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G = C42.72D6  order 192 = 26·3

72nd non-split extension by C42 of D6 acting via D6/C3=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.72D6, (C2xD4).54D6, C4:1D4.5S3, (C2xC12).291D4, C12.75(C4oD4), D4:Dic3:21C2, C6.93(C8:C22), C12.6Q8:13C2, (C6xD4).70C22, C4.23(D4:2S3), (C4xC12).119C22, (C2xC12).389C23, C42.S3:12C2, C6.44(C4.4D4), C2.14(D12:6C22), C4:Dic3.155C22, C2.11(C23.12D6), C3:4(C42.29C22), (C2xC6).520(C2xD4), (C3xC4:1D4).3C2, (C2xC4).69(C3:D4), (C2xC3:C8).129C22, (C2xC4).487(C22xS3), C22.193(C2xC3:D4), SmallGroup(192,630)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2xC12 — C42.72D6
Chief series
C1 — C3 — C6 — C2xC6 — C2xC12 — C2xC3:C8 — C42.S3 — C42.72D6
Lower central
C3 — C6 — C2xC12 — C42.72D6
Upper central
C1 — C22 — C42 — C4:1D4

Generators and relations for C42.72D6
 G = < a,b,c,d | a4=b4=c6=1, d2=a2, ab=ba, cac-1=a-1, dad-1=a-1b2, cbc-1=dbd-1=b-1, dcd-1=a2bc-1 >

Subgroups: 304 in 110 conjugacy classes, 39 normal (15 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C6, C6, C6, C8, C2xC4, C2xC4, C2xC4, D4, C23, Dic3, C12, C12, C2xC6, C2xC6, C42, C4:C4, C2xC8, C2xD4, C2xD4, C3:C8, C2xDic3, C2xC12, C2xC12, C3xD4, C22xC6, C8:C4, D4:C4, C42.C2, C4:1D4, C2xC3:C8, Dic3:C4, C4:Dic3, C4xC12, C6xD4, C6xD4, C42.29C22, C42.S3, D4:Dic3, C12.6Q8, C3xC4:1D4, C42.72D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, C4oD4, C3:D4, C22xS3, C4.4D4, C8:C22, D4:2S3, C2xC3:D4, C42.29C22, D12:6C22, C23.12D6, C42.72D6

Character table of C42.72D6

 class 12A2B2C2D2E34A4B4C4D4E4F6A6B6C6D6E6F6G8A8B8C8D12A12B12C12D12E12F
 size 111188222442424222888812121212444444
ρ1111111111111111111111111111111    trivial
ρ211111111111-1-11111111-1-1-1-1111111    linear of order 2
ρ31111-1-11111111111-1-1-1-1-1-1-1-1111111    linear of order 2
ρ41111-1-111111-1-1111-1-1-1-11111111111    linear of order 2
ρ511111-1111-1-11-11111-11-1-1-111-11-1-11-1    linear of order 2
ρ611111-1111-1-1-111111-11-111-1-1-11-1-11-1    linear of order 2
ρ71111-11111-1-11-1111-11-1111-1-1-11-1-11-1    linear of order 2
ρ81111-11111-1-1-11111-11-11-1-111-11-1-11-1    linear of order 2
ρ92222-2-2-1222200-1-1-111110000-1-1-1-1-1-1    orthogonal lifted from D6
ρ1022222-2-122-2-200-1-1-1-11-1100001-111-11    orthogonal lifted from D6
ρ112222002-2-2-2200222000000002-2-2-2-22    orthogonal lifted from D4
ρ122222002-2-22-20022200000000-2-222-2-2    orthogonal lifted from D4
ρ132222-22-122-2-200-1-1-11-11-100001-111-11    orthogonal lifted from D6
ρ14222222-1222200-1-1-1-1-1-1-10000-1-1-1-1-1-1    orthogonal lifted from S3
ρ15222200-1-2-22-200-1-1-1-√-3-√-3√-3√-3000011-1-111    complex lifted from C3:D4
ρ16222200-1-2-2-2200-1-1-1-√-3√-3√-3-√-30000-11111-1    complex lifted from C3:D4
ρ17222200-1-2-2-2200-1-1-1√-3-√-3-√-3√-30000-11111-1    complex lifted from C3:D4
ρ18222200-1-2-22-200-1-1-1√-3√-3-√-3-√-3000011-1-111    complex lifted from C3:D4
ρ192-2-22002-220000-22-20000002i-2i0-20020    complex lifted from C4oD4
ρ202-2-220022-20000-22-20000-2i2i000200-20    complex lifted from C4oD4
ρ212-2-22002-220000-22-2000000-2i2i0-20020    complex lifted from C4oD4
ρ222-2-220022-20000-22-200002i-2i000200-20    complex lifted from C4oD4
ρ234-44-40040000004-4-400000000000000    orthogonal lifted from C8:C22
ρ2444-4-4004000000-4-4400000000000000    orthogonal lifted from C8:C22
ρ254-4-4400-24-400002-22000000000-20020    symplectic lifted from D4:2S3, Schur index 2
ρ264-4-4400-2-4400002-22000000000200-20    symplectic lifted from D4:2S3, Schur index 2
ρ2744-4-400-200000022-20000000000-2√-32√-300    complex lifted from D12:6C22
ρ2844-4-400-200000022-200000000002√-3-2√-300    complex lifted from D12:6C22
ρ294-44-400-2000000-222000000002√-30000-2√-3    complex lifted from D12:6C22
ρ304-44-400-2000000-22200000000-2√-300002√-3    complex lifted from D12:6C22

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

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

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

Matrix representation of C42.72D6 ►in GL6(F73)

0720000
100000
00003013
00006043
00436000
00133000
,
100000
010000
000010
000001
0072000
0007200
,
010000
100000
00006043
00003030
00604300
00303000
,
4600000
0270000
008306543
003865358
0065436543
00358358

G:=sub<GL(6,GF(73))| [0,1,0,0,0,0,72,0,0,0,0,0,0,0,0,0,43,13,0,0,0,0,60,30,0,0,30,60,0,0,0,0,13,43,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,1,0,0,0,0,0,0,1,0,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,60,30,0,0,0,0,43,30,0,0,60,30,0,0,0,0,43,30,0,0],[46,0,0,0,0,0,0,27,0,0,0,0,0,0,8,38,65,35,0,0,30,65,43,8,0,0,65,35,65,35,0,0,43,8,43,8] >;
 

C42.72D6 in GAP, Magma, Sage, TeX 

C_4^2._{72}D_6
 
% in TeX
 
G:=Group("C4^2.72D6");
 
// GroupNames label
 
G:=SmallGroup(192,630);
 
// by ID
 
G=gap.SmallGroup(192,630);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,477,64,590,135,438,102,6278]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^4=c^6=1,d^2=a^2,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a^-1*b^2,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=a^2*b*c^-1>;
 
// generators/relations
 

Export

Character table of C42.72D6 in TeX

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