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G = C42.216D6order 192 = 26·3

36th non-split extension by C42 of D6 acting via D6/S3=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.216D6, C4⋊C4.77D6, C42.C23S3, (C2×C12).276D4, C6.110(C4○D8), C12.72(C4○D4), C6.SD1641C2, C427S3.7C2, C6.D8.13C2, (C4×C12).116C22, (C2×C12).386C23, C4.14(Q83S3), C6.56(C4.4D4), C2.29(Q8.13D6), C2.9(C12.23D4), (C2×D12).104C22, C35(C42.78C22), (C2×Dic6).109C22, (C4×C3⋊C8)⋊13C2, (C2×C6).517(C2×D4), (C3×C42.C2)⋊3C2, (C2×C3⋊C8).255C22, (C2×C4).112(C3⋊D4), (C3×C4⋊C4).124C22, (C2×C4).484(C22×S3), C22.190(C2×C3⋊D4), SmallGroup(192,627)

Series: Derived Chief Lower central Upper central

C1C2×C12 — C42.216D6
C1C3C6C12C2×C12C2×D12C427S3 — C42.216D6
C3C6C2×C12 — C42.216D6
C1C22C42C42.C2

Generators and relations for C42.216D6
 G = < a,b,c,d | a4=b4=1, c6=a2b2, d2=a2b, ab=ba, cac-1=a-1b2, ad=da, cbc-1=b-1, bd=db, dcd-1=b-1c5 >

Subgroups: 288 in 96 conjugacy classes, 39 normal (17 characteristic)
C1, C2, C2 [×2], C2, C3, C4 [×2], C4 [×5], C22, C22 [×3], S3, C6, C6 [×2], C8 [×2], C2×C4, C2×C4 [×2], C2×C4 [×3], D4 [×2], Q8 [×2], C23, Dic3, C12 [×2], C12 [×4], D6 [×3], C2×C6, C42, C22⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], C2×D4, C2×Q8, C3⋊C8 [×2], Dic6 [×2], D12 [×2], C2×Dic3, C2×C12, C2×C12 [×2], C2×C12 [×2], C22×S3, C4×C8, D4⋊C4 [×2], Q8⋊C4 [×2], C4.4D4, C42.C2, C2×C3⋊C8 [×2], D6⋊C4 [×2], C4×C12, C3×C4⋊C4 [×2], C3×C4⋊C4 [×2], C2×Dic6, C2×D12, C42.78C22, C4×C3⋊C8, C6.D8 [×2], C6.SD16 [×2], C427S3, C3×C42.C2, C42.216D6
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], C2×D4, C4○D4 [×2], C3⋊D4 [×2], C22×S3, C4.4D4, C4○D8 [×2], Q83S3 [×2], C2×C3⋊D4, C42.78C22, C12.23D4, Q8.13D6 [×2], C42.216D6

Smallest permutation representation of C42.216D6
On 96 points
Generators in S96
(1 89 23 57)(2 96 24 52)(3 91 13 59)(4 86 14 54)(5 93 15 49)(6 88 16 56)(7 95 17 51)(8 90 18 58)(9 85 19 53)(10 92 20 60)(11 87 21 55)(12 94 22 50)(25 77 46 63)(26 84 47 70)(27 79 48 65)(28 74 37 72)(29 81 38 67)(30 76 39 62)(31 83 40 69)(32 78 41 64)(33 73 42 71)(34 80 43 66)(35 75 44 61)(36 82 45 68)
(1 44 17 29)(2 30 18 45)(3 46 19 31)(4 32 20 47)(5 48 21 33)(6 34 22 37)(7 38 23 35)(8 36 24 39)(9 40 13 25)(10 26 14 41)(11 42 15 27)(12 28 16 43)(49 79 87 71)(50 72 88 80)(51 81 89 61)(52 62 90 82)(53 83 91 63)(54 64 92 84)(55 73 93 65)(56 66 94 74)(57 75 95 67)(58 68 96 76)(59 77 85 69)(60 70 86 78)
(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)(49 50 51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80 81 82 83 84)(85 86 87 88 89 90 91 92 93 94 95 96)
(1 6 35 43 17 22 38 28)(2 27 39 21 18 42 36 5)(3 4 25 41 19 20 40 26)(7 12 29 37 23 16 44 34)(8 33 45 15 24 48 30 11)(9 10 31 47 13 14 46 32)(49 52 65 76 87 90 73 68)(50 67 74 89 88 75 66 51)(53 60 69 84 91 86 77 64)(54 63 78 85 92 83 70 59)(55 58 71 82 93 96 79 62)(56 61 80 95 94 81 72 57)

G:=sub<Sym(96)| (1,89,23,57)(2,96,24,52)(3,91,13,59)(4,86,14,54)(5,93,15,49)(6,88,16,56)(7,95,17,51)(8,90,18,58)(9,85,19,53)(10,92,20,60)(11,87,21,55)(12,94,22,50)(25,77,46,63)(26,84,47,70)(27,79,48,65)(28,74,37,72)(29,81,38,67)(30,76,39,62)(31,83,40,69)(32,78,41,64)(33,73,42,71)(34,80,43,66)(35,75,44,61)(36,82,45,68), (1,44,17,29)(2,30,18,45)(3,46,19,31)(4,32,20,47)(5,48,21,33)(6,34,22,37)(7,38,23,35)(8,36,24,39)(9,40,13,25)(10,26,14,41)(11,42,15,27)(12,28,16,43)(49,79,87,71)(50,72,88,80)(51,81,89,61)(52,62,90,82)(53,83,91,63)(54,64,92,84)(55,73,93,65)(56,66,94,74)(57,75,95,67)(58,68,96,76)(59,77,85,69)(60,70,86,78), (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)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96), (1,6,35,43,17,22,38,28)(2,27,39,21,18,42,36,5)(3,4,25,41,19,20,40,26)(7,12,29,37,23,16,44,34)(8,33,45,15,24,48,30,11)(9,10,31,47,13,14,46,32)(49,52,65,76,87,90,73,68)(50,67,74,89,88,75,66,51)(53,60,69,84,91,86,77,64)(54,63,78,85,92,83,70,59)(55,58,71,82,93,96,79,62)(56,61,80,95,94,81,72,57)>;

G:=Group( (1,89,23,57)(2,96,24,52)(3,91,13,59)(4,86,14,54)(5,93,15,49)(6,88,16,56)(7,95,17,51)(8,90,18,58)(9,85,19,53)(10,92,20,60)(11,87,21,55)(12,94,22,50)(25,77,46,63)(26,84,47,70)(27,79,48,65)(28,74,37,72)(29,81,38,67)(30,76,39,62)(31,83,40,69)(32,78,41,64)(33,73,42,71)(34,80,43,66)(35,75,44,61)(36,82,45,68), (1,44,17,29)(2,30,18,45)(3,46,19,31)(4,32,20,47)(5,48,21,33)(6,34,22,37)(7,38,23,35)(8,36,24,39)(9,40,13,25)(10,26,14,41)(11,42,15,27)(12,28,16,43)(49,79,87,71)(50,72,88,80)(51,81,89,61)(52,62,90,82)(53,83,91,63)(54,64,92,84)(55,73,93,65)(56,66,94,74)(57,75,95,67)(58,68,96,76)(59,77,85,69)(60,70,86,78), (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)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96), (1,6,35,43,17,22,38,28)(2,27,39,21,18,42,36,5)(3,4,25,41,19,20,40,26)(7,12,29,37,23,16,44,34)(8,33,45,15,24,48,30,11)(9,10,31,47,13,14,46,32)(49,52,65,76,87,90,73,68)(50,67,74,89,88,75,66,51)(53,60,69,84,91,86,77,64)(54,63,78,85,92,83,70,59)(55,58,71,82,93,96,79,62)(56,61,80,95,94,81,72,57) );

G=PermutationGroup([(1,89,23,57),(2,96,24,52),(3,91,13,59),(4,86,14,54),(5,93,15,49),(6,88,16,56),(7,95,17,51),(8,90,18,58),(9,85,19,53),(10,92,20,60),(11,87,21,55),(12,94,22,50),(25,77,46,63),(26,84,47,70),(27,79,48,65),(28,74,37,72),(29,81,38,67),(30,76,39,62),(31,83,40,69),(32,78,41,64),(33,73,42,71),(34,80,43,66),(35,75,44,61),(36,82,45,68)], [(1,44,17,29),(2,30,18,45),(3,46,19,31),(4,32,20,47),(5,48,21,33),(6,34,22,37),(7,38,23,35),(8,36,24,39),(9,40,13,25),(10,26,14,41),(11,42,15,27),(12,28,16,43),(49,79,87,71),(50,72,88,80),(51,81,89,61),(52,62,90,82),(53,83,91,63),(54,64,92,84),(55,73,93,65),(56,66,94,74),(57,75,95,67),(58,68,96,76),(59,77,85,69),(60,70,86,78)], [(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),(49,50,51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80,81,82,83,84),(85,86,87,88,89,90,91,92,93,94,95,96)], [(1,6,35,43,17,22,38,28),(2,27,39,21,18,42,36,5),(3,4,25,41,19,20,40,26),(7,12,29,37,23,16,44,34),(8,33,45,15,24,48,30,11),(9,10,31,47,13,14,46,32),(49,52,65,76,87,90,73,68),(50,67,74,89,88,75,66,51),(53,60,69,84,91,86,77,64),(54,63,78,85,92,83,70,59),(55,58,71,82,93,96,79,62),(56,61,80,95,94,81,72,57)])

36 conjugacy classes

class 1 2A2B2C2D 3 4A···4F4G4H4I6A6B6C8A···8H12A···12F12G12H12I12J
order1222234···44446668···812···1212121212
size11112422···288242226···64···48888

36 irreducible representations

dim111111222222244
type+++++++++++
imageC1C2C2C2C2C2S3D4D6D6C4○D4C3⋊D4C4○D8Q83S3Q8.13D6
kernelC42.216D6C4×C3⋊C8C6.D8C6.SD16C427S3C3×C42.C2C42.C2C2×C12C42C4⋊C4C12C2×C4C6C4C2
# reps112211121244824

Matrix representation of C42.216D6 in GL6(𝔽73)

100000
010000
0046000
0004600
0000278
00005546
,
7200000
0720000
0072300
0048100
000013
00004872
,
30430000
30600000
0004800
0035000
00001218
00006961
,
43300000
60300000
00322500
0035000
0000055
0000412

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,46,0,0,0,0,0,0,46,0,0,0,0,0,0,27,55,0,0,0,0,8,46],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,48,0,0,0,0,3,1,0,0,0,0,0,0,1,48,0,0,0,0,3,72],[30,30,0,0,0,0,43,60,0,0,0,0,0,0,0,35,0,0,0,0,48,0,0,0,0,0,0,0,12,69,0,0,0,0,18,61],[43,60,0,0,0,0,30,30,0,0,0,0,0,0,32,35,0,0,0,0,25,0,0,0,0,0,0,0,0,4,0,0,0,0,55,12] >;

C42.216D6 in GAP, Magma, Sage, TeX

C_4^2._{216}D_6
% in TeX

G:=Group("C4^2.216D6");
// GroupNames label

G:=SmallGroup(192,627);
// by ID

G=gap.SmallGroup(192,627);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,344,254,219,100,1123,297,136,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=1,c^6=a^2*b^2,d^2=a^2*b,a*b=b*a,c*a*c^-1=a^-1*b^2,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=b^-1*c^5>;
// generators/relations

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