(C3×D4), C22.12(C3×C4⋊C4), (C3×C8.C4)⋊12C2, (C2×C12).199(C2×C4), (C22×C4).39(C2×C6), (C2×C4).78(C22×C6), SmallGroup(192,863)

Series: Derived Chief Lower central Upper central

C1C4 — C3×M4(2).C4
C1C2C4C2×C4C2×C12C3×M4(2)C3×C8.C4 — C3×M4(2).C4
C1C2C4 — C3×M4(2).C4
C1C12C22×C12 — C3×M4(2).C4

Generators and relations for C3×M4(2).C4
 G = < a,b,c,d | a3=b8=c2=1, d4=b4, ab=ba, ac=ca, ad=da, cbc=b5, dbd-1=b-1, dcd-1=b4c >

Subgroups: 130 in 102 conjugacy classes, 78 normal (22 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C8, C8, C2×C4, C2×C4, C23, C12, C12, C2×C6, C2×C6, C2×C6, C2×C8, C2×C8, M4(2), M4(2), C22×C4, C24, C24, C2×C12, C2×C12, C22×C6, C8.C4, C2×M4(2), C2×M4(2), C2×C24, C2×C24, C3×M4(2), C3×M4(2), C22×C12, M4(2).C4, C3×C8.C4, C6×M4(2), C6×M4(2), C3×M4(2).C4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, Q8, C23, C12, C2×C6, C4⋊C4, C22×C4, C2×D4, C2×Q8, C2×C12, C3×D4, C3×Q8, C22×C6, C2×C4⋊C4, C3×C4⋊C4, C22×C12, C6×D4, C6×Q8, M4(2).C4, C6×C4⋊C4, C3×M4(2).C4

Smallest permutation representation of C3×M4(2).C4
On 48 points
Generators in S48
(1 29 21)(2 30 22)(3 31 23)(4 32 24)(5 25 17)(6 26 18)(7 27 19)(8 28 20)(9 39 44)(10 40 45)(11 33 46)(12 34 47)(13 35 48)(14 36 41)(15 37 42)(16 38 43)
(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 5)(3 7)(9 13)(11 15)(17 21)(19 23)(25 29)(27 31)(33 37)(35 39)(42 46)(44 48)
(1 41 7 43 5 45 3 47)(2 48 8 42 6 44 4 46)(9 32 11 30 13 28 15 26)(10 31 12 29 14 27 16 25)(17 40 23 34 21 36 19 38)(18 39 24 33 22 35 20 37)

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

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

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

66 conjugacy classes

class 1 2A2B2C2D3A3B4A4B4C4D4E6A6B6C···6H8A···8L12A12B12C12D12E···12J24A···24X
order122223344444666···68···81212121212···1224···24
size112221111222112···24···411112···24···4

66 irreducible representations

dim1111111122222244
type++++--
imageC1C2C2C3C4C6C6C12D4Q8Q8C3×D4C3×Q8C3×Q8M4(2).C4C3×M4(2).C4
kernelC3×M4(2).C4C3×C8.C4C6×M4(2)M4(2).C4C3×M4(2)C8.C4C2×M4(2)M4(2)C2×C12C2×C12C22×C6C2×C4C2×C4C23C3C1
# reps14328861621142224

Matrix representation of C3×M4(2).C4 in GL4(𝔽73) generated by

8000
0800
0080
0008
,
465400
592700
592701
4627270
,
72000
72100
72010
00072
,
10710
00721
600720
6027720
G:=sub<GL(4,GF(73))| [8,0,0,0,0,8,0,0,0,0,8,0,0,0,0,8],[46,59,59,46,54,27,27,27,0,0,0,27,0,0,1,0],[72,72,72,0,0,1,0,0,0,0,1,0,0,0,0,72],[1,0,60,60,0,0,0,27,71,72,72,72,0,1,0,0] >;

C3×M4(2).C4 in GAP, Magma, Sage, TeX

C_3\times M_4(2).C_4
% in TeX

G:=Group("C3xM4(2).C4");
// GroupNames label

G:=SmallGroup(192,863);
// by ID

G=gap.SmallGroup(192,863);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,336,365,176,1059,4204,172,124]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^8=c^2=1,d^4=b^4,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^5,d*b*d^-1=b^-1,d*c*d^-1=b^4*c>;
// generators/relations

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