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G = (C2×Q8).49D6order 192 = 26·3

25th non-split extension by C2×Q8 of D6 acting via D6/C3=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C4⋊C4.63D6, (C2×C12).75D4, (C2×Q8).49D6, C22⋊Q8.2S3, (C2×C6).16SD16, C6.71(C2×SD16), (C22×C6).89D4, Q82Dic312C2, C12.Q837C2, (C22×C4).140D6, (C6×Q8).43C22, C12.187(C4○D4), C4.93(D42S3), (C2×C12).362C23, C12.55D4.6C2, C23.67(C3⋊D4), C35(C23.47D4), C2.14(Q8.14D6), C6.116(C8.C22), C4⋊Dic3.337C22, C22.6(Q82S3), (C22×C12).166C22, C6.80(C22.D4), C2.14(C23.23D6), (C2×C6).493(C2×D4), C2.8(C2×Q82S3), (C3×C22⋊Q8).1C2, (C2×C4).53(C3⋊D4), (C2×C3⋊C8).112C22, (C2×C4⋊Dic3).37C2, (C3×C4⋊C4).110C22, (C2×C4).462(C22×S3), C22.168(C2×C3⋊D4), SmallGroup(192,602)

Series: Derived Chief Lower central Upper central

C1C2×C12 — (C2×Q8).49D6
C1C3C6C12C2×C12C4⋊Dic3C2×C4⋊Dic3 — (C2×Q8).49D6
C3C6C2×C12 — (C2×Q8).49D6
C1C22C22×C4C22⋊Q8

Generators and relations for (C2×Q8).49D6
 G = < a,b,c,d,e | a2=b4=d6=1, c2=b2, e2=ab2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe-1=b-1, bd=db, dcd-1=ab2c, ece-1=b-1c, ede-1=d-1 >

Subgroups: 256 in 104 conjugacy classes, 43 normal (27 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C4 [×5], C22, C22 [×2], C22 [×2], C6 [×3], C6 [×2], C8 [×2], C2×C4 [×2], C2×C4 [×8], Q8 [×2], C23, Dic3 [×2], C12 [×2], C12 [×3], C2×C6, C2×C6 [×2], C2×C6 [×2], C22⋊C4, C4⋊C4, C4⋊C4 [×4], C2×C8 [×2], C22×C4, C22×C4, C2×Q8, C3⋊C8 [×2], C2×Dic3 [×4], C2×C12 [×2], C2×C12 [×4], C3×Q8 [×2], C22×C6, C22⋊C8, Q8⋊C4 [×2], C4.Q8 [×2], C2×C4⋊C4, C22⋊Q8, C2×C3⋊C8 [×2], C4⋊Dic3 [×2], C4⋊Dic3, C3×C22⋊C4, C3×C4⋊C4, C3×C4⋊C4, C22×Dic3, C22×C12, C6×Q8, C23.47D4, C12.Q8 [×2], C12.55D4, Q82Dic3 [×2], C2×C4⋊Dic3, C3×C22⋊Q8, (C2×Q8).49D6
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], SD16 [×2], C2×D4, C4○D4 [×2], C3⋊D4 [×2], C22×S3, C22.D4, C2×SD16, C8.C22, Q82S3 [×2], D42S3 [×2], C2×C3⋊D4, C23.47D4, C23.23D6, C2×Q82S3, Q8.14D6, (C2×Q8).49D6

Smallest permutation representation of (C2×Q8).49D6
On 96 points
Generators in S96
(1 10)(2 11)(3 12)(4 7)(5 8)(6 9)(13 22)(14 23)(15 24)(16 45)(17 43)(18 44)(19 26)(20 27)(21 25)(28 36)(29 34)(30 35)(31 39)(32 37)(33 38)(40 48)(41 46)(42 47)(49 81)(50 82)(51 83)(52 84)(53 79)(54 80)(55 65)(56 66)(57 61)(58 62)(59 63)(60 64)(67 74)(68 75)(69 76)(70 77)(71 78)(72 73)(85 91)(86 92)(87 93)(88 94)(89 95)(90 96)
(1 22 4 19)(2 23 5 20)(3 24 6 21)(7 26 10 13)(8 27 11 14)(9 25 12 15)(16 46 29 33)(17 47 30 31)(18 48 28 32)(34 38 45 41)(35 39 43 42)(36 37 44 40)(49 64 84 57)(50 65 79 58)(51 66 80 59)(52 61 81 60)(53 62 82 55)(54 63 83 56)(67 88 77 91)(68 89 78 92)(69 90 73 93)(70 85 74 94)(71 86 75 95)(72 87 76 96)
(1 52 4 81)(2 50 5 79)(3 54 6 83)(7 49 10 84)(8 53 11 82)(9 51 12 80)(13 64 26 57)(14 62 27 55)(15 66 25 59)(16 89 29 92)(17 87 30 96)(18 85 28 94)(19 61 22 60)(20 65 23 58)(21 63 24 56)(31 76 47 72)(32 74 48 70)(33 78 46 68)(34 86 45 95)(35 90 43 93)(36 88 44 91)(37 67 40 77)(38 71 41 75)(39 69 42 73)
(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 34 7 16)(2 36 8 18)(3 35 9 17)(4 45 10 29)(5 44 11 28)(6 43 12 30)(13 46 19 38)(14 48 20 37)(15 47 21 39)(22 41 26 33)(23 40 27 32)(24 42 25 31)(49 78 52 75)(50 77 53 74)(51 76 54 73)(55 94 58 91)(56 93 59 96)(57 92 60 95)(61 86 64 89)(62 85 65 88)(63 90 66 87)(67 82 70 79)(68 81 71 84)(69 80 72 83)

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

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

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

33 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F4G4H4I6A6B6C6D6E8A8B8C8D12A12B12C12D12E12F12G12H
order12222234444444446666688881212121212121212
size11112222248812121212222441212121244448888

33 irreducible representations

dim11111122222222224444
type++++++++++++--+-
imageC1C2C2C2C2C2S3D4D4D6D6D6C4○D4SD16C3⋊D4C3⋊D4C8.C22D42S3Q82S3Q8.14D6
kernel(C2×Q8).49D6C12.Q8C12.55D4Q82Dic3C2×C4⋊Dic3C3×C22⋊Q8C22⋊Q8C2×C12C22×C6C4⋊C4C22×C4C2×Q8C12C2×C6C2×C4C23C6C4C22C2
# reps12121111111144221222

Matrix representation of (C2×Q8).49D6 in GL6(𝔽73)

7200000
0720000
001000
000100
0000720
0000072
,
7200000
0720000
0072000
0007200
0000213
00002352
,
7220000
010000
00436000
00133000
00001816
00003955
,
100000
1720000
0007200
0017200
000010
000001
,
2700000
27460000
0018500
00235500
00006137
0000612

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,21,23,0,0,0,0,3,52],[72,0,0,0,0,0,2,1,0,0,0,0,0,0,43,13,0,0,0,0,60,30,0,0,0,0,0,0,18,39,0,0,0,0,16,55],[1,1,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,72,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[27,27,0,0,0,0,0,46,0,0,0,0,0,0,18,23,0,0,0,0,5,55,0,0,0,0,0,0,61,6,0,0,0,0,37,12] >;

(C2×Q8).49D6 in GAP, Magma, Sage, TeX

(C_2\times Q_8)._{49}D_6
% in TeX

G:=Group("(C2xQ8).49D6");
// GroupNames label

G:=SmallGroup(192,602);
// by ID

G=gap.SmallGroup(192,602);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,254,219,184,1123,297,136,6278]);
// Polycyclic

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

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