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G = C42:24D6  order 192 = 26·3

22nd semidirect product of C42 and D6 acting via D6/C3=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42:24D6, C6.1292+ 1+4, (C2xQ8):13D6, D6:C4:6C22, C22:C4:21D6, C23:2D6:25C2, D6:D4:26C2, (C4xC12):29C22, (C2xD4).112D6, C4.4D4:16S3, (C6xQ8):16C22, C42:7S3:30C2, C2.53(D4oD12), (C2xC12).83C23, (C2xC6).227C24, C2.77(D4:6D6), C12.23D4:24C2, (S3xC23):12C22, C3:2(C24:C22), (C4xDic3):37C22, (C2xDic6):10C22, (C6xD4).212C22, (C2xD12).34C22, (C22xC6).57C23, C23.59(C22xS3), C23.11D6:43C2, C6.D4:35C22, (C22xS3).99C23, C22.248(S3xC23), (C2xDic3).117C23, (C3xC4.4D4):19C2, (C3xC22:C4):32C22, (C2xC4).200(C22xS3), (C2xC3:D4).65C22, SmallGroup(192,1242)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2xC6 — C42:24D6
Chief series
C1 — C3 — C6 — C2xC6 — C22xS3 — S3xC23 — C23:2D6 — C42:24D6
Lower central
C3 — C2xC6 — C42:24D6
Upper central
C1 — C22 — C4.4D4

Generators and relations for C42:24D6
 G = < a,b,c,d | a4=b4=c6=d2=1, ab=ba, cac-1=a-1, dad=ab2, cbc-1=a2b-1, dbd=a2b, dcd=c-1 >

Subgroups: 896 in 260 conjugacy classes, 91 normal (17 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, S3, C6, C6, C6, C2xC4, C2xC4, C2xC4, D4, Q8, C23, C23, Dic3, C12, D6, C2xC6, C2xC6, C42, C42, C22:C4, C22:C4, C2xD4, C2xD4, C2xQ8, C2xQ8, C24, Dic6, D12, C2xDic3, C3:D4, C2xC12, C2xC12, C3xD4, C3xQ8, C22xS3, C22xS3, C22xC6, C22wrC2, C4.4D4, C4.4D4, C4xDic3, D6:C4, C6.D4, C4xC12, C3xC22:C4, C2xDic6, C2xD12, C2xC3:D4, C6xD4, C6xQ8, S3xC23, C24:C22, C42:7S3, D6:D4, C23.11D6, C23:2D6, C12.23D4, C3xC4.4D4, C42:24D6
Quotients: C1, C2, C22, S3, C23, D6, C24, C22xS3, 2+ 1+4, S3xC23, C24:C22, D4:6D6, D4oD12, C42:24D6

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

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

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

33 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I 3 4A···4E4F4G4H4I6A6B6C6D6E12A···12F12G12H
order122222222234···444446666612···121212
size1111441212121224···412121212222884···488

33 irreducible representations

dim111111122222444
type++++++++++++++
imageC1C2C2C2C2C2C2S3D6D6D6D62+ 1+4D4:6D6D4oD12
kernelC42:24D6C42:7S3D6:D4C23.11D6C23:2D6C12.23D4C3xC4.4D4C4.4D4C42C22:C4C2xD4C2xQ8C6C2C2
# reps124422111411324

Matrix representation of C42:24D6 ►in GL8(F13)

00100000
000120000
10000000
012000000
00000010
00000001
000012000
000001200
,
01000000
120000000
000120000
00100000
000010600
00007300
000000106
00000073
,
10000000
012000000
00100000
000120000
000001200
000011200
00000001
000000121
,
10000000
01000000
001200000
000120000
000011200
000001200
000000121
00000001

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

C42:24D6 in GAP, Magma, Sage, TeX 

C_4^2\rtimes_{24}D_6
 
% in TeX
 
G:=Group("C4^2:24D6");
 
// GroupNames label
 
G:=SmallGroup(192,1242);
 
// by ID
 
G=gap.SmallGroup(192,1242);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,758,219,1571,570,297,192,6278]);
 
// 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^-1,d*a*d=a*b^2,c*b*c^-1=a^2*b^-1,d*b*d=a^2*b,d*c*d=c^-1>;
 
// generators/relations
 

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