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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, (C2×Q8)⋊13D6, D6⋊C4⋊6C22, C22⋊C4⋊21D6, C23⋊2D6⋊25C2, D6⋊D4⋊26C2, (C4×C12)⋊29C22, (C2×D4).112D6, C4.4D4⋊16S3, (C6×Q8)⋊16C22, C42⋊7S3⋊30C2, C2.53(D4○D12), (C2×C12).83C23, (C2×C6).227C24, C2.77(D4⋊6D6), C12.23D4⋊24C2, (S3×C23)⋊12C22, C3⋊2(C24⋊C22), (C4×Dic3)⋊37C22, (C2×Dic6)⋊10C22, (C6×D4).212C22, (C2×D12).34C22, (C22×C6).57C23, C23.59(C22×S3), C23.11D6⋊43C2, C6.D4⋊35C22, (C22×S3).99C23, C22.248(S3×C23), (C2×Dic3).117C23, (C3×C4.4D4)⋊19C2, (C3×C22⋊C4)⋊32C22, (C2×C4).200(C22×S3), (C2×C3⋊D4).65C22, SmallGroup(192,1242)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C6 — C42⋊24D6
Chief series
C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C23 — C23⋊2D6 — C42⋊24D6
Lower central
C3 — C2×C6 — 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, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, C12, D6, C2×C6, C2×C6, C42, C42, C22⋊C4, C22⋊C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C24, Dic6, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×Q8, C22×S3, C22×S3, C22×C6, C22≀C2, C4.4D4, C4.4D4, C4×Dic3, D6⋊C4, C6.D4, C4×C12, C3×C22⋊C4, C2×Dic6, C2×D12, C2×C3⋊D4, C6×D4, C6×Q8, S3×C23, C24⋊C22, C42⋊7S3, D6⋊D4, C23.11D6, C23⋊2D6, C12.23D4, C3×C4.4D4, C42⋊24D6
Quotients: C1, C2, C22, S3, C23, D6, C24, C22×S3, 2+ 1+4, S3×C23, C24⋊C22, D4⋊6D6, D4○D12, 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⋊6D6D4○D12
kernelC42⋊24D6C42⋊7S3D6⋊D4C23.11D6C23⋊2D6C12.23D4C3×C4.4D4C4.4D4C42C22⋊C4C2×D4C2×Q8C6C2C2
# 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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