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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, D6⋊D4⋊26C2, C23⋊2D6⋊25C2, (C4×C12)⋊29C22, (C2×D4).112D6, C4.4D4⋊16S3, (C6×Q8)⋊16C22, C42⋊7S3⋊30C2, C2.53(D4○D12), (C2×C6).227C24, (C2×C12).83C23, 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

Subgroups: 896 in 260 conjugacy classes, 91 normal (17 characteristic)
C1, C2, C2 [×2], C2 [×6], C3, C4 [×9], C22, C22 [×26], S3 [×4], C6, C6 [×2], C6 [×2], C2×C4, C2×C4 [×4], C2×C4 [×4], D4 [×9], Q8 [×3], C23 [×2], C23 [×10], Dic3 [×4], C12 [×5], D6 [×20], C2×C6, C2×C6 [×6], C42, C42 [×2], C22⋊C4 [×4], C22⋊C4 [×14], C2×D4, C2×D4 [×8], C2×Q8, C2×Q8 [×2], C24 [×2], Dic6 [×2], D12 [×4], C2×Dic3 [×4], C3⋊D4 [×4], C2×C12, C2×C12 [×4], C3×D4, C3×Q8, C22×S3 [×4], C22×S3 [×6], C22×C6 [×2], C22≀C2 [×6], C4.4D4, C4.4D4 [×8], C4×Dic3 [×2], D6⋊C4 [×12], C6.D4 [×2], C4×C12, C3×C22⋊C4 [×4], C2×Dic6 [×2], C2×D12 [×4], C2×C3⋊D4 [×4], C6×D4, C6×Q8, S3×C23 [×2], C24⋊C22, C42⋊7S3 [×2], D6⋊D4 [×4], C23.11D6 [×4], C23⋊2D6 [×2], C12.23D4 [×2], C3×C4.4D4, C42⋊24D6

Quotients:
C1, C2 [×15], C22 [×35], S3, C23 [×15], D6 [×7], C24, C22×S3 [×7], 2+ (1+4) [×3], S3×C23, C24⋊C22, D4⋊6D6, D4○D12 [×2], C42⋊24D6

Generators and relations
 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 >

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

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

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

Matrix representation ►G ⊆ 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] >;
 

33 conjugacy classes

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

33 irreducible representations

dim111111122222444
type++++++++++++++
imageC1C2C2C2C2C2C2S3D6D6D6D62+ (1+4)D4⋊6D6D4○D12
kernelC42⋊24D6C42⋊7S3D6⋊D4C23.11D6C23⋊2D6C12.23D4C3×C4.4D4C4.4D4C42C22⋊C4C2×D4C2×Q8C6C2C2
# reps124422111411324

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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