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G = C42.100D6order 192 = 26·3

100th non-split extension by C42 of D6 acting via D6/C3=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.100D6, C6.1002+ 1+4, (C4xD12):12C2, Dic3:D4:5C2, C4:D12:4C2, C4:C4.275D6, C12:7D4:42C2, D6.D4:5C2, (C2xC6).79C24, C12.6Q8:5C2, C42:C2:19S3, C2.12(D4oD12), (C4xC12).30C22, D6:C4.64C22, C22:C4.103D6, (C22xC4).216D6, C12.237(C4oD4), C4.121(C4oD12), (C2xC12).152C23, (C2xD12).25C22, Dic3:C4.4C22, (C22xS3).27C23, C4:Dic3.294C22, C22.108(S3xC23), (C22xC6).149C23, C23.100(C22xS3), (C2xDic3).32C23, (C22xC12).309C22, C3:1(C22.34C24), C6.35(C2xC4oD4), C2.38(C2xC4oD12), (S3xC2xC4).196C22, (C3xC42:C2):21C2, (C3xC4:C4).315C22, (C2xC4).280(C22xS3), (C2xC3:D4).12C22, (C3xC22:C4).118C22, SmallGroup(192,1094)

Series: Derived Chief Lower central Upper central

C1C2xC6 — C42.100D6
C1C3C6C2xC6C22xS3C2xD12C4xD12 — C42.100D6
C3C2xC6 — C42.100D6
C1C22C42:C2

Generators and relations for C42.100D6
 G = < a,b,c,d | a4=b4=1, c6=a2, d2=a2b2, ab=ba, ac=ca, dad-1=a-1, cbc-1=dbd-1=a2b, dcd-1=b2c5 >

Subgroups: 712 in 240 conjugacy classes, 95 normal (23 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C6, C2xC4, C2xC4, C2xC4, D4, C23, C23, Dic3, C12, C12, D6, C2xC6, C2xC6, C42, C22:C4, C22:C4, C4:C4, C4:C4, C22xC4, C22xC4, C2xD4, C4xS3, D12, C2xDic3, C3:D4, C2xC12, C2xC12, C2xC12, C22xS3, C22xC6, C42:C2, C4xD4, C4:D4, C22.D4, C42.C2, C4:1D4, Dic3:C4, C4:Dic3, D6:C4, C4xC12, C3xC22:C4, C3xC4:C4, S3xC2xC4, C2xD12, C2xC3:D4, C22xC12, C22.34C24, C12.6Q8, C4xD12, C4:D12, Dic3:D4, D6.D4, C12:7D4, C3xC42:C2, C42.100D6
Quotients: C1, C2, C22, S3, C23, D6, C4oD4, C24, C22xS3, C2xC4oD4, 2+ 1+4, C4oD12, S3xC23, C22.34C24, C2xC4oD12, D4oD12, C42.100D6

Smallest permutation representation of C42.100D6
On 96 points
Generators in S96
(1 58 7 52)(2 59 8 53)(3 60 9 54)(4 49 10 55)(5 50 11 56)(6 51 12 57)(13 63 19 69)(14 64 20 70)(15 65 21 71)(16 66 22 72)(17 67 23 61)(18 68 24 62)(25 47 31 41)(26 48 32 42)(27 37 33 43)(28 38 34 44)(29 39 35 45)(30 40 36 46)(73 85 79 91)(74 86 80 92)(75 87 81 93)(76 88 82 94)(77 89 83 95)(78 90 84 96)
(1 27 94 18)(2 34 95 13)(3 29 96 20)(4 36 85 15)(5 31 86 22)(6 26 87 17)(7 33 88 24)(8 28 89 19)(9 35 90 14)(10 30 91 21)(11 25 92 16)(12 32 93 23)(37 76 68 58)(38 83 69 53)(39 78 70 60)(40 73 71 55)(41 80 72 50)(42 75 61 57)(43 82 62 52)(44 77 63 59)(45 84 64 54)(46 79 65 49)(47 74 66 56)(48 81 67 51)
(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 88 81)(2 80 89 56)(3 55 90 79)(4 78 91 54)(5 53 92 77)(6 76 93 52)(7 51 94 75)(8 74 95 50)(9 49 96 73)(10 84 85 60)(11 59 86 83)(12 82 87 58)(13 47 28 72)(14 71 29 46)(15 45 30 70)(16 69 31 44)(17 43 32 68)(18 67 33 42)(19 41 34 66)(20 65 35 40)(21 39 36 64)(22 63 25 38)(23 37 26 62)(24 61 27 48)

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

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

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

42 conjugacy classes

class 1 2A2B2C2D2E2F2G2H 3 4A···4F4G4H4I4J4K4L4M6A6B6C6D6E12A12B12C12D12E···12N
order12222222234···44444444666661212121212···12
size111141212121222···2444121212122224422224···4

42 irreducible representations

dim11111111222222244
type+++++++++++++++
imageC1C2C2C2C2C2C2C2S3D6D6D6D6C4oD4C4oD122+ 1+4D4oD12
kernelC42.100D6C12.6Q8C4xD12C4:D12Dic3:D4D6.D4C12:7D4C3xC42:C2C42:C2C42C22:C4C4:C4C22xC4C12C4C6C2
# reps11214421122214824

Matrix representation of C42.100D6 in GL8(F13)

10000000
01000000
001200000
000120000
00000207
000011070
00000602
000060110
,
80000000
08000000
001200000
000120000
00000010
00000001
000012000
000001200
,
1211000000
01000000
00010000
001210000
00000100
000012000
000000012
00000010
,
12000000
1212000000
001120000
000120000
00002060
000001106
000060110
00000602

G:=sub<GL(8,GF(13))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,11,0,6,0,0,0,0,2,0,6,0,0,0,0,0,0,7,0,11,0,0,0,0,7,0,2,0],[8,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[12,0,0,0,0,0,0,0,11,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0],[1,12,0,0,0,0,0,0,2,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,2,0,6,0,0,0,0,0,0,11,0,6,0,0,0,0,6,0,11,0,0,0,0,0,0,6,0,2] >;

C42.100D6 in GAP, Magma, Sage, TeX

C_4^2._{100}D_6
% in TeX

G:=Group("C4^2.100D6");
// GroupNames label

G:=SmallGroup(192,1094);
// by ID

G=gap.SmallGroup(192,1094);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,232,100,675,297,136,6278]);
// Polycyclic

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

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