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G = C42⋊23D6  order 192 = 26·3

21st semidirect product of C42 and D6 acting via D6/C3=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42⋊23D6, C6.732+ 1+4, (C2×Q8)⋊12D6, C22⋊C4⋊35D6, (C4×C12)⋊33C22, D6⋊3Q8⋊31C2, (C2×D4).111D6, C23⋊2D6.6C2, C4.4D4⋊13S3, (C6×Q8)⋊15C22, C42⋊2S3⋊36C2, D6.23(C4○D4), C23.9D6⋊44C2, (C2×C6).223C24, C4⋊Dic3⋊42C22, Dic3⋊4D4⋊32C2, C2.76(D4⋊6D6), (C2×C12).632C23, Dic3⋊C4⋊67C22, D6⋊C4.136C22, (C4×Dic3)⋊57C22, (C6×D4).211C22, C23.8D6⋊40C2, (C22×C6).53C23, C23.55(C22×S3), C3⋊8(C22.45C24), C6.D4⋊34C22, C23.23D6⋊25C2, (S3×C23).66C22, C22.244(S3×C23), (C22×S3).217C23, (C2×Dic3).255C23, (C22×Dic3)⋊28C22, C2.79(S3×C4○D4), (S3×C22⋊C4)⋊19C2, C6.190(C2×C4○D4), (C3×C4.4D4)⋊15C2, (S3×C2×C4).215C22, (C2×C4).74(C22×S3), (C3×C22⋊C4)⋊31C22, (C2×C3⋊D4).61C22, SmallGroup(192,1238)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C6 — C42⋊23D6
Chief series
C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C23 — S3×C22⋊C4 — C42⋊23D6
Lower central
C3 — C2×C6 — C42⋊23D6
Upper central
C1 — C22 — C4.4D4

Generators and relations for C42⋊23D6
 G = < a,b,c,d | a4=b4=c6=d2=1, ab=ba, cac-1=ab2, ad=da, cbc-1=dbd=a2b, dcd=c-1 >

Subgroups: 672 in 248 conjugacy classes, 95 normal (27 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, C12, D6, D6, C2×C6, C2×C6, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, C2×Q8, C24, C4×S3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×Q8, C22×S3, C22×S3, C22×C6, C2×C22⋊C4, C42⋊C2, C4×D4, C22≀C2, C22⋊Q8, C22.D4, C4.4D4, C42⋊2C2, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C6.D4, C4×C12, C3×C22⋊C4, S3×C2×C4, C22×Dic3, C2×C3⋊D4, C6×D4, C6×Q8, S3×C23, C22.45C24, C42⋊2S3, C23.8D6, S3×C22⋊C4, Dic3⋊4D4, C23.9D6, C23.23D6, C23⋊2D6, D6⋊3Q8, C3×C4.4D4, C42⋊23D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, 2+ 1+4, S3×C23, C22.45C24, D4⋊6D6, S3×C4○D4, C42⋊23D6

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

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

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

39 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I 3 4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O6A6B6C6D6E12A···12F12G12H
order122222222234444444444444446666612···121212
size111144666622222444666612121212222884···488

39 irreducible representations

dim1111111111222222444
type++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2S3D6D6D6D6C4○D42+ 1+4D4⋊6D6S3×C4○D4
kernelC42⋊23D6C42⋊2S3C23.8D6S3×C22⋊C4Dic3⋊4D4C23.9D6C23.23D6C23⋊2D6D6⋊3Q8C3×C4.4D4C4.4D4C42C22⋊C4C2×D4C2×Q8D6C6C2C2
# reps1222221121114118124

Matrix representation of C42⋊23D6 ►in GL6(𝔽13)

010000
100000
001000
000100
000050
000005
,
800000
080000
0012000
0001200
0000111
0000012
,
100000
0120000
0001200
0011200
000010
0000112
,
100000
010000
0011200
0001200
0000120
0000121

G:=sub<GL(6,GF(13))| [0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,5,0,0,0,0,0,0,5],[8,0,0,0,0,0,0,8,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,11,12],[1,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,12,12,0,0,0,0,0,0,1,1,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,12,12,0,0,0,0,0,0,12,12,0,0,0,0,0,1] >;
 

C42⋊23D6 in GAP, Magma, Sage, TeX 

C_4^2\rtimes_{23}D_6
 
% in TeX
 
G:=Group("C4^2:23D6");
 
// GroupNames label
 
G:=SmallGroup(192,1238);
 
// by ID
 
G=gap.SmallGroup(192,1238);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,758,387,100,346,136,6278]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^4=c^6=d^2=1,a*b=b*a,c*a*c^-1=a*b^2,a*d=d*a,c*b*c^-1=d*b*d=a^2*b,d*c*d=c^-1>;
 
// generators/relations
 

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