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G = D12.32C23order 192 = 26·3

13rd non-split extension by D12 of C23 acting via C23/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C12.37C24, D12.32C23, 2+ 1+4:8S3, Dic6.32C23, C4oD4:8D6, C3:5(D4oD8), D4oD12:9C2, (C2xD4):16D6, (C3xD4).36D4, C3:C8.16C23, (C3xQ8).36D4, D4:D6:11C2, D4:S3:20C22, Q8.13D6:8C2, C12.269(C2xD4), (C6xD4):24C22, C4.37(S3xC23), D12:6C22:11C2, D4.18(C3:D4), C4oD12:10C22, (C2xD12):39C22, D4.Dic3:10C2, D4.S3:19C22, Q8.25(C3:D4), D4.25(C22xS3), C3:Q16:21C22, (C3xD4).25C23, C6.171(C22xD4), (C3xQ8).25C23, Q8.35(C22xS3), (C2xC12).118C23, Q8:2S3:19C22, C4.Dic3:16C22, (C3x2+ 1+4):2C2, (C2xD4:S3):32C2, (C2xC3:C8):24C22, (C2xC6).85(C2xD4), C4.75(C2xC3:D4), (C3xC4oD4):8C22, C22.6(C2xC3:D4), C2.44(C22xC3:D4), (C2xC4).102(C22xS3), SmallGroup(192,1394)

Series: Derived Chief Lower central Upper central

C1C12 — D12.32C23
C1C3C6C12D12C2xD12D4oD12 — D12.32C23
C3C6C12 — D12.32C23
C1C2C4oD42+ 1+4

Generators and relations for D12.32C23
 G = < a,b,c,d,e | a12=b2=c2=d2=e2=1, bab=dad=a-1, ac=ca, eae=a7, cbc=a6b, dbd=a10b, ebe=a3b, cd=dc, ce=ec, ede=a9d >

Subgroups: 728 in 268 conjugacy classes, 107 normal (20 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, S3, C6, C6, C8, C2xC4, C2xC4, D4, D4, Q8, Q8, C23, Dic3, C12, C12, C12, D6, C2xC6, C2xC6, C2xC8, M4(2), D8, SD16, Q16, C2xD4, C2xD4, C4oD4, C4oD4, C4oD4, C3:C8, C3:C8, Dic6, C4xS3, D12, D12, C3:D4, C2xC12, C2xC12, C3xD4, C3xD4, C3xQ8, C22xS3, C22xC6, C8oD4, C2xD8, C4oD8, C8:C22, 2+ 1+4, 2+ 1+4, C2xC3:C8, C4.Dic3, D4:S3, D4.S3, Q8:2S3, C3:Q16, C2xD12, C4oD12, S3xD4, Q8:3S3, C6xD4, C6xD4, C3xC4oD4, C3xC4oD4, C3xC4oD4, D4oD8, C2xD4:S3, D12:6C22, D4.Dic3, D4:D6, Q8.13D6, D4oD12, C3x2+ 1+4, D12.32C23
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, C24, C3:D4, C22xS3, C22xD4, C2xC3:D4, S3xC23, D4oD8, C22xC3:D4, D12.32C23

Smallest permutation representation of D12.32C23
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 33)(2 32)(3 31)(4 30)(5 29)(6 28)(7 27)(8 26)(9 25)(10 36)(11 35)(12 34)(13 45)(14 44)(15 43)(16 42)(17 41)(18 40)(19 39)(20 38)(21 37)(22 48)(23 47)(24 46)
(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)
(2 12)(3 11)(4 10)(5 9)(6 8)(13 22)(14 21)(15 20)(16 19)(17 18)(23 24)(25 31)(26 30)(27 29)(32 36)(33 35)(37 46)(38 45)(39 44)(40 43)(41 42)(47 48)
(1 19)(2 14)(3 21)(4 16)(5 23)(6 18)(7 13)(8 20)(9 15)(10 22)(11 17)(12 24)(25 40)(26 47)(27 42)(28 37)(29 44)(30 39)(31 46)(32 41)(33 48)(34 43)(35 38)(36 45)

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

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

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

39 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J 3 4A4B4C4D4E4F6A6B···6J8A8B8C8D8E12A···12F
order12222222222344444466···68888812···12
size112224441212122222241224···4661212124···4

39 irreducible representations

dim11111111222222248
type+++++++++++++++
imageC1C2C2C2C2C2C2C2S3D4D4D6D6C3:D4C3:D4D4oD8D12.32C23
kernelD12.32C23C2xD4:S3D12:6C22D4.Dic3D4:D6Q8.13D6D4oD12C3x2+ 1+42+ 1+4C3xD4C3xQ8C2xD4C4oD4D4Q8C3C1
# reps13313311131346221

Matrix representation of D12.32C23 in GL6(F73)

72720000
100000
0007200
001000
00727212
00107272
,
72720000
010000
000010
00727212
001000
0072011
,
7200000
0720000
001000
000100
0000720
0011072
,
7200000
110000
000100
001000
00117271
000001
,
7200000
0720000
00575700
00571600
005757032
00570160

G:=sub<GL(6,GF(73))| [72,1,0,0,0,0,72,0,0,0,0,0,0,0,0,1,72,1,0,0,72,0,72,0,0,0,0,0,1,72,0,0,0,0,2,72],[72,0,0,0,0,0,72,1,0,0,0,0,0,0,0,72,1,72,0,0,0,72,0,0,0,0,1,1,0,1,0,0,0,2,0,1],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,1,0,0,0,1,0,1,0,0,0,0,72,0,0,0,0,0,0,72],[72,1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,1,0,0,0,1,0,1,0,0,0,0,0,72,0,0,0,0,0,71,1],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,57,57,57,57,0,0,57,16,57,0,0,0,0,0,0,16,0,0,0,0,32,0] >;

D12.32C23 in GAP, Magma, Sage, TeX

D_{12}._{32}C_2^3
% in TeX

G:=Group("D12.32C2^3");
// GroupNames label

G:=SmallGroup(192,1394);
// by ID

G=gap.SmallGroup(192,1394);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,387,675,1684,235,102,6278]);
// Polycyclic

G:=Group<a,b,c,d,e|a^12=b^2=c^2=d^2=e^2=1,b*a*b=d*a*d=a^-1,a*c=c*a,e*a*e=a^7,c*b*c=a^6*b,d*b*d=a^10*b,e*b*e=a^3*b,c*d=d*c,c*e=e*c,e*d*e=a^9*d>;
// generators/relations

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