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G = C24.38D6order 192 = 26·3

27th non-split extension by C24 of D6 acting via D6/C3=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C24.38D6, C6.32+ 1+4, C3⋊D48D4, Dic3⋊D41C2, D6⋊D42C2, C31(D45D4), C22⋊C440D6, D6.12(C2×D4), (C22×C4)⋊11D6, C127D417C2, C244S32C2, D6⋊C446C22, C23.9D61C2, (C2×D12)⋊2C22, (C2×C6).34C24, C4⋊Dic34C22, C6.37(C22×D4), C22.18(S3×D4), C2.7(D46D6), C225(C4○D12), Dic3.14(C2×D4), Dic34D440C2, (C2×C12).128C23, Dic3⋊C449C22, (C22×C12)⋊14C22, Dic3.D42C2, C23.11D61C2, (C4×Dic3)⋊47C22, (C2×Dic6)⋊48C22, C23.28D69C2, C6.D48C22, (C23×C6).60C22, C22.73(S3×C23), (S3×C23).31C22, (C22×C6).387C23, C23.231(C22×S3), (C22×S3).152C23, (C2×Dic3).180C23, (C22×Dic3).78C22, C2.11(C2×S3×D4), (C4×C3⋊D4)⋊1C2, (C2×C4○D12)⋊3C2, (C2×C6)⋊8(C4○D4), (S3×C2×C4)⋊40C22, C6.14(C2×C4○D4), (C6×C22⋊C4)⋊18C2, (C2×C22⋊C4)⋊13S3, (S3×C22⋊C4)⋊24C2, C2.16(C2×C4○D12), (C2×C6).383(C2×D4), (C22×C3⋊D4)⋊5C2, (C2×C3⋊D4)⋊1C22, (C3×C22⋊C4)⋊53C22, (C2×C4).259(C22×S3), SmallGroup(192,1049)

Series: Derived Chief Lower central Upper central

C1C2×C6 — C24.38D6
C1C3C6C2×C6C22×S3S3×C23S3×C22⋊C4 — C24.38D6
C3C2×C6 — C24.38D6
C1C22C2×C22⋊C4

Generators and relations for C24.38D6
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e6=f2=c, ab=ba, ac=ca, eae-1=faf-1=ad=da, fbf-1=bc=cb, bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e5 >

Subgroups: 952 in 334 conjugacy classes, 107 normal (91 characteristic)
C1, C2 [×3], C2 [×9], C3, C4 [×10], C22, C22 [×4], C22 [×25], S3 [×4], C6 [×3], C6 [×5], C2×C4 [×4], C2×C4 [×15], D4 [×18], Q8 [×2], C23 [×3], C23 [×13], Dic3 [×2], Dic3 [×4], C12 [×4], D6 [×2], D6 [×12], C2×C6, C2×C6 [×4], C2×C6 [×11], C42, C22⋊C4 [×4], C22⋊C4 [×8], C4⋊C4 [×4], C22×C4 [×2], C22×C4 [×4], C2×D4 [×13], C2×Q8, C4○D4 [×4], C24, C24, Dic6 [×2], C4×S3 [×5], D12 [×3], C2×Dic3 [×5], C2×Dic3 [×2], C3⋊D4 [×4], C3⋊D4 [×11], C2×C12 [×4], C2×C12 [×3], C22×S3 [×3], C22×S3 [×5], C22×C6 [×3], C22×C6 [×5], C2×C22⋊C4, C2×C22⋊C4, C4×D4 [×2], C22≀C2 [×2], C4⋊D4 [×3], C22⋊Q8, C22.D4 [×2], C4.4D4, C22×D4, C2×C4○D4, C4×Dic3, Dic3⋊C4 [×3], C4⋊Dic3, D6⋊C4 [×5], C6.D4 [×3], C3×C22⋊C4 [×4], C2×Dic6, S3×C2×C4 [×3], C2×D12 [×2], C4○D12 [×4], C22×Dic3, C2×C3⋊D4 [×7], C2×C3⋊D4 [×4], C22×C12 [×2], S3×C23, C23×C6, D45D4, Dic3.D4, S3×C22⋊C4, Dic34D4, D6⋊D4, C23.9D6, Dic3⋊D4 [×2], C23.11D6, C4×C3⋊D4, C23.28D6, C127D4, C244S3, C6×C22⋊C4, C2×C4○D12, C22×C3⋊D4, C24.38D6
Quotients: C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C4○D4 [×2], C24, C22×S3 [×7], C22×D4, C2×C4○D4, 2+ 1+4, C4○D12 [×2], S3×D4 [×2], S3×C23, D45D4, C2×C4○D12, C2×S3×D4, D46D6, C24.38D6

Smallest permutation representation of C24.38D6
On 48 points
Generators in S48
(1 7)(2 41)(3 9)(4 43)(5 11)(6 45)(8 47)(10 37)(12 39)(13 19)(14 34)(15 21)(16 36)(17 23)(18 26)(20 28)(22 30)(24 32)(25 31)(27 33)(29 35)(38 44)(40 46)(42 48)
(1 35)(2 36)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 31)(10 32)(11 33)(12 34)(13 38)(14 39)(15 40)(16 41)(17 42)(18 43)(19 44)(20 45)(21 46)(22 47)(23 48)(24 37)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)
(1 46)(2 47)(3 48)(4 37)(5 38)(6 39)(7 40)(8 41)(9 42)(10 43)(11 44)(12 45)(13 27)(14 28)(15 29)(16 30)(17 31)(18 32)(19 33)(20 34)(21 35)(22 36)(23 25)(24 26)
(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 7 12)(2 11 8 5)(3 4 9 10)(13 16 19 22)(14 21 20 15)(17 24 23 18)(25 32 31 26)(27 30 33 36)(28 35 34 29)(37 42 43 48)(38 47 44 41)(39 40 45 46)

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

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

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

45 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J2K2L 3 4A4B4C4D4E4F4G4H4I4J4K4L6A···6G6H6I6J6K12A···12H
order122222222222234444444444446···6666612···12
size111122224661212222224466121212122···244444···4

45 irreducible representations

dim1111111111111112222222444
type++++++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2C2C2C2S3D4D6D6D6C4○D4C4○D122+ 1+4S3×D4D46D6
kernelC24.38D6Dic3.D4S3×C22⋊C4Dic34D4D6⋊D4C23.9D6Dic3⋊D4C23.11D6C4×C3⋊D4C23.28D6C127D4C244S3C6×C22⋊C4C2×C4○D12C22×C3⋊D4C2×C22⋊C4C3⋊D4C22⋊C4C22×C4C24C2×C6C22C6C22C2
# reps1111112111111111442148122

Matrix representation of C24.38D6 in GL4(𝔽13) generated by

12000
01200
0010
00512
,
2900
41100
00120
00012
,
12000
01200
0010
0001
,
1000
0100
00120
00012
,
0500
8500
00123
0001
,
5000
5800
00123
0001
G:=sub<GL(4,GF(13))| [12,0,0,0,0,12,0,0,0,0,1,5,0,0,0,12],[2,4,0,0,9,11,0,0,0,0,12,0,0,0,0,12],[12,0,0,0,0,12,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,12,0,0,0,0,12],[0,8,0,0,5,5,0,0,0,0,12,0,0,0,3,1],[5,5,0,0,0,8,0,0,0,0,12,0,0,0,3,1] >;

C24.38D6 in GAP, Magma, Sage, TeX

C_2^4._{38}D_6
% in TeX

G:=Group("C2^4.38D6");
// GroupNames label

G:=SmallGroup(192,1049);
// by ID

G=gap.SmallGroup(192,1049);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,387,100,675,6278]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^6=f^2=c,a*b=b*a,a*c=c*a,e*a*e^-1=f*a*f^-1=a*d=d*a,f*b*f^-1=b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^5>;
// generators/relations

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