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G = C24:9D6order 192 = 26·3

4th semidirect product of C24 and D6 acting via D6/C3=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C24:9D6, C6.302+ 1+4, C22:C4:8D6, C22wrC2:8S3, C23:2D6:7C2, (C2xD4).87D6, C24:4S3:9C2, D6:C4:15C22, (C2xC6).138C24, (C2xC12).32C23, (S3xC23):8C22, (C23xC6):11C22, C2.32(D4:6D6), C23.12D6:13C2, C3:1(C24:C22), (C4xDic3):18C22, (C2xDic6):23C22, (C6xD4).112C22, C23.11D6:15C2, C6.D4:18C22, (C22xS3).57C23, C23.120(C22xS3), C22.159(S3xC23), (C22xC6).183C23, (C2xDic3).63C23, (C3xC22wrC2):9C2, (C3xC22:C4):8C22, (C2xC4).32(C22xS3), (C2xC3:D4).22C22, SmallGroup(192,1153)

Series: Derived Chief Lower central Upper central

C1C2xC6 — C24:9D6
C1C3C6C2xC6C22xS3S3xC23C23:2D6 — C24:9D6
C3C2xC6 — C24:9D6
C1C22C22wrC2

Generators and relations for C24:9D6
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e6=f2=1, ab=ba, eae-1=ac=ca, ad=da, faf=acd, fbf=bc=cb, ebe-1=bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=e-1 >

Subgroups: 800 in 260 conjugacy classes, 91 normal (12 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C6, C2xC4, C2xC4, D4, Q8, C23, C23, C23, Dic3, C12, D6, C2xC6, C2xC6, C42, C22:C4, C22:C4, C2xD4, C2xD4, C2xQ8, C24, C24, Dic6, C2xDic3, C3:D4, C2xC12, C3xD4, C22xS3, C22xS3, C22xC6, C22xC6, C22xC6, C22wrC2, C22wrC2, C4.4D4, C4xDic3, D6:C4, C6.D4, C3xC22:C4, C2xDic6, C2xC3:D4, C6xD4, S3xC23, C23xC6, C24:C22, C23.11D6, C23.12D6, C23:2D6, C24:4S3, C3xC22wrC2, C24:9D6
Quotients: C1, C2, C22, S3, C23, D6, C24, C22xS3, 2+ 1+4, S3xC23, C24:C22, D4:6D6, C24:9D6

Smallest permutation representation of C24:9D6
On 48 points
Generators in S48
(1 40)(2 38)(3 42)(4 34)(5 32)(6 36)(7 37)(8 41)(9 39)(10 31)(11 35)(12 33)(13 25)(14 44)(15 27)(16 46)(17 29)(18 48)(19 43)(20 26)(21 45)(22 28)(23 47)(24 30)
(1 13)(2 17)(3 15)(4 16)(5 14)(6 18)(7 19)(8 23)(9 21)(10 22)(11 20)(12 24)(25 40)(26 35)(27 42)(28 31)(29 38)(30 33)(32 44)(34 46)(36 48)(37 43)(39 45)(41 47)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(25 43)(26 44)(27 45)(28 46)(29 47)(30 48)(31 34)(32 35)(33 36)(37 40)(38 41)(39 42)
(1 4)(2 5)(3 6)(7 10)(8 11)(9 12)(13 16)(14 17)(15 18)(19 22)(20 23)(21 24)(25 46)(26 47)(27 48)(28 43)(29 44)(30 45)(31 37)(32 38)(33 39)(34 40)(35 41)(36 42)
(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 6)(2 5)(3 4)(7 12)(8 11)(9 10)(13 24)(14 23)(15 22)(16 21)(17 20)(18 19)(25 27)(28 30)(31 36)(32 35)(33 34)(37 42)(38 41)(39 40)(43 45)(46 48)

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

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

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

33 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I 3 4A4B4C4D···4I6A6B6C6D···6I6J12A12B12C
order122222222234444···46666···66121212
size111144441212244412···122224···48888

33 irreducible representations

dim111111222244
type+++++++++++
imageC1C2C2C2C2C2S3D6D6D62+ 1+4D4:6D6
kernelC24:9D6C23.11D6C23.12D6C23:2D6C24:4S3C3xC22wrC2C22wrC2C22:C4C2xD4C24C6C2
# reps163321133136

Matrix representation of C24:9D6 in GL8(F13)

00100000
00010000
10000000
01000000
00000010
000011021
00001000
000011120
,
119000000
42000000
001190000
00420000
00001300
000001200
00007073
000036106
,
120000000
012000000
001200000
000120000
00001000
00000100
00000010
00000001
,
10000000
01000000
00100000
00010000
000012000
000001200
000000120
000000012
,
1212000000
10000000
00110000
001200000
00001000
000081200
00000010
000040412
,
1212000000
01000000
00110000
000120000
000012000
000001200
00000010
00009001

G:=sub<GL(8,GF(13))| [0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,11,1,11,0,0,0,0,0,0,0,1,0,0,0,0,1,2,0,2,0,0,0,0,0,1,0,0],[11,4,0,0,0,0,0,0,9,2,0,0,0,0,0,0,0,0,11,4,0,0,0,0,0,0,9,2,0,0,0,0,0,0,0,0,1,0,7,3,0,0,0,0,3,12,0,6,0,0,0,0,0,0,7,10,0,0,0,0,0,0,3,6],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12],[12,1,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,8,0,4,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,4,0,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,12,0,0,9,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1] >;

C24:9D6 in GAP, Magma, Sage, TeX

C_2^4\rtimes_9D_6
% in TeX

G:=Group("C2^4:9D6");
// GroupNames label

G:=SmallGroup(192,1153);
// by ID

G=gap.SmallGroup(192,1153);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,758,219,1571,570,6278]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^6=f^2=1,a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*a,f*a*f=a*c*d,f*b*f=b*c=c*b,e*b*e^-1=b*d=d*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f=e^-1>;
// generators/relations

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