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G = C42⋊25D6  order 192 = 26·3

23rd semidirect product of C42 and D6 acting via D6/C3=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42⋊25D6, C6.1372+ 1+4, C4⋊C4⋊33D6, (C4×D12)⋊13C2, (C4×C12)⋊7C22, D6⋊C4⋊7C22, Dic3⋊D4⋊44C2, D6⋊D4⋊27C2, C12⋊D4⋊36C2, C42⋊7S3⋊8C2, C42⋊2C2⋊2S3, D6⋊Q8⋊40C2, C22⋊C4.40D6, Dic3⋊5D4⋊39C2, D6.12(C4○D4), C23.9D6⋊48C2, D6.D4⋊38C2, C2.62(D4○D12), (C2×D12)⋊29C22, (C2×C6).248C24, C4⋊Dic3⋊61C22, (C2×C12).193C23, Dic3⋊C4⋊27C22, C3⋊9(C22.32C24), (C4×Dic3)⋊38C22, (C2×Dic6)⋊11C22, (C22×C6).62C23, C23.64(C22×S3), C23.11D6⋊44C2, (S3×C23).68C22, C22.269(S3×C23), (C22×S3).111C23, (C2×Dic3).264C23, C6.D4.64C22, (S3×C2×C4)⋊27C22, C4⋊C4⋊S3⋊41C2, C2.95(S3×C4○D4), (S3×C22⋊C4)⋊20C2, (C3×C4⋊C4)⋊32C22, C6.206(C2×C4○D4), (C3×C42⋊2C2)⋊3C2, (C2×C4).85(C22×S3), (C2×C3⋊D4).68C22, (C3×C22⋊C4).73C22, SmallGroup(192,1263)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C6 — C42⋊25D6
Chief series
C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C23 — S3×C22⋊C4 — C42⋊25D6
Lower central
C3 — C2×C6 — C42⋊25D6
Upper central
C1 — C22 — C42⋊2C2

Generators and relations for C42⋊25D6
 G = < a,b,c,d | a4=b4=c6=d2=1, ab=ba, cac-1=dad=a-1b2, cbc-1=a2b, dbd=b-1, dcd=c-1 >

Subgroups: 784 in 250 conjugacy classes, 93 normal (91 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, C12, D6, D6, C2×C6, C2×C6, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C24, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C22×S3, C22×S3, C22×C6, C2×C22⋊C4, C4×D4, C22≀C2, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42⋊2C2, C42⋊2C2, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, C2×D12, C2×C3⋊D4, S3×C23, C22.32C24, C4×D12, C42⋊7S3, S3×C22⋊C4, D6⋊D4, C23.9D6, Dic3⋊D4, C23.11D6, Dic3⋊5D4, D6.D4, C12⋊D4, D6⋊Q8, C4⋊C4⋊S3, C3×C42⋊2C2, C42⋊25D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, 2+ 1+4, S3×C23, C22.32C24, S3×C4○D4, D4○D12, C42⋊25D6

Smallest permutation representation of C42⋊25D6
►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 46 22 38)(20 42 23 44)(21 48 24 40)

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

36 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I 3 4A4B4C···4G4H4I4J4K4L6A6B6C6D12A···12F12G12H12I
order12222222223444···444444666612···12121212
size11114661212122224···46612121222284···4888

36 irreducible representations

dim1111111111111122222444
type++++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2C2C2S3D6D6D6C4○D42+ 1+4S3×C4○D4D4○D12
kernelC42⋊25D6C4×D12C42⋊7S3S3×C22⋊C4D6⋊D4C23.9D6Dic3⋊D4C23.11D6Dic3⋊5D4D6.D4C12⋊D4D6⋊Q8C4⋊C4⋊S3C3×C42⋊2C2C42⋊2C2C42C22⋊C4C4⋊C4D6C6C2C2
# reps1111211111211111334224

Matrix representation of C42⋊25D6 ►in GL6(𝔽13)

800000
080000
0012071
00012126
0010710
006301
,
930000
340000
0036110
00710011
00120107
0001263
,
100000
7120000
001100
0012000
001031212
0010710
,
100000
7120000
00121200
000100
001031212
006301

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

C42⋊25D6 in GAP, Magma, Sage, TeX 

C_4^2\rtimes_{25}D_6
 
% in TeX
 
G:=Group("C4^2:25D6");
 
// GroupNames label
 
G:=SmallGroup(192,1263);
 
// by ID
 
G=gap.SmallGroup(192,1263);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,219,184,675,570,192,6278]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^4=c^6=d^2=1,a*b=b*a,c*a*c^-1=d*a*d=a^-1*b^2,c*b*c^-1=a^2*b,d*b*d=b^-1,d*c*d=c^-1>;
 
// generators/relations
 

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