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G = C42.7D6order 192 = 26·3

7th non-split extension by C42 of D6 acting via D6/C3=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.7D6, C6.18C4≀C2, (C6×D4).1C4, (C6×Q8).1C4, (C2×C12).231D4, (C2×Q8).4Dic3, (C2×D4).1Dic3, C4.4D4.1S3, C6.8(C4.D4), C42.S323C2, (C4×C12).235C22, C2.3(C12.D4), C2.6(Q83Dic3), C32(C42.C22), C22.39(C6.D4), (C2×C4).9(C2×Dic3), (C2×C12).169(C2×C4), (C3×C4.4D4).8C2, (C2×C4).165(C3⋊D4), (C2×C6).98(C22⋊C4), SmallGroup(192,99)

Series: Derived Chief Lower central Upper central

C1C2×C12 — C42.7D6
C1C3C6C2×C6C2×C12C4×C12C42.S3 — C42.7D6
C3C2×C6C2×C12 — C42.7D6
C1C22C42C4.4D4

Generators and relations for C42.7D6
 G = < a,b,c,d | a4=b4=c6=1, d2=ab-1, ab=ba, cac-1=dad-1=a-1b2, cbc-1=b-1, dbd-1=a2b-1, dcd-1=a2bc-1 >

Subgroups: 176 in 70 conjugacy classes, 27 normal (17 characteristic)
C1, C2, C2 [×2], C2, C3, C4 [×4], C22, C22 [×3], C6, C6 [×2], C6, C8 [×4], C2×C4, C2×C4 [×2], C2×C4, D4, Q8, C23, C12 [×4], C2×C6, C2×C6 [×3], C42, C22⋊C4 [×2], C2×C8 [×2], C2×D4, C2×Q8, C3⋊C8 [×4], C2×C12, C2×C12 [×2], C2×C12, C3×D4, C3×Q8, C22×C6, C8⋊C4 [×2], C4.4D4, C2×C3⋊C8 [×2], C4×C12, C3×C22⋊C4 [×2], C6×D4, C6×Q8, C42.C22, C42.S3 [×2], C3×C4.4D4, C42.7D6
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], Dic3 [×2], D6, C22⋊C4, C2×Dic3, C3⋊D4 [×2], C4.D4, C4≀C2 [×2], C6.D4, C42.C22, C12.D4, Q83Dic3 [×2], C42.7D6

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

33 conjugacy classes

class 1 2A2B2C2D 3 4A4B4C4D4E4F6A6B6C6D6E8A···8H12A···12F12G12H
order122223444444666668···812···121212
size1111822222482228812···124···488

33 irreducible representations

dim111112222222444
type++++++--+
imageC1C2C2C4C4S3D4D6Dic3Dic3C3⋊D4C4≀C2C4.D4C12.D4Q83Dic3
kernelC42.7D6C42.S3C3×C4.4D4C6×D4C6×Q8C4.4D4C2×C12C42C2×D4C2×Q8C2×C4C6C6C2C2
# reps121221211148124

Matrix representation of C42.7D6 in GL6(𝔽73)

100000
010000
0004600
0027000
0000460
0000046
,
7200000
0720000
000100
0072000
000001
0000720
,
6400000
0650000
001000
0007200
000010
0000072
,
080000
900000
00131300
00136000
00006013
00006060

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,27,0,0,0,0,46,0,0,0,0,0,0,0,46,0,0,0,0,0,0,46],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,1,0,0,0,0,0,0,0,0,72,0,0,0,0,1,0],[64,0,0,0,0,0,0,65,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,72],[0,9,0,0,0,0,8,0,0,0,0,0,0,0,13,13,0,0,0,0,13,60,0,0,0,0,0,0,60,60,0,0,0,0,13,60] >;

C42.7D6 in GAP, Magma, Sage, TeX

C_4^2._7D_6
% in TeX

G:=Group("C4^2.7D6");
// GroupNames label

G:=SmallGroup(192,99);
// by ID

G=gap.SmallGroup(192,99);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,141,219,268,1571,570,136,6278]);
// Polycyclic

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

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