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G = C12⋊Q8⋊C2order 192 = 26·3

4th semidirect product of C12⋊Q8 and C2 acting faithfully

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C12⋊Q84C2, C4⋊C4.6D6, C24⋊C417C2, (C2×D4).20D6, (C2×C8).167D6, D4⋊C4.8S3, C12.4(C4○D4), C6.SD161C2, C4.22(C4○D12), C2.12(D8⋊S3), C6.28(C8⋊C22), C2.Dic1221C2, (C2×Dic3).16D4, D4⋊Dic3.4C2, C2.9(D4.D6), (C6×D4).26C22, C22.167(S3×D4), C4.48(D42S3), (C2×C24).223C22, (C2×C12).205C23, C23.12D6.2C2, C6.24(C4.4D4), C6.26(C8.C22), C4⋊Dic3.64C22, (C4×Dic3).9C22, (C2×Dic6).52C22, C32(C42.28C22), C2.14(C23.11D6), (C2×C6).218(C2×D4), (C2×C3⋊C8).11C22, (C3×C4⋊C4).10C22, (C3×D4⋊C4).12C2, (C2×C4).312(C22×S3), SmallGroup(192,324)

Series: Derived Chief Lower central Upper central

C1C2×C12 — C12⋊Q8⋊C2
C1C3C6C12C2×C12C4×Dic3C12⋊Q8 — C12⋊Q8⋊C2
C3C6C2×C12 — C12⋊Q8⋊C2
C1C22C2×C4D4⋊C4

Generators and relations for C12⋊Q8⋊C2
 G = < a,b,c,d | a12=b4=d2=1, c2=b2, bab-1=dad=a7, cac-1=a5, cbc-1=b-1, dbd=a3b-1, dcd=a6b2c >

Subgroups: 296 in 100 conjugacy classes, 37 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, C12, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C2×D4, C2×Q8, C3⋊C8, C24, Dic6, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×D4, C22×C6, C8⋊C4, D4⋊C4, D4⋊C4, Q8⋊C4, C4.4D4, C4⋊Q8, C2×C3⋊C8, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C6.D4, C3×C4⋊C4, C2×C24, C2×Dic6, C2×Dic6, C6×D4, C42.28C22, C6.SD16, C24⋊C4, C2.Dic12, D4⋊Dic3, C3×D4⋊C4, C12⋊Q8, C23.12D6, C12⋊Q8⋊C2
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C22×S3, C4.4D4, C8⋊C22, C8.C22, C4○D12, S3×D4, D42S3, C42.28C22, C23.11D6, D8⋊S3, D4.D6, C12⋊Q8⋊C2

Character table of C12⋊Q8⋊C2

 class 12A2B2C2D34A4B4C4D4E4F4G6A6B6C6D6E8A8B8C8D12A12B12C12D24A24B24C24D
 size 111182228121224242228844121244884444
ρ1111111111111111111111111111111    trivial
ρ21111-1111-1-1-111111-1-111-1-111-1-11111    linear of order 2
ρ31111-11111111-1111-1-1-1-1-1-11111-1-1-1-1    linear of order 2
ρ411111111-1-1-11-111111-1-11111-1-1-1-1-1-1    linear of order 2
ρ5111111111-1-1-1-11111111-1-111111111    linear of order 2
ρ61111-1111-111-1-1111-1-1111111-1-11111    linear of order 2
ρ71111-11111-1-1-11111-1-1-1-1111111-1-1-1-1    linear of order 2
ρ811111111-111-1111111-1-1-1-111-1-1-1-1-1-1    linear of order 2
ρ92222-2-12220000-1-1-111-2-200-1-1-1-11111    orthogonal lifted from D6
ρ102222-2-122-20000-1-1-1112200-1-111-1-1-1-1    orthogonal lifted from D6
ρ11222202-2-202-200222000000-2-2000000    orthogonal lifted from D4
ρ12222202-2-20-2200222000000-2-2000000    orthogonal lifted from D4
ρ1322222-12220000-1-1-1-1-12200-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ1422222-122-20000-1-1-1-1-1-2-200-1-1111111    orthogonal lifted from D6
ρ152-2-2202-2200000-2-2200-2i2i002-2002i2i-2i-2i    complex lifted from C4○D4
ρ162-2-2202-2200000-2-22002i-2i002-200-2i-2i2i2i    complex lifted from C4○D4
ρ172-2-22022-200000-2-2200002i-2i-22000000    complex lifted from C4○D4
ρ182-2-22022-200000-2-220000-2i2i-22000000    complex lifted from C4○D4
ρ192-2-220-1-220000011-1--3-32i-2i00-113-3ii-i-i    complex lifted from C4○D12
ρ202-2-220-1-220000011-1-3--3-2i2i00-113-3-i-iii    complex lifted from C4○D12
ρ212-2-220-1-220000011-1--3-3-2i2i00-11-33-i-iii    complex lifted from C4○D12
ρ222-2-220-1-220000011-1-3--32i-2i00-11-33ii-i-i    complex lifted from C4○D12
ρ234-44-4040000000-44-400000000000000    orthogonal lifted from C8⋊C22
ρ2444440-2-4-400000-2-2-200000022000000    orthogonal lifted from S3×D4
ρ254-4-440-24-40000022-20000002-2000000    symplectic lifted from D42S3, Schur index 2
ρ2644-4-40400000004-4-400000000000000    symplectic lifted from C8.C22, Schur index 2
ρ2744-4-40-20000000-2220000000000-66-66    symplectic lifted from D4.D6, Schur index 2
ρ2844-4-40-20000000-22200000000006-66-6    symplectic lifted from D4.D6, Schur index 2
ρ294-44-40-200000002-220000000000-6--6--6-6    complex lifted from D8⋊S3
ρ304-44-40-200000002-220000000000--6-6-6--6    complex lifted from D8⋊S3

Smallest permutation representation of C12⋊Q8⋊C2
On 96 points
Generators in S96
(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 40 89 60)(2 47 90 55)(3 42 91 50)(4 37 92 57)(5 44 93 52)(6 39 94 59)(7 46 95 54)(8 41 96 49)(9 48 85 56)(10 43 86 51)(11 38 87 58)(12 45 88 53)(13 62 74 29)(14 69 75 36)(15 64 76 31)(16 71 77 26)(17 66 78 33)(18 61 79 28)(19 68 80 35)(20 63 81 30)(21 70 82 25)(22 65 83 32)(23 72 84 27)(24 67 73 34)
(1 22 89 83)(2 15 90 76)(3 20 91 81)(4 13 92 74)(5 18 93 79)(6 23 94 84)(7 16 95 77)(8 21 96 82)(9 14 85 75)(10 19 86 80)(11 24 87 73)(12 17 88 78)(25 49 70 41)(26 54 71 46)(27 59 72 39)(28 52 61 44)(29 57 62 37)(30 50 63 42)(31 55 64 47)(32 60 65 40)(33 53 66 45)(34 58 67 38)(35 51 68 43)(36 56 69 48)
(2 8)(4 10)(6 12)(13 74)(14 81)(15 76)(16 83)(17 78)(18 73)(19 80)(20 75)(21 82)(22 77)(23 84)(24 79)(25 34)(26 29)(27 36)(28 31)(30 33)(32 35)(37 60)(38 55)(39 50)(40 57)(41 52)(42 59)(43 54)(44 49)(45 56)(46 51)(47 58)(48 53)(61 64)(62 71)(63 66)(65 68)(67 70)(69 72)(86 92)(88 94)(90 96)

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

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

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

Matrix representation of C12⋊Q8⋊C2 in GL6(𝔽73)

100000
010000
0000072
0000172
000100
0072100
,
46710000
0270000
0042624262
0011311131
0042623111
0011316242
,
1190000
46720000
00003465
00002639
0039800
00473400
,
100000
46720000
001000
000100
0000720
0000072

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,72,0,0,0,0,1,1,0,0,0,1,0,0,0,0,72,72,0,0],[46,0,0,0,0,0,71,27,0,0,0,0,0,0,42,11,42,11,0,0,62,31,62,31,0,0,42,11,31,62,0,0,62,31,11,42],[1,46,0,0,0,0,19,72,0,0,0,0,0,0,0,0,39,47,0,0,0,0,8,34,0,0,34,26,0,0,0,0,65,39,0,0],[1,46,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72] >;

C12⋊Q8⋊C2 in GAP, Magma, Sage, TeX

C_{12}\rtimes Q_8\rtimes C_2
% in TeX

G:=Group("C12:Q8:C2");
// GroupNames label

G:=SmallGroup(192,324);
// by ID

G=gap.SmallGroup(192,324);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,253,120,1094,135,570,297,136,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^12=b^4=d^2=1,c^2=b^2,b*a*b^-1=d*a*d=a^7,c*a*c^-1=a^5,c*b*c^-1=b^-1,d*b*d=a^3*b^-1,d*c*d=a^6*b^2*c>;
// generators/relations

Export

Character table of C12⋊Q8⋊C2 in TeX

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