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G = D12:21D4order 192 = 26·3

9th semidirect product of D12 and D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D12:21D4, C6.1172+ 1+4, C4:C4:10D6, (C2xQ8):19D6, C22:Q8:7S3, C3:7(D4:5D4), D6.20(C2xD4), C4.111(S3xD4), C12:D4:25C2, D6:D4:16C2, (C6xQ8):7C22, D6:C4:20C22, C12.234(C2xD4), C22:C4.57D6, Dic3:5D4:25C2, C6.76(C22xD4), D6.D4:17C2, C2.34(D4oD12), (C2xD12):25C22, (C22xD12):16C2, (C2xC6).174C24, (C2xC12).54C23, (C22xC4).252D6, C12.23D4:12C2, Dic3:C4:53C22, C22:3(Q8:3S3), (C4xDic3):28C22, (S3xC23).52C22, (C22xC6).202C23, C23.199(C22xS3), C22.195(S3xC23), (C22xS3).196C23, (C22xC12).254C22, (C2xDic3).233C23, C6.D4.115C22, C2.49(C2xS3xD4), (C2xC6):7(C4oD4), (C4xC3:D4):22C2, (S3xC22:C4):8C2, (S3xC2xC4):18C22, (C3xC4:C4):19C22, (C2xQ8:3S3):7C2, C6.114(C2xC4oD4), (C3xC22:Q8):10C2, (C2xC4).47(C22xS3), C2.17(C2xQ8:3S3), (C2xC3:D4).122C22, (C3xC22:C4).29C22, SmallGroup(192,1189)

Series: Derived Chief Lower central Upper central

C1C2xC6 — D12:21D4
C1C3C6C2xC6C22xS3S3xC23S3xC22:C4 — D12:21D4
C3C2xC6 — D12:21D4
C1C22C22:Q8

Generators and relations for D12:21D4
 G = < a,b,c,d | a12=b2=c4=d2=1, bab=a-1, cac-1=dad=a5, cbc-1=dbd=a10b, dcd=c-1 >

Subgroups: 1040 in 334 conjugacy classes, 107 normal (43 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C2xC4, C2xC4, C2xC4, D4, Q8, C23, C23, Dic3, C12, C12, D6, D6, C2xC6, C2xC6, C2xC6, C42, C22:C4, C22:C4, C4:C4, C4:C4, C4:C4, C22xC4, C22xC4, C2xD4, C2xQ8, C4oD4, C24, C4xS3, D12, D12, C2xDic3, C3:D4, C2xC12, C2xC12, C2xC12, C3xQ8, C22xS3, C22xS3, C22xS3, C22xC6, C2xC22:C4, C4xD4, C22wrC2, C4:D4, C22:Q8, C22.D4, C4.4D4, C22xD4, C2xC4oD4, C4xDic3, Dic3:C4, D6:C4, D6:C4, C6.D4, C3xC22:C4, C3xC4:C4, C3xC4:C4, S3xC2xC4, S3xC2xC4, C2xD12, C2xD12, C2xD12, Q8:3S3, C2xC3:D4, C22xC12, C6xQ8, S3xC23, D4:5D4, S3xC22:C4, D6:D4, Dic3:5D4, D6.D4, C12:D4, C12:D4, C4xC3:D4, C12.23D4, C3xC22:Q8, C22xD12, C2xQ8:3S3, D12:21D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, C4oD4, C24, C22xS3, C22xD4, C2xC4oD4, 2+ 1+4, S3xD4, Q8:3S3, S3xC23, D4:5D4, C2xS3xD4, C2xQ8:3S3, D4oD12, D12:21D4

Smallest permutation representation of D12:21D4
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 3)(4 12)(5 11)(6 10)(7 9)(13 23)(14 22)(15 21)(16 20)(17 19)(25 31)(26 30)(27 29)(32 36)(33 35)(37 45)(38 44)(39 43)(40 42)(46 48)
(1 36 40 14)(2 29 41 19)(3 34 42 24)(4 27 43 17)(5 32 44 22)(6 25 45 15)(7 30 46 20)(8 35 47 13)(9 28 48 18)(10 33 37 23)(11 26 38 16)(12 31 39 21)
(1 20)(2 13)(3 18)(4 23)(5 16)(6 21)(7 14)(8 19)(9 24)(10 17)(11 22)(12 15)(25 39)(26 44)(27 37)(28 42)(29 47)(30 40)(31 45)(32 38)(33 43)(34 48)(35 41)(36 46)

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

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

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

39 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J2K2L 3 4A4B4C···4G4H4I4J4K4L6A6B6C6D6E12A12B12C12D12E12F12G12H
order12222222222223444···444444666661212121212121212
size11112266661212122224···46666122224444448888

39 irreducible representations

dim1111111111122222224444
type+++++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2S3D4D6D6D6D6C4oD42+ 1+4S3xD4Q8:3S3D4oD12
kernelD12:21D4S3xC22:C4D6:D4Dic3:5D4D6.D4C12:D4C4xC3:D4C12.23D4C3xC22:Q8C22xD12C2xQ8:3S3C22:Q8D12C22:C4C4:C4C22xC4C2xQ8C2xC6C6C4C22C2
# reps1221231111114231141222

Matrix representation of D12:21D4 in GL6(F13)

010000
1200000
001100
0012000
0000120
0000012
,
1200000
010000
00121200
000100
000010
000001
,
080000
500000
0012000
001100
000001
0000120
,
050000
800000
0012000
001100
000001
000010

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

D12:21D4 in GAP, Magma, Sage, TeX

D_{12}\rtimes_{21}D_4
% in TeX

G:=Group("D12:21D4");
// GroupNames label

G:=SmallGroup(192,1189);
// by ID

G=gap.SmallGroup(192,1189);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,219,184,1571,297,192,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^12=b^2=c^4=d^2=1,b*a*b=a^-1,c*a*c^-1=d*a*d=a^5,c*b*c^-1=d*b*d=a^10*b,d*c*d=c^-1>;
// generators/relations

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