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G = C42.74D6order 192 = 26·3

74th non-split extension by C42 of D6 acting via D6/C3=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.74D6, C3⋊C88D4, C41D44S3, C33(C83D4), C4.15(S3×D4), (C2×D4).57D6, C12.32(C2×D4), (C2×C12).292D4, C427S315C2, C6.21(C41D4), C6.95(C8⋊C22), (C6×D4).73C22, C2.12(C123D4), C42.S313C2, (C4×C12).122C22, (C2×C12).392C23, C2.16(D126C22), (C2×D12).106C22, (C2×Dic6).111C22, (C2×D4⋊S3)⋊15C2, (C3×C41D4)⋊3C2, (C2×D4.S3)⋊13C2, (C2×C6).523(C2×D4), (C2×C4).70(C3⋊D4), (C2×C3⋊C8).131C22, (C2×C4).490(C22×S3), C22.196(C2×C3⋊D4), SmallGroup(192,633)

Series: Derived Chief Lower central Upper central

C1C2×C12 — C42.74D6
C1C3C6C12C2×C12C2×D12C427S3 — C42.74D6
C3C6C2×C12 — C42.74D6
C1C22C42C41D4

Generators and relations for C42.74D6
 G = < a,b,c,d | a4=b4=c6=1, d2=cbc-1=b-1, ab=ba, cac-1=a-1, dad-1=ab2, bd=db, dcd-1=b-1c-1 >

Subgroups: 464 in 144 conjugacy classes, 43 normal (17 characteristic)
C1, C2, C2 [×2], C2 [×3], C3, C4 [×2], C4 [×3], C22, C22 [×9], S3, C6, C6 [×2], C6 [×2], C8 [×4], C2×C4, C2×C4 [×2], C2×C4, D4 [×10], Q8 [×2], C23 [×3], Dic3, C12 [×2], C12 [×2], D6 [×3], C2×C6, C2×C6 [×6], C42, C22⋊C4 [×2], C2×C8 [×2], D8 [×4], SD16 [×4], C2×D4 [×2], C2×D4 [×3], C2×Q8, C3⋊C8 [×4], Dic6 [×2], D12 [×2], C2×Dic3, C2×C12, C2×C12 [×2], C3×D4 [×8], C22×S3, C22×C6 [×2], C8⋊C4, C4.4D4, C41D4, C2×D8 [×2], C2×SD16 [×2], C2×C3⋊C8 [×2], D6⋊C4 [×2], D4⋊S3 [×4], D4.S3 [×4], C4×C12, C2×Dic6, C2×D12, C6×D4 [×2], C6×D4 [×2], C83D4, C42.S3, C427S3, C2×D4⋊S3 [×2], C2×D4.S3 [×2], C3×C41D4, C42.74D6
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×6], C23, D6 [×3], C2×D4 [×3], C3⋊D4 [×2], C22×S3, C41D4, C8⋊C22 [×2], S3×D4 [×2], C2×C3⋊D4, C83D4, D126C22 [×2], C123D4, C42.74D6

Character table of C42.74D6

 class 12A2B2C2D2E2F34A4B4C4D4E6A6B6C6D6E6F6G8A8B8C8D12A12B12C12D12E12F
 size 111188242224424222888812121212444444
ρ1111111111111111111111111111111    trivial
ρ2111111-111111-11111111-1-1-1-1111111    linear of order 2
ρ31111-1-1-111111-1111-1-1-1-11111111111    linear of order 2
ρ41111-1-11111111111-1-1-1-1-1-1-1-1111111    linear of order 2
ρ511111-11111-1-1-111111-1-1-111-11-1-11-1-1    linear of order 2
ρ611111-1-1111-1-1111111-1-11-1-111-1-11-1-1    linear of order 2
ρ71111-11-1111-1-11111-1-111-111-11-1-11-1-1    linear of order 2
ρ81111-111111-1-1-1111-1-1111-1-111-1-11-1-1    linear of order 2
ρ922220002-2-2-22022200000000-22-2-22-2    orthogonal lifted from D4
ρ102222-2-20-122220-1-1-111110000-1-1-1-1-1-1    orthogonal lifted from D6
ρ112222-220-122-2-20-1-1-111-1-10000-111-111    orthogonal lifted from D6
ρ122-2-2200022-2000-22-2000002-20-200200    orthogonal lifted from D4
ρ132-2-220002-22000-22-20000200-2200-200    orthogonal lifted from D4
ρ142222220-122220-1-1-1-1-1-1-10000-1-1-1-1-1-1    orthogonal lifted from S3
ρ152-2-220002-22000-22-20000-2002200-200    orthogonal lifted from D4
ρ162-2-2200022-2000-22-200000-220-200200    orthogonal lifted from D4
ρ1722220002-2-22-2022200000000-2-22-2-22    orthogonal lifted from D4
ρ1822222-20-122-2-20-1-1-1-1-1110000-111-111    orthogonal lifted from D6
ρ192222000-1-2-22-20-1-1-1-3--3-3--3000011-111-1    complex lifted from C3⋊D4
ρ202222000-1-2-2-220-1-1-1-3--3--3-300001-111-11    complex lifted from C3⋊D4
ρ212222000-1-2-22-20-1-1-1--3-3--3-3000011-111-1    complex lifted from C3⋊D4
ρ222222000-1-2-2-220-1-1-1--3-3-3--300001-111-11    complex lifted from C3⋊D4
ρ234-4-44000-24-40002-2200000000200-200    orthogonal lifted from S3×D4
ρ244-44-4000400000-4-4400000000000000    orthogonal lifted from C8⋊C22
ρ254-4-44000-2-440002-2200000000-200200    orthogonal lifted from S3×D4
ρ2644-4-40004000004-4-400000000000000    orthogonal lifted from C8⋊C22
ρ2744-4-4000-200000-2220000000000-2-3002-3    complex lifted from D126C22
ρ2844-4-4000-200000-22200000000002-300-2-3    complex lifted from D126C22
ρ294-44-4000-20000022-2000000000-2-3002-30    complex lifted from D126C22
ρ304-44-4000-20000022-20000000002-300-2-30    complex lifted from D126C22

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

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

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

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

Matrix representation of C42.74D6 in GL6(𝔽73)

72710000
110000
00431300
00603000
00004313
00006030
,
100000
010000
00720710
00072071
001010
000101
,
100000
72720000
004627054
0046191919
002702746
000272754
,
100000
010000
004646190
000275454
002702746
0046461946

G:=sub<GL(6,GF(73))| [72,1,0,0,0,0,71,1,0,0,0,0,0,0,43,60,0,0,0,0,13,30,0,0,0,0,0,0,43,60,0,0,0,0,13,30],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,1,0,0,0,0,72,0,1,0,0,71,0,1,0,0,0,0,71,0,1],[1,72,0,0,0,0,0,72,0,0,0,0,0,0,46,46,27,0,0,0,27,19,0,27,0,0,0,19,27,27,0,0,54,19,46,54],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,46,0,27,46,0,0,46,27,0,46,0,0,19,54,27,19,0,0,0,54,46,46] >;

C42.74D6 in GAP, Magma, Sage, TeX

C_4^2._{74}D_6
% in TeX

G:=Group("C4^2.74D6");
// GroupNames label

G:=SmallGroup(192,633);
// by ID

G=gap.SmallGroup(192,633);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,344,254,555,1123,297,136,6278]);
// Polycyclic

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

Export

Character table of C42.74D6 in TeX

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