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G = C22⋊C4⋊D6order 192 = 26·3

4th semidirect product of C22⋊C4 and D6 acting via D6/C3=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C22⋊C44D6, (C22×C4)⋊5D6, (C2×D4).43D6, (C2×C12).10D4, C6.50C22≀C2, D46D6.3C2, (C2×Dic3).4D4, (C22×S3).4D4, (C22×C6).21D4, C22.34(S3×D4), (C22×C12)⋊2C22, (C6×D4).59C22, C22.D41S3, C23.7D66C2, C23.6D66C2, C32(C23.7D4), C23.9(C3⋊D4), C2.18(C232D6), C23.28D61C2, C6.D45C22, C23.85(C22×S3), (C22×C6).114C23, (C2×C6).31(C2×D4), (C2×C4).9(C3⋊D4), (C2×C3⋊D4).6C22, C22.30(C2×C3⋊D4), (C3×C22⋊C4)⋊35C22, (C3×C22.D4)⋊1C2, SmallGroup(192,612)

Series: Derived Chief Lower central Upper central

C1C22×C6 — C22⋊C4⋊D6
C1C3C6C2×C6C22×C6C2×C3⋊D4D46D6 — C22⋊C4⋊D6
C3C6C22×C6 — C22⋊C4⋊D6
C1C2C23C22.D4

Generators and relations for C22⋊C4⋊D6
 G = < a,b,c,d,e | a2=b2=c4=d6=e2=1, cac-1=dad-1=ab=ba, ae=ea, bc=cb, bd=db, be=eb, dcd-1=bc-1, ece=abc, ede=d-1 >

Subgroups: 528 in 160 conjugacy classes, 39 normal (23 characteristic)
C1, C2, C2 [×6], C3, C4 [×7], C22, C22 [×2], C22 [×8], S3 [×2], C6, C6 [×4], C2×C4, C2×C4 [×11], D4 [×9], Q8, C23 [×2], C23 [×3], Dic3 [×4], C12 [×3], D6 [×5], C2×C6, C2×C6 [×2], C2×C6 [×3], C22⋊C4, C22⋊C4 [×5], C4⋊C4 [×3], C22×C4, C2×D4, C2×D4 [×5], C4○D4 [×3], Dic6, C4×S3 [×2], D12, C2×Dic3 [×2], C2×Dic3 [×3], C3⋊D4 [×6], C2×C12, C2×C12 [×4], C3×D4 [×2], C22×S3 [×2], C22×S3, C22×C6 [×2], C23⋊C4 [×3], C22.D4, C22.D4 [×2], 2+ 1+4, Dic3⋊C4 [×2], D6⋊C4 [×2], C6.D4 [×2], C3×C22⋊C4, C3×C22⋊C4, C3×C4⋊C4, C4○D12, S3×D4 [×2], D42S3 [×2], C2×C3⋊D4 [×2], C2×C3⋊D4, C22×C12, C6×D4, C23.7D4, C23.6D6 [×2], C23.7D6, C23.28D6 [×2], C3×C22.D4, D46D6, C22⋊C4⋊D6
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×6], C23, D6 [×3], C2×D4 [×3], C3⋊D4 [×2], C22×S3, C22≀C2, S3×D4 [×2], C2×C3⋊D4, C23.7D4, C232D6, C22⋊C4⋊D6

Character table of C22⋊C4⋊D6

 class 12A2B2C2D2E2F2G34A4B4C4D4E4F4G4H6A6B6C6D6E6F12A12B12C12D12E12F12G
 size 112224121224448121224242224484444888
ρ1111111111111111111111111111111    trivial
ρ211111-11-1111-1-1-111-111111-11111-1-1-1    linear of order 2
ρ3111111-1-111111-1-1-1-11111111111111    linear of order 2
ρ411111-1-11111-1-11-1-1111111-11111-1-1-1    linear of order 2
ρ5111111111-1-11-111-1-1111111-1-1-1-11-1-1    linear of order 2
ρ611111-11-11-1-1-11-11-1111111-1-1-1-1-1-111    linear of order 2
ρ7111111-1-11-1-11-1-1-111111111-1-1-1-11-1-1    linear of order 2
ρ811111-1-111-1-1-111-11-111111-1-1-1-1-1-111    linear of order 2
ρ922-2-22-200200200000222-2-2-20000200    orthogonal lifted from D4
ρ1022222200-122220000-1-1-1-1-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ1122222-200-1-2-2-220000-1-1-1-1-1111111-1-1    orthogonal lifted from D6
ρ1222-22-20-202000002002-2-22-200000000    orthogonal lifted from D4
ρ1322222200-1-2-22-20000-1-1-1-1-1-11111-111    orthogonal lifted from D6
ρ14222-2-200220000-20002-2-2-2200000000    orthogonal lifted from D4
ρ1522-22-2020200000-2002-2-22-200000000    orthogonal lifted from D4
ρ1622222-200-122-2-20000-1-1-1-1-11-1-1-1-1111    orthogonal lifted from D6
ρ17222-2-200-22000020002-2-2-2200000000    orthogonal lifted from D4
ρ1822-2-22200200-200000222-2-220000-200    orthogonal lifted from D4
ρ1922-2-22-200-100200000-1-1-1111--3-3-3--3-1--3-3    complex lifted from C3⋊D4
ρ2022-2-22200-100-200000-1-1-111-1-3--3--3-31--3-3    complex lifted from C3⋊D4
ρ2122-2-22-200-100200000-1-1-1111-3--3--3-3-1-3--3    complex lifted from C3⋊D4
ρ2222-2-22200-100-200000-1-1-111-1--3-3-3--31-3--3    complex lifted from C3⋊D4
ρ2344-44-4000-200000000-222-2200000000    orthogonal lifted from S3×D4
ρ24444-4-4000-200000000-2222-200000000    orthogonal lifted from S3×D4
ρ254-400000042i-2i000000-4000002i-2i2i-2i000    complex lifted from C23.7D4
ρ264-40000004-2i2i000000-400000-2i2i-2i2i000    complex lifted from C23.7D4
ρ274-4000000-22i-2i0000002-2-32-30004ζ343ζ324ζ3243ζ3000    complex faithful
ρ284-4000000-2-2i2i00000022-3-2-300043ζ324ζ343ζ34ζ32000    complex faithful
ρ294-4000000-2-2i2i0000002-2-32-300043ζ34ζ3243ζ324ζ3000    complex faithful
ρ304-4000000-22i-2i00000022-3-2-30004ζ3243ζ34ζ343ζ32000    complex faithful

Smallest permutation representation of C22⋊C4⋊D6
On 48 points
Generators in S48
(1 22)(2 20)(3 24)(4 19)(5 23)(6 21)(7 16)(8 14)(9 18)(10 13)(11 17)(12 15)(25 43)(26 40)(27 45)(28 42)(29 47)(30 38)(31 44)(32 41)(33 46)(34 37)(35 48)(36 39)
(1 4)(2 5)(3 6)(7 10)(8 11)(9 12)(13 16)(14 17)(15 18)(19 22)(20 23)(21 24)(25 36)(26 31)(27 32)(28 33)(29 34)(30 35)(37 47)(38 48)(39 43)(40 44)(41 45)(42 46)
(1 31 10 29)(2 35 11 27)(3 33 12 25)(4 26 7 34)(5 30 8 32)(6 28 9 36)(13 37 22 40)(14 45 23 48)(15 39 24 42)(16 47 19 44)(17 41 20 38)(18 43 21 46)
(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)
(1 3)(4 6)(7 12)(8 11)(9 10)(13 18)(14 17)(15 16)(19 21)(22 24)(25 37)(26 42)(27 41)(28 40)(29 39)(30 38)(31 46)(32 45)(33 44)(34 43)(35 48)(36 47)

G:=sub<Sym(48)| (1,22)(2,20)(3,24)(4,19)(5,23)(6,21)(7,16)(8,14)(9,18)(10,13)(11,17)(12,15)(25,43)(26,40)(27,45)(28,42)(29,47)(30,38)(31,44)(32,41)(33,46)(34,37)(35,48)(36,39), (1,4)(2,5)(3,6)(7,10)(8,11)(9,12)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24)(25,36)(26,31)(27,32)(28,33)(29,34)(30,35)(37,47)(38,48)(39,43)(40,44)(41,45)(42,46), (1,31,10,29)(2,35,11,27)(3,33,12,25)(4,26,7,34)(5,30,8,32)(6,28,9,36)(13,37,22,40)(14,45,23,48)(15,39,24,42)(16,47,19,44)(17,41,20,38)(18,43,21,46), (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), (1,3)(4,6)(7,12)(8,11)(9,10)(13,18)(14,17)(15,16)(19,21)(22,24)(25,37)(26,42)(27,41)(28,40)(29,39)(30,38)(31,46)(32,45)(33,44)(34,43)(35,48)(36,47)>;

G:=Group( (1,22)(2,20)(3,24)(4,19)(5,23)(6,21)(7,16)(8,14)(9,18)(10,13)(11,17)(12,15)(25,43)(26,40)(27,45)(28,42)(29,47)(30,38)(31,44)(32,41)(33,46)(34,37)(35,48)(36,39), (1,4)(2,5)(3,6)(7,10)(8,11)(9,12)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24)(25,36)(26,31)(27,32)(28,33)(29,34)(30,35)(37,47)(38,48)(39,43)(40,44)(41,45)(42,46), (1,31,10,29)(2,35,11,27)(3,33,12,25)(4,26,7,34)(5,30,8,32)(6,28,9,36)(13,37,22,40)(14,45,23,48)(15,39,24,42)(16,47,19,44)(17,41,20,38)(18,43,21,46), (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), (1,3)(4,6)(7,12)(8,11)(9,10)(13,18)(14,17)(15,16)(19,21)(22,24)(25,37)(26,42)(27,41)(28,40)(29,39)(30,38)(31,46)(32,45)(33,44)(34,43)(35,48)(36,47) );

G=PermutationGroup([(1,22),(2,20),(3,24),(4,19),(5,23),(6,21),(7,16),(8,14),(9,18),(10,13),(11,17),(12,15),(25,43),(26,40),(27,45),(28,42),(29,47),(30,38),(31,44),(32,41),(33,46),(34,37),(35,48),(36,39)], [(1,4),(2,5),(3,6),(7,10),(8,11),(9,12),(13,16),(14,17),(15,18),(19,22),(20,23),(21,24),(25,36),(26,31),(27,32),(28,33),(29,34),(30,35),(37,47),(38,48),(39,43),(40,44),(41,45),(42,46)], [(1,31,10,29),(2,35,11,27),(3,33,12,25),(4,26,7,34),(5,30,8,32),(6,28,9,36),(13,37,22,40),(14,45,23,48),(15,39,24,42),(16,47,19,44),(17,41,20,38),(18,43,21,46)], [(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)], [(1,3),(4,6),(7,12),(8,11),(9,10),(13,18),(14,17),(15,16),(19,21),(22,24),(25,37),(26,42),(27,41),(28,40),(29,39),(30,38),(31,46),(32,45),(33,44),(34,43),(35,48),(36,47)])

Matrix representation of C22⋊C4⋊D6 in GL6(𝔽13)

100000
010000
0000012
0000120
0001200
0012000
,
100000
010000
0012000
0001200
0000120
0000012
,
100000
010000
009949
009994
009444
004944
,
110000
1200000
001000
000100
0000120
0000012
,
12120000
010000
001000
0001200
0000120
000001

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,12,0,0,0,0,12,0,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,9,9,4,0,0,9,9,4,9,0,0,4,9,4,4,0,0,9,4,4,4],[1,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,0,0,0,0,0,12,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1] >;

C22⋊C4⋊D6 in GAP, Magma, Sage, TeX

C_2^2\rtimes C_4\rtimes D_6
% in TeX

G:=Group("C2^2:C4:D6");
// GroupNames label

G:=SmallGroup(192,612);
// by ID

G=gap.SmallGroup(192,612);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,254,219,184,570,1684,6278]);
// Polycyclic

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

Export

Character table of C22⋊C4⋊D6 in TeX

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