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G = D12.37D4order 192 = 26·3

7th non-split extension by D12 of D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D12.37D4, Dic6.36D4, C4:C4.66D6, C22:Q8:2S3, C4.102(S3xD4), (C2xQ8).52D6, C6.48C22wrC2, C6.D8:38C2, C12.153(C2xD4), (C2xC12).266D4, C3:4(D4.7D4), (C22xC6).93D4, C6.100(C4oD8), C6.SD16:37C2, (C22xC4).144D6, (C6xQ8).46C22, C12.55D4:14C2, C2.16(C23:2D6), (C2xC12).366C23, C6.90(C8.C22), C23.34(C3:D4), C2.19(Q8.13D6), (C2xD12).243C22, C2.11(Q8.11D6), (C22xC12).170C22, (C2xDic6).270C22, (C2xC3:Q16):8C2, (C3xC22:Q8):2C2, (C2xQ8:2S3):9C2, (C2xC6).497(C2xD4), (C2xC4oD12).10C2, (C2xC3:C8).115C22, (C2xC4).173(C3:D4), (C3xC4:C4).113C22, (C2xC4).466(C22xS3), C22.172(C2xC3:D4), SmallGroup(192,606)

Series: Derived Chief Lower central Upper central

C1C2xC12 — D12.37D4
C1C3C6C12C2xC12C2xD12C2xC4oD12 — D12.37D4
C3C6C2xC12 — D12.37D4
C1C22C22xC4C22:Q8

Generators and relations for D12.37D4
 G = < a,b,c,d | a12=b2=c4=1, d2=a6, bab=a-1, cac-1=a7, ad=da, cbc-1=a3b, bd=db, dcd-1=c-1 >

Subgroups: 448 in 152 conjugacy classes, 43 normal (39 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2xC4, C2xC4, D4, Q8, C23, C23, Dic3, C12, C12, D6, C2xC6, C2xC6, C22:C4, C4:C4, C4:C4, C2xC8, SD16, Q16, C22xC4, C22xC4, C2xD4, C2xQ8, C2xQ8, C4oD4, C3:C8, Dic6, Dic6, C4xS3, D12, D12, C2xDic3, C3:D4, C2xC12, C2xC12, C3xQ8, C22xS3, C22xC6, C22:C8, D4:C4, Q8:C4, C22:Q8, C2xSD16, C2xQ16, C2xC4oD4, C2xC3:C8, Q8:2S3, C3:Q16, C3xC22:C4, C3xC4:C4, C3xC4:C4, C2xDic6, S3xC2xC4, C2xD12, C4oD12, C2xC3:D4, C22xC12, C6xQ8, D4.7D4, C6.D8, C6.SD16, C12.55D4, C2xQ8:2S3, C2xC3:Q16, C3xC22:Q8, C2xC4oD12, D12.37D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, C3:D4, C22xS3, C22wrC2, C4oD8, C8.C22, S3xD4, C2xC3:D4, D4.7D4, C23:2D6, Q8.11D6, Q8.13D6, D12.37D4

Smallest permutation representation of D12.37D4
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 12)(2 11)(3 10)(4 9)(5 8)(6 7)(13 20)(14 19)(15 18)(16 17)(21 24)(22 23)(25 26)(27 36)(28 35)(29 34)(30 33)(31 32)(37 44)(38 43)(39 42)(40 41)(45 48)(46 47)(49 55)(50 54)(51 53)(56 60)(57 59)(61 65)(62 64)(66 72)(67 71)(68 70)(73 77)(74 76)(78 84)(79 83)(80 82)(85 93)(86 92)(87 91)(88 90)(94 96)
(1 85 41 71)(2 92 42 66)(3 87 43 61)(4 94 44 68)(5 89 45 63)(6 96 46 70)(7 91 47 65)(8 86 48 72)(9 93 37 67)(10 88 38 62)(11 95 39 69)(12 90 40 64)(13 79 34 56)(14 74 35 51)(15 81 36 58)(16 76 25 53)(17 83 26 60)(18 78 27 55)(19 73 28 50)(20 80 29 57)(21 75 30 52)(22 82 31 59)(23 77 32 54)(24 84 33 49)
(1 32 7 26)(2 33 8 27)(3 34 9 28)(4 35 10 29)(5 36 11 30)(6 25 12 31)(13 37 19 43)(14 38 20 44)(15 39 21 45)(16 40 22 46)(17 41 23 47)(18 42 24 48)(49 72 55 66)(50 61 56 67)(51 62 57 68)(52 63 58 69)(53 64 59 70)(54 65 60 71)(73 87 79 93)(74 88 80 94)(75 89 81 95)(76 90 82 96)(77 91 83 85)(78 92 84 86)

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

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

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

33 conjugacy classes

class 1 2A2B2C2D2E2F 3 4A4B4C4D4E4F4G4H6A6B6C6D6E8A8B8C8D12A12B12C12D12E12F12G12H
order12222223444444446666688881212121212121212
size11114121222222881212222441212121244448888

33 irreducible representations

dim11111111222222222224444
type++++++++++++++++-+
imageC1C2C2C2C2C2C2C2S3D4D4D4D4D6D6D6C3:D4C3:D4C4oD8C8.C22S3xD4Q8.11D6Q8.13D6
kernelD12.37D4C6.D8C6.SD16C12.55D4C2xQ8:2S3C2xC3:Q16C3xC22:Q8C2xC4oD12C22:Q8Dic6D12C2xC12C22xC6C4:C4C22xC4C2xQ8C2xC4C23C6C6C4C2C2
# reps11111111122111112241222

Matrix representation of D12.37D4 in GL6(F73)

110000
7200000
0017100
0017200
0000720
0000072
,
110000
0720000
0017100
0007200
0000720
0000561
,
43130000
60300000
00413200
00573200
00001771
00007256
,
7200000
0720000
0046000
0004600
0000720
0000561

G:=sub<GL(6,GF(73))| [1,72,0,0,0,0,1,0,0,0,0,0,0,0,1,1,0,0,0,0,71,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[1,0,0,0,0,0,1,72,0,0,0,0,0,0,1,0,0,0,0,0,71,72,0,0,0,0,0,0,72,56,0,0,0,0,0,1],[43,60,0,0,0,0,13,30,0,0,0,0,0,0,41,57,0,0,0,0,32,32,0,0,0,0,0,0,17,72,0,0,0,0,71,56],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,46,0,0,0,0,0,0,46,0,0,0,0,0,0,72,56,0,0,0,0,0,1] >;

D12.37D4 in GAP, Magma, Sage, TeX

D_{12}._{37}D_4
% in TeX

G:=Group("D12.37D4");
// GroupNames label

G:=SmallGroup(192,606);
// by ID

G=gap.SmallGroup(192,606);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,254,219,184,1123,297,136,6278]);
// Polycyclic

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

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