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G = C42:23D6order 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, (C2xQ8):12D6, C22:C4:35D6, (C4xC12):33C22, D6:3Q8:31C2, (C2xD4).111D6, C23:2D6.6C2, C4.4D4:13S3, (C6xQ8):15C22, C42:2S3:36C2, D6.23(C4oD4), C23.9D6:44C2, (C2xC6).223C24, C4:Dic3:42C22, Dic3:4D4:32C2, C2.76(D4:6D6), (C2xC12).632C23, Dic3:C4:67C22, D6:C4.136C22, (C4xDic3):57C22, (C6xD4).211C22, C23.8D6:40C2, (C22xC6).53C23, C23.55(C22xS3), C3:8(C22.45C24), C6.D4:34C22, C23.23D6:25C2, (S3xC23).66C22, C22.244(S3xC23), (C22xS3).217C23, (C2xDic3).255C23, (C22xDic3):28C22, C2.79(S3xC4oD4), (S3xC22:C4):19C2, C6.190(C2xC4oD4), (C3xC4.4D4):15C2, (S3xC2xC4).215C22, (C2xC4).74(C22xS3), (C3xC22:C4):31C22, (C2xC3:D4).61C22, SmallGroup(192,1238)

Series: Derived Chief Lower central Upper central

C1C2xC6 — C42:23D6
C1C3C6C2xC6C22xS3S3xC23S3xC22:C4 — C42:23D6
C3C2xC6 — C42:23D6
C1C22C4.4D4

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, C2xC4, C2xC4, C2xC4, D4, Q8, C23, C23, Dic3, C12, D6, D6, C2xC6, C2xC6, C42, C42, C22:C4, C22:C4, C4:C4, C22xC4, C2xD4, C2xD4, C2xQ8, C24, C4xS3, C2xDic3, C2xDic3, C3:D4, C2xC12, C2xC12, C3xD4, C3xQ8, C22xS3, C22xS3, C22xC6, C2xC22:C4, C42:C2, C4xD4, C22wrC2, C22:Q8, C22.D4, C4.4D4, C42:2C2, C4xDic3, Dic3:C4, C4:Dic3, D6:C4, C6.D4, C6.D4, C4xC12, C3xC22:C4, S3xC2xC4, C22xDic3, C2xC3:D4, C6xD4, C6xQ8, S3xC23, C22.45C24, C42:2S3, C23.8D6, S3xC22:C4, Dic3:4D4, C23.9D6, C23.23D6, C23:2D6, D6:3Q8, C3xC4.4D4, C42:23D6
Quotients: C1, C2, C22, S3, C23, D6, C4oD4, C24, C22xS3, C2xC4oD4, 2+ 1+4, S3xC23, C22.45C24, D4:6D6, S3xC4oD4, 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++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2S3D6D6D6D6C4oD42+ 1+4D4:6D6S3xC4oD4
kernelC42:23D6C42:2S3C23.8D6S3xC22:C4Dic3:4D4C23.9D6C23.23D6C23:2D6D6:3Q8C3xC4.4D4C4.4D4C42C22:C4C2xD4C2xQ8D6C6C2C2
# 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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