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G = C3⋊C8.6D4order 192 = 26·3

6th non-split extension by C3⋊C8 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C3⋊C8.6D4, C4⋊C4.71D6, C4.177(S3×D4), (C2×C12).80D4, C33(C8.D4), (C2×Q8).57D6, C22⋊Q8.7S3, C12.158(C2×D4), (C22×C6).98D4, Q82Dic318C2, C6.SD1640C2, C12.Q840C2, (C22×C4).147D6, C12.193(C4○D4), (C6×Q8).51C22, C4.66(D42S3), C6.100(C4⋊D4), (C2×C12).371C23, C6.92(C8.C22), C23.36(C3⋊D4), C12.48D4.13C2, C2.16(Q8.14D6), C4⋊Dic3.149C22, C2.21(C23.14D6), C2.13(Q8.11D6), (C22×C12).175C22, (C2×Dic6).106C22, (C2×C3⋊Q16)⋊11C2, (C2×C6).502(C2×D4), (C3×C22⋊Q8).6C2, (C2×C4).58(C3⋊D4), (C2×C3⋊C8).118C22, (C3×C4⋊C4).118C22, (C2×C4).471(C22×S3), C22.177(C2×C3⋊D4), (C2×C4.Dic3).20C2, SmallGroup(192,611)

Series: Derived Chief Lower central Upper central

C1C2×C12 — C3⋊C8.6D4
C1C3C6C12C2×C12C2×Dic6C12.48D4 — C3⋊C8.6D4
C3C6C2×C12 — C3⋊C8.6D4
C1C22C22×C4C22⋊Q8

Generators and relations for C3⋊C8.6D4
 G = < a,b,c,d | a3=b8=c4=d2=1, bab-1=cac-1=a-1, ad=da, cbc-1=b3, dbd=b5, dcd=b4c-1 >

Subgroups: 272 in 110 conjugacy classes, 41 normal (39 characteristic)
C1, C2 [×3], C2, C3, C4 [×2], C4 [×5], C22, C22 [×3], C6 [×3], C6, C8 [×3], C2×C4 [×2], C2×C4 [×6], Q8 [×4], C23, Dic3 [×2], C12 [×2], C12 [×3], C2×C6, C2×C6 [×3], C22⋊C4 [×2], C4⋊C4, C4⋊C4 [×3], C2×C8 [×2], M4(2) [×2], Q16 [×2], C22×C4, C2×Q8, C2×Q8, C3⋊C8 [×2], C3⋊C8, Dic6 [×2], C2×Dic3 [×2], C2×C12 [×2], C2×C12 [×4], C3×Q8 [×2], C22×C6, Q8⋊C4 [×2], C4.Q8, C22⋊Q8, C22⋊Q8, C2×M4(2), C2×Q16, C2×C3⋊C8 [×2], C4.Dic3 [×2], Dic3⋊C4, C4⋊Dic3, C3⋊Q16 [×2], C6.D4, C3×C22⋊C4, C3×C4⋊C4, C3×C4⋊C4, C2×Dic6, C22×C12, C6×Q8, C8.D4, C12.Q8, C6.SD16, Q82Dic3, C2×C4.Dic3, C12.48D4, C2×C3⋊Q16, C3×C22⋊Q8, C3⋊C8.6D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×4], C23, D6 [×3], C2×D4 [×2], C4○D4, C3⋊D4 [×2], C22×S3, C4⋊D4, C8.C22 [×2], S3×D4, D42S3, C2×C3⋊D4, C8.D4, C23.14D6, Q8.11D6, Q8.14D6, C3⋊C8.6D4

Character table of C3⋊C8.6D4

 class 12A2B2C2D34A4B4C4D4E4F4G6A6B6C6D6E8A8B8C8D12A12B12C12D12E12F12G12H
 size 111142224882424222441212121244448888
ρ1111111111111111111111111111111    trivial
ρ2111111111-1-1-1-11111111111111-1-1-1-1    linear of order 2
ρ311111111111-1-111111-1-1-1-111111111    linear of order 2
ρ4111111111-1-11111111-1-1-1-11111-1-1-1-1    linear of order 2
ρ51111-1111-11-11-1111-1-11-1-111-11-1-111-1    linear of order 2
ρ61111-1111-1-11-11111-1-11-1-111-11-11-1-11    linear of order 2
ρ71111-1111-1-111-1111-1-1-111-11-11-11-1-11    linear of order 2
ρ81111-1111-11-1-11111-1-1-111-11-11-1-111-1    linear of order 2
ρ922-2-202-22000002-2-2000-22020-200000    orthogonal lifted from D4
ρ102222-2-122-2-2200-1-1-1110000-11-11-111-1    orthogonal lifted from D6
ρ1122222-12222200-1-1-1-1-10000-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ122222-2-122-22-200-1-1-1110000-11-111-1-11    orthogonal lifted from D6
ρ1322222-1222-2-200-1-1-1-1-10000-1-1-1-11111    orthogonal lifted from D6
ρ142222-22-2-220000222-2-20000-22-220000    orthogonal lifted from D4
ρ1522-2-202-22000002-2-20002-2020-200000    orthogonal lifted from D4
ρ16222222-2-2-20000222220000-2-2-2-20000    orthogonal lifted from D4
ρ1722222-1-2-2-20000-1-1-1-1-100001111--3-3--3-3    complex lifted from C3⋊D4
ρ182222-2-1-2-220000-1-1-11100001-11-1--3--3-3-3    complex lifted from C3⋊D4
ρ1922222-1-2-2-20000-1-1-1-1-100001111-3--3-3--3    complex lifted from C3⋊D4
ρ202222-2-1-2-220000-1-1-11100001-11-1-3-3--3--3    complex lifted from C3⋊D4
ρ2122-2-2022-2000002-2-2002i00-2i-20200000    complex lifted from C4○D4
ρ2222-2-2022-2000002-2-200-2i002i-20200000    complex lifted from C4○D4
ρ2344-4-40-2-4400000-222000000-20200000    orthogonal lifted from S3×D4
ρ244-44-4040000000-4-4400000000000000    symplectic lifted from C8.C22, Schur index 2
ρ254-4-44040000000-44-400000000000000    symplectic lifted from C8.C22, Schur index 2
ρ2644-4-40-24-400000-22200000020-200000    symplectic lifted from D42S3, Schur index 2
ρ274-44-40-2000000022-20000000230-230000    symplectic lifted from Q8.14D6, Schur index 2
ρ284-44-40-2000000022-20000000-230230000    symplectic lifted from Q8.14D6, Schur index 2
ρ294-4-440-200000002-222-3-2-3000000000000    complex lifted from Q8.11D6
ρ304-4-440-200000002-22-2-32-3000000000000    complex lifted from Q8.11D6

Smallest permutation representation of C3⋊C8.6D4
On 96 points
Generators in S96
(1 83 94)(2 95 84)(3 85 96)(4 89 86)(5 87 90)(6 91 88)(7 81 92)(8 93 82)(9 45 55)(10 56 46)(11 47 49)(12 50 48)(13 41 51)(14 52 42)(15 43 53)(16 54 44)(17 31 33)(18 34 32)(19 25 35)(20 36 26)(21 27 37)(22 38 28)(23 29 39)(24 40 30)(57 73 68)(58 69 74)(59 75 70)(60 71 76)(61 77 72)(62 65 78)(63 79 66)(64 67 80)
(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 57 12 24)(2 60 13 19)(3 63 14 22)(4 58 15 17)(5 61 16 20)(6 64 9 23)(7 59 10 18)(8 62 11 21)(25 84 71 51)(26 87 72 54)(27 82 65 49)(28 85 66 52)(29 88 67 55)(30 83 68 50)(31 86 69 53)(32 81 70 56)(33 89 74 43)(34 92 75 46)(35 95 76 41)(36 90 77 44)(37 93 78 47)(38 96 79 42)(39 91 80 45)(40 94 73 48)
(2 6)(4 8)(9 13)(11 15)(17 58)(18 63)(19 60)(20 57)(21 62)(22 59)(23 64)(24 61)(25 71)(26 68)(27 65)(28 70)(29 67)(30 72)(31 69)(32 66)(33 74)(34 79)(35 76)(36 73)(37 78)(38 75)(39 80)(40 77)(41 45)(43 47)(49 53)(51 55)(82 86)(84 88)(89 93)(91 95)

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

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

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

Matrix representation of C3⋊C8.6D4 in GL6(𝔽73)

100000
010000
00727200
001000
00007272
000010
,
100000
010000
00006225
00003611
00113600
00256200
,
3430000
28390000
00623700
00481100
00001148
00003762
,
7200000
4710000
001000
000100
0000720
0000072

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,1,0,0,0,0,72,0,0,0,0,0,0,0,72,1,0,0,0,0,72,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,11,25,0,0,0,0,36,62,0,0,62,36,0,0,0,0,25,11,0,0],[34,28,0,0,0,0,3,39,0,0,0,0,0,0,62,48,0,0,0,0,37,11,0,0,0,0,0,0,11,37,0,0,0,0,48,62],[72,47,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72] >;

C3⋊C8.6D4 in GAP, Magma, Sage, TeX

C_3\rtimes C_8._6D_4
% in TeX

G:=Group("C3:C8.6D4");
// GroupNames label

G:=SmallGroup(192,611);
// by ID

G=gap.SmallGroup(192,611);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,253,254,555,184,1123,297,136,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^8=c^4=d^2=1,b*a*b^-1=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=b^3,d*b*d=b^5,d*c*d=b^4*c^-1>;
// generators/relations

Export

Character table of C3⋊C8.6D4 in TeX

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