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G = C42.56D6order 192 = 26·3

56th non-split extension by C42 of D6 acting via D6/C3=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.56D6, Q85(C4×S3), (C4×Q8)⋊6S3, (Q8×C12)⋊2C2, C6.74(C4×D4), C4⋊C4.252D6, Q82S35C4, (C4×D12).15C2, D12.17(C2×C4), (C2×C12).258D4, C6.Q1633C2, (C2×Q8).183D6, C4.41(C4○D12), C12.59(C4○D4), C2.4(D4⋊D6), Q82Dic311C2, C36(SD16⋊C4), C42.S37C2, C12.26(C22×C4), (C4×C12).97C22, C6.D8.10C2, C6.111(C8⋊C22), (C2×C12).346C23, C6.87(C8.C22), (C6×Q8).194C22, C2.3(Q8.11D6), (C2×D12).239C22, C4⋊Dic3.331C22, C3⋊C810(C2×C4), C4.26(S3×C2×C4), (C3×Q8)⋊9(C2×C4), C2.20(C4×C3⋊D4), (C2×C6).477(C2×D4), (C2×C3⋊C8).100C22, (C2×Q82S3).4C2, C22.80(C2×C3⋊D4), (C2×C4).221(C3⋊D4), (C3×C4⋊C4).283C22, (C2×C4).446(C22×S3), SmallGroup(192,585)

Series: Derived Chief Lower central Upper central

C1C12 — C42.56D6
C1C3C6C2×C6C2×C12C2×D12C2×Q82S3 — C42.56D6
C3C6C12 — C42.56D6
C1C22C42C4×Q8

Generators and relations for C42.56D6
 G = < a,b,c,d | a4=b4=1, c6=b2, d2=b, ab=ba, cac-1=dad-1=ab2, cbc-1=b-1, bd=db, dcd-1=b-1c5 >

Subgroups: 328 in 120 conjugacy classes, 51 normal (39 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C4 [×6], C22, C22 [×4], S3 [×2], C6 [×3], C8 [×3], C2×C4 [×3], C2×C4 [×5], D4 [×3], Q8 [×2], Q8, C23, Dic3, C12 [×2], C12 [×5], D6 [×4], C2×C6, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4 [×2], C2×C8 [×2], SD16 [×4], C22×C4, C2×D4, C2×Q8, C3⋊C8 [×2], C3⋊C8, C4×S3 [×2], D12 [×2], D12, C2×Dic3, C2×C12 [×3], C2×C12 [×2], C3×Q8 [×2], C3×Q8, C22×S3, C8⋊C4, D4⋊C4, Q8⋊C4, C2.D8, C4×D4, C4×Q8, C2×SD16, C2×C3⋊C8 [×2], C4⋊Dic3, D6⋊C4, Q82S3 [×4], C4×C12, C4×C12, C3×C4⋊C4, C3×C4⋊C4, S3×C2×C4, C2×D12, C6×Q8, SD16⋊C4, C42.S3, C6.Q16, C6.D8, Q82Dic3, C4×D12, C2×Q82S3, Q8×C12, C42.56D6
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×2], C23, D6 [×3], C22×C4, C2×D4, C4○D4, C4×S3 [×2], C3⋊D4 [×2], C22×S3, C4×D4, C8⋊C22, C8.C22, S3×C2×C4, C4○D12, C2×C3⋊D4, SD16⋊C4, C4×C3⋊D4, Q8.11D6, D4⋊D6, C42.56D6

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

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

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

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

42 conjugacy classes

class 1 2A2B2C2D2E 3 4A···4F4G4H4I4J4K4L6A6B6C8A8B8C8D12A12B12C12D12E···12P
order12222234···444444466688881212121212···12
size1111121222···2444412122221212121222224···4

42 irreducible representations

dim1111111112222222224444
type++++++++++++++-+
imageC1C2C2C2C2C2C2C2C4S3D4D6D6D6C4○D4C3⋊D4C4×S3C4○D12C8⋊C22C8.C22Q8.11D6D4⋊D6
kernelC42.56D6C42.S3C6.Q16C6.D8Q82Dic3C4×D12C2×Q82S3Q8×C12Q82S3C4×Q8C2×C12C42C4⋊C4C2×Q8C12C2×C4Q8C4C6C6C2C2
# reps1111111181211124441122

Matrix representation of C42.56D6 in GL8(𝔽73)

460000000
046000000
00100000
00010000
0000710670
00001446271
000037020
00005372627
,
720000000
072000000
007200000
000720000
00001300
0000487200
0000664901
00006470720
,
6867000000
45000000
0013300000
0043430000
000044604141
0000833070
000020561717
000035611252
,
56000000
2068000000
0030300000
0060430000
000068134132
0000654003
000046171756
000012121221

G:=sub<GL(8,GF(73))| [46,0,0,0,0,0,0,0,0,46,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,71,14,37,53,0,0,0,0,0,46,0,72,0,0,0,0,67,2,2,6,0,0,0,0,0,71,0,27],[72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,48,66,64,0,0,0,0,3,72,49,70,0,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0],[68,4,0,0,0,0,0,0,67,5,0,0,0,0,0,0,0,0,13,43,0,0,0,0,0,0,30,43,0,0,0,0,0,0,0,0,44,8,20,35,0,0,0,0,60,33,56,61,0,0,0,0,41,0,17,12,0,0,0,0,41,70,17,52],[5,20,0,0,0,0,0,0,6,68,0,0,0,0,0,0,0,0,30,60,0,0,0,0,0,0,30,43,0,0,0,0,0,0,0,0,68,65,46,12,0,0,0,0,13,40,17,12,0,0,0,0,41,0,17,12,0,0,0,0,32,3,56,21] >;

C42.56D6 in GAP, Magma, Sage, TeX

C_4^2._{56}D_6
% in TeX

G:=Group("C4^2.56D6");
// GroupNames label

G:=SmallGroup(192,585);
// by ID

G=gap.SmallGroup(192,585);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,232,387,58,1684,851,102,6278]);
// Polycyclic

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

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