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G = C23.39D12order 192 = 26·3

5th non-split extension by C23 of D12 acting via D12/D6=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C23.39D12, C8⋊Dic36C2, (C2×C12).40D4, (C2×C8).107D6, (C2×C4).29D12, C22⋊C8.6S3, C6.6(C2×SD16), C2.Dic129C2, (C2×C6).13SD16, (C22×C6).48D4, (C22×C4).91D6, C6.7(C8.C22), C12.279(C4○D4), (C2×C24).118C22, (C2×C12).738C23, C22.8(C24⋊C2), C12.48D4.2C2, C2.10(C8.D6), C22.101(C2×D12), C31(C23.47D4), C4.103(D42S3), C4⋊Dic3.268C22, (C22×C12).48C22, (C2×Dic6).10C22, C6.14(C22.D4), C2.10(C23.21D6), C2.9(C2×C24⋊C2), (C2×C6).121(C2×D4), (C3×C22⋊C8).8C2, (C2×C4⋊Dic3).11C2, (C2×C4).683(C22×S3), SmallGroup(192,280)

Series: Derived Chief Lower central Upper central

C1C2×C12 — C23.39D12
C1C3C6C12C2×C12C4⋊Dic3C2×C4⋊Dic3 — C23.39D12
C3C6C2×C12 — C23.39D12
C1C22C22×C4C22⋊C8

Generators and relations for C23.39D12
 G = < a,b,c,d,e | a2=b2=c2=1, d12=c, e2=cb=bc, dad-1=eae-1=ab=ba, ac=ca, bd=db, be=eb, cd=dc, ce=ec, ede-1=d11 >

Subgroups: 288 in 104 conjugacy classes, 43 normal (25 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C4 [×5], C22, C22 [×2], C22 [×2], C6 [×3], C6 [×2], C8 [×2], C2×C4 [×2], C2×C4 [×8], Q8 [×2], C23, Dic3 [×4], C12 [×2], C12, C2×C6, C2×C6 [×2], C2×C6 [×2], C22⋊C4, C4⋊C4 [×5], C2×C8 [×2], C22×C4, C22×C4, C2×Q8, C24 [×2], Dic6 [×2], C2×Dic3 [×6], C2×C12 [×2], C2×C12 [×2], C22×C6, C22⋊C8, Q8⋊C4 [×2], C4.Q8 [×2], C2×C4⋊C4, C22⋊Q8, Dic3⋊C4, C4⋊Dic3, C4⋊Dic3 [×2], C4⋊Dic3, C6.D4, C2×C24 [×2], C2×Dic6, C22×Dic3, C22×C12, C23.47D4, C2.Dic12 [×2], C8⋊Dic3 [×2], C3×C22⋊C8, C12.48D4, C2×C4⋊Dic3, C23.39D12
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], SD16 [×2], C2×D4, C4○D4 [×2], D12 [×2], C22×S3, C22.D4, C2×SD16, C8.C22, C24⋊C2 [×2], C2×D12, D42S3 [×2], C23.47D4, C23.21D6, C2×C24⋊C2, C8.D6, C23.39D12

Smallest permutation representation of C23.39D12
On 96 points
Generators in S96
(2 64)(4 66)(6 68)(8 70)(10 72)(12 50)(14 52)(16 54)(18 56)(20 58)(22 60)(24 62)(26 80)(28 82)(30 84)(32 86)(34 88)(36 90)(38 92)(40 94)(42 96)(44 74)(46 76)(48 78)
(1 63)(2 64)(3 65)(4 66)(5 67)(6 68)(7 69)(8 70)(9 71)(10 72)(11 49)(12 50)(13 51)(14 52)(15 53)(16 54)(17 55)(18 56)(19 57)(20 58)(21 59)(22 60)(23 61)(24 62)(25 79)(26 80)(27 81)(28 82)(29 83)(30 84)(31 85)(32 86)(33 87)(34 88)(35 89)(36 90)(37 91)(38 92)(39 93)(40 94)(41 95)(42 96)(43 73)(44 74)(45 75)(46 76)(47 77)(48 78)
(1 13)(2 14)(3 15)(4 16)(5 17)(6 18)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)(25 37)(26 38)(27 39)(28 40)(29 41)(30 42)(31 43)(32 44)(33 45)(34 46)(35 47)(36 48)(49 61)(50 62)(51 63)(52 64)(53 65)(54 66)(55 67)(56 68)(57 69)(58 70)(59 71)(60 72)(73 85)(74 86)(75 87)(76 88)(77 89)(78 90)(79 91)(80 92)(81 93)(82 94)(83 95)(84 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 42 51 84)(2 29 52 95)(3 40 53 82)(4 27 54 93)(5 38 55 80)(6 25 56 91)(7 36 57 78)(8 47 58 89)(9 34 59 76)(10 45 60 87)(11 32 61 74)(12 43 62 85)(13 30 63 96)(14 41 64 83)(15 28 65 94)(16 39 66 81)(17 26 67 92)(18 37 68 79)(19 48 69 90)(20 35 70 77)(21 46 71 88)(22 33 72 75)(23 44 49 86)(24 31 50 73)

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

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

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

39 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F4G4H4I6A6B6C6D6E8A8B8C8D12A12B12C12D12E12F24A···24H
order122222344444444466666888812121212121224···24
size11112222241212121224242224444442222444···4

39 irreducible representations

dim1111112222222222444
type+++++++++++++---
imageC1C2C2C2C2C2S3D4D4D6D6C4○D4SD16D12D12C24⋊C2C8.C22D42S3C8.D6
kernelC23.39D12C2.Dic12C8⋊Dic3C3×C22⋊C8C12.48D4C2×C4⋊Dic3C22⋊C8C2×C12C22×C6C2×C8C22×C4C12C2×C6C2×C4C23C22C6C4C2
# reps1221111112144228122

Matrix representation of C23.39D12 in GL6(𝔽73)

100000
0720000
001000
000100
000010
000001
,
7200000
0720000
001000
000100
000010
000001
,
100000
010000
0072000
0007200
000010
000001
,
010000
100000
00121200
0067000
000001
0000721
,
0270000
2700000
00565700
0091700
00006011
00007113

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,12,67,0,0,0,0,12,0,0,0,0,0,0,0,0,72,0,0,0,0,1,1],[0,27,0,0,0,0,27,0,0,0,0,0,0,0,56,9,0,0,0,0,57,17,0,0,0,0,0,0,60,71,0,0,0,0,11,13] >;

C23.39D12 in GAP, Magma, Sage, TeX

C_2^3._{39}D_{12}
% in TeX

G:=Group("C2^3.39D12");
// GroupNames label

G:=SmallGroup(192,280);
// by ID

G=gap.SmallGroup(192,280);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,254,219,58,1123,136,6278]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^12=c,e^2=c*b=b*c,d*a*d^-1=e*a*e^-1=a*b=b*a,a*c=c*a,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^11>;
// generators/relations

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