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G = C12.D8order 192 = 26·3

19th non-split extension by C12 of D8 acting via D8/C4=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C12.19D8, C12.19SD16, C42.224D6, C4⋊Q86S3, C4⋊C4.84D6, C6.61(C2×D8), C4.7(D4⋊S3), C34(C4.4D8), C6.D842C2, (C2×C12).158D4, C4⋊D12.9C2, C6.78(C2×SD16), C12.83(C4○D4), C4.5(Q82S3), (C2×C12).406C23, (C4×C12).135C22, C4.16(Q83S3), C6.58(C4.4D4), (C2×D12).109C22, C2.11(C12.23D4), (C4×C3⋊C8)⋊18C2, (C3×C4⋊Q8)⋊6C2, C2.16(C2×D4⋊S3), (C2×C6).537(C2×D4), (C2×C3⋊C8).263C22, C2.16(C2×Q82S3), (C2×C4).137(C3⋊D4), (C3×C4⋊C4).131C22, (C2×C4).503(C22×S3), C22.209(C2×C3⋊D4), SmallGroup(192,647)

Series: Derived Chief Lower central Upper central

C1C2×C12 — C12.D8
C1C3C6C12C2×C12C2×D12C4⋊D12 — C12.D8
C3C6C2×C12 — C12.D8
C1C22C42C4⋊Q8

Generators and relations for C12.D8
 G = < a,b,c | a12=b8=c2=1, bab-1=a5, cac=a-1, cbc=a6b-1 >

Subgroups: 432 in 118 conjugacy classes, 47 normal (23 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×6], C4 [×2], C22, C22 [×6], S3 [×2], C6 [×3], C8 [×2], C2×C4 [×3], C2×C4 [×2], D4 [×8], Q8 [×2], C23 [×2], C12 [×6], C12 [×2], D6 [×6], C2×C6, C42, C4⋊C4 [×2], C4⋊C4, C2×C8 [×2], C2×D4 [×4], C2×Q8, C3⋊C8 [×2], D12 [×8], C2×C12 [×3], C2×C12 [×2], C3×Q8 [×2], C22×S3 [×2], C4×C8, D4⋊C4 [×4], C41D4, C4⋊Q8, C2×C3⋊C8 [×2], C4×C12, C3×C4⋊C4 [×2], C3×C4⋊C4, C2×D12 [×2], C2×D12 [×2], C6×Q8, C4.4D8, C4×C3⋊C8, C6.D8 [×4], C4⋊D12, C3×C4⋊Q8, C12.D8
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], D8 [×2], SD16 [×2], C2×D4, C4○D4 [×2], C3⋊D4 [×2], C22×S3, C4.4D4, C2×D8, C2×SD16, D4⋊S3 [×2], Q82S3 [×2], Q83S3 [×2], C2×C3⋊D4, C4.4D8, C2×D4⋊S3, C2×Q82S3, C12.23D4, C12.D8

Smallest permutation representation of C12.D8
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 31 64 94 38 73 54 23)(2 36 65 87 39 78 55 16)(3 29 66 92 40 83 56 21)(4 34 67 85 41 76 57 14)(5 27 68 90 42 81 58 19)(6 32 69 95 43 74 59 24)(7 25 70 88 44 79 60 17)(8 30 71 93 45 84 49 22)(9 35 72 86 46 77 50 15)(10 28 61 91 47 82 51 20)(11 33 62 96 48 75 52 13)(12 26 63 89 37 80 53 18)
(2 12)(3 11)(4 10)(5 9)(6 8)(13 35)(14 34)(15 33)(16 32)(17 31)(18 30)(19 29)(20 28)(21 27)(22 26)(23 25)(24 36)(37 39)(40 48)(41 47)(42 46)(43 45)(49 69)(50 68)(51 67)(52 66)(53 65)(54 64)(55 63)(56 62)(57 61)(58 72)(59 71)(60 70)(73 88)(74 87)(75 86)(76 85)(77 96)(78 95)(79 94)(80 93)(81 92)(82 91)(83 90)(84 89)

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

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

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

36 conjugacy classes

class 1 2A2B2C2D2E 3 4A···4F4G4H6A6B6C8A···8H12A···12F12G12H12I12J
order12222234···4446668···812···1212121212
size1111242422···2882226···64···48888

36 irreducible representations

dim1111122222222444
type+++++++++++++
imageC1C2C2C2C2S3D4D6D6D8SD16C4○D4C3⋊D4D4⋊S3Q82S3Q83S3
kernelC12.D8C4×C3⋊C8C6.D8C4⋊D12C3×C4⋊Q8C4⋊Q8C2×C12C42C4⋊C4C12C12C12C2×C4C4C4C4
# reps1141112124444222

Matrix representation of C12.D8 in GL6(𝔽73)

0270000
2700000
000100
00727200
0000720
0000072
,
0720000
7200000
0072000
001100
00001657
00001616
,
100000
0720000
001000
00727200
000010
0000072

G:=sub<GL(6,GF(73))| [0,27,0,0,0,0,27,0,0,0,0,0,0,0,0,72,0,0,0,0,1,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[0,72,0,0,0,0,72,0,0,0,0,0,0,0,72,1,0,0,0,0,0,1,0,0,0,0,0,0,16,16,0,0,0,0,57,16],[1,0,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,72] >;

C12.D8 in GAP, Magma, Sage, TeX

C_{12}.D_8
% in TeX

G:=Group("C12.D8");
// GroupNames label

G:=SmallGroup(192,647);
// by ID

G=gap.SmallGroup(192,647);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,120,254,219,100,1123,297,136,6278]);
// Polycyclic

G:=Group<a,b,c|a^12=b^8=c^2=1,b*a*b^-1=a^5,c*a*c=a^-1,c*b*c=a^6*b^-1>;
// generators/relations

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