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G = C42⋊27D6  order 192 = 26·3

25th semidirect product of C42 and D6 acting via D6/C3=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42⋊27D6, C6.1422+ (1+4), C4⋊C4⋊17D6, C4⋊D12⋊5C2, (C4×C12)⋊2C22, D6⋊D4⋊28C2, Dic3⋊D4⋊46C2, C12⋊D4⋊38C2, C42⋊2C2⋊8S3, C42⋊3S3⋊1C2, D6⋊C4⋊24C22, C22⋊C4.41D6, D6.D4⋊44C2, C2.67(D4○D12), (C2×D12)⋊10C22, (C2×C6).255C24, Dic3⋊C4⋊5C22, (C2×C12).195C23, (C22×C6).69C23, C23.71(C22×S3), C3⋊4(C22.54C24), (S3×C23).70C22, C22.276(S3×C23), (C22×S3).114C23, (C2×Dic3).131C23, (S3×C2×C4)⋊28C22, (C3×C4⋊C4)⋊34C22, (C3×C42⋊2C2)⋊10C2, (C2×C4).211(C22×S3), (C2×C3⋊D4).75C22, (C3×C22⋊C4).80C22, SmallGroup(192,1270)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C6 — C42⋊27D6
Chief series
C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C23 — D6⋊D4 — C42⋊27D6
Lower central
C3 — C2×C6 — C42⋊27D6

Subgroups: 864 in 252 conjugacy classes, 91 normal (16 characteristic)
C1, C2 [×3], C2 [×6], C3, C4 [×9], C22, C22 [×22], S3 [×5], C6 [×3], C6, C2×C4 [×6], C2×C4 [×6], D4 [×12], C23, C23 [×8], Dic3 [×3], C12 [×6], D6 [×19], C2×C6, C2×C6 [×3], C42, C22⋊C4 [×3], C22⋊C4 [×9], C4⋊C4 [×3], C4⋊C4 [×3], C22×C4 [×3], C2×D4 [×12], C24, C4×S3 [×3], D12 [×9], C2×Dic3 [×3], C3⋊D4 [×3], C2×C12 [×6], C22×S3 [×2], C22×S3 [×3], C22×S3 [×3], C22×C6, C22≀C2 [×3], C4⋊D4 [×6], C22.D4 [×3], C42⋊2C2, C42⋊2C2, C4⋊1D4, Dic3⋊C4 [×3], D6⋊C4 [×9], C4×C12, C3×C22⋊C4 [×3], C3×C4⋊C4 [×3], S3×C2×C4 [×3], C2×D12 [×9], C2×C3⋊D4 [×3], S3×C23, C22.54C24, C4⋊D12, C42⋊3S3, D6⋊D4 [×3], Dic3⋊D4 [×3], D6.D4 [×3], C12⋊D4 [×3], C3×C42⋊2C2, C42⋊27D6

Quotients:
C1, C2 [×15], C22 [×35], S3, C23 [×15], D6 [×7], C24, C22×S3 [×7], 2+ (1+4) [×3], S3×C23, C22.54C24, D4○D12 [×3], C42⋊27D6

Generators and relations
 G = < a,b,c,d | a4=b4=c6=d2=1, ab=ba, cac-1=a-1b2, dad=ab2, cbc-1=a2b, dbd=a2b-1, dcd=c-1 >

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

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

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

Matrix representation ►G ⊆ GL8(𝔽13)

00370000
006100000
37000000
610000000
00000010
0000112111
00001000
00000001
,
00100000
00010000
10000000
01000000
00000100
000012000
0000112111
000010112
,
012000000
112000000
00010000
001210000
00001000
00000100
000000120
0000112012
,
121000000
01000000
001120000
000120000
00001000
000001200
000000120
000001121

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

33 conjugacy classes

class 1 2A2B2C2D2E···2I 3 4A···4F4G4H4I6A6B6C6D12A···12F12G12H12I
order122222···234···4444666612···12121212
size1111412···1224···412121222284···4888

33 irreducible representations

dim11111111222244
type++++++++++++++
imageC1C2C2C2C2C2C2C2S3D6D6D62+ (1+4)D4○D12
kernelC42⋊27D6C4⋊D12C42⋊3S3D6⋊D4Dic3⋊D4D6.D4C12⋊D4C3×C42⋊2C2C42⋊2C2C42C22⋊C4C4⋊C4C6C2
# reps11133331113336

In GAP, Magma, Sage, TeX 

C_4^2\rtimes_{27}D_6
 
% in TeX
 
G:=Group("C4^2:27D6");
 
// GroupNames label
 
G:=SmallGroup(192,1270);
 
// by ID
 
G=gap.SmallGroup(192,1270);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,758,219,184,1571,570,192,6278]);
 
// 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*b^2,d*a*d=a*b^2,c*b*c^-1=a^2*b,d*b*d=a^2*b^-1,d*c*d=c^-1>;
 
// generators/relations
 

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