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G = (C2×Dic3)⋊C8order 192 = 26·3

The semidirect product of C2×Dic3 and C8 acting via C8/C2=C4

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C2×Dic3)⋊C8, C2.6(D6⋊C8), C22⋊C8.2S3, C22.4(S3×C8), (C2×C4).108D12, (C2×C12).439D4, C6.4(C22⋊C8), C6.5(C23⋊C4), C23.46(C4×S3), (C22×C4).18D6, (C2×C6).2M4(2), C22.4(C8⋊S3), C22.33(D6⋊C4), C6.2(C4.10D4), (C22×Dic3).2C4, C12.55D4.11C2, C2.1(C12.47D4), C2.2(C23.6D6), (C22×C12).323C22, C31(C22.M4(2)), (C2×C6).2(C2×C8), (C3×C22⋊C8).2C2, (C22×C6).27(C2×C4), (C2×C4).210(C3⋊D4), (C2×C6).41(C22⋊C4), (C2×Dic3⋊C4).24C2, SmallGroup(192,28)

Series: Derived Chief Lower central Upper central

C1C2×C6 — (C2×Dic3)⋊C8
C1C3C6C2×C6C2×C12C22×C12C2×Dic3⋊C4 — (C2×Dic3)⋊C8
C3C6C2×C6 — (C2×Dic3)⋊C8
C1C22C22×C4C22⋊C8

Generators and relations for (C2×Dic3)⋊C8
 G = < a,b,c,d | a2=b6=d8=1, c2=b3, ab=ba, ac=ca, dad-1=ab3, cbc-1=b-1, bd=db, dcd-1=ab3c >

Subgroups: 208 in 78 conjugacy classes, 31 normal (29 characteristic)
C1, C2, C2, C3, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, C23, Dic3, C12, C2×C6, C2×C6, C4⋊C4, C2×C8, C22×C4, C22×C4, C3⋊C8, C24, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×C6, C22⋊C8, C22⋊C8, C2×C4⋊C4, C2×C3⋊C8, Dic3⋊C4, C2×C24, C22×Dic3, C22×C12, C22.M4(2), C12.55D4, C3×C22⋊C8, C2×Dic3⋊C4, (C2×Dic3)⋊C8
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, D4, D6, C22⋊C4, C2×C8, M4(2), C4×S3, D12, C3⋊D4, C22⋊C8, C23⋊C4, C4.10D4, S3×C8, C8⋊S3, D6⋊C4, C22.M4(2), C23.6D6, D6⋊C8, C12.47D4, (C2×Dic3)⋊C8

Smallest permutation representation of (C2×Dic3)⋊C8
On 96 points
Generators in S96
(2 55)(4 49)(6 51)(8 53)(9 43)(11 45)(13 47)(15 41)(17 92)(19 94)(21 96)(23 90)(25 35)(27 37)(29 39)(31 33)(57 81)(59 83)(61 85)(63 87)(66 73)(68 75)(70 77)(72 79)
(1 62 78 54 86 71)(2 63 79 55 87 72)(3 64 80 56 88 65)(4 57 73 49 81 66)(5 58 74 50 82 67)(6 59 75 51 83 68)(7 60 76 52 84 69)(8 61 77 53 85 70)(9 17 39 43 92 29)(10 18 40 44 93 30)(11 19 33 45 94 31)(12 20 34 46 95 32)(13 21 35 47 96 25)(14 22 36 48 89 26)(15 23 37 41 90 27)(16 24 38 42 91 28)
(1 38 54 28)(2 29 55 39)(3 30 56 40)(4 33 49 31)(5 34 50 32)(6 25 51 35)(7 26 52 36)(8 37 53 27)(9 79 43 72)(10 80 44 65)(11 66 45 73)(12 67 46 74)(13 75 47 68)(14 76 48 69)(15 70 41 77)(16 71 42 78)(17 63 92 87)(18 64 93 88)(19 81 94 57)(20 82 95 58)(21 59 96 83)(22 60 89 84)(23 85 90 61)(24 86 91 62)
(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)

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

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

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

42 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F4G4H6A6B6C6D6E8A8B8C8D8E8F8G8H12A12B12C12D12E12F24A···24H
order122222344444444666668888888812121212121224···24
size1111222222212121212222444444121212122222444···4

42 irreducible representations

dim1111112222222224444
type+++++++++--
imageC1C2C2C2C4C8S3D4D6M4(2)D12C3⋊D4C4×S3S3×C8C8⋊S3C23⋊C4C4.10D4C23.6D6C12.47D4
kernel(C2×Dic3)⋊C8C12.55D4C3×C22⋊C8C2×Dic3⋊C4C22×Dic3C2×Dic3C22⋊C8C2×C12C22×C4C2×C6C2×C4C2×C4C23C22C22C6C6C2C2
# reps1111481212222441122

Matrix representation of (C2×Dic3)⋊C8 in GL6(𝔽73)

7200000
0720000
001000
000100
0000720
0000072
,
100000
010000
000100
0072100
000001
0000721
,
7200000
010000
00195900
0055400
00001959
0000554
,
0220000
5100000
000010
000001
0075900
00146600

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,1,1,0,0,0,0,0,0,0,72,0,0,0,0,1,1],[72,0,0,0,0,0,0,1,0,0,0,0,0,0,19,5,0,0,0,0,59,54,0,0,0,0,0,0,19,5,0,0,0,0,59,54],[0,51,0,0,0,0,22,0,0,0,0,0,0,0,0,0,7,14,0,0,0,0,59,66,0,0,1,0,0,0,0,0,0,1,0,0] >;

(C2×Dic3)⋊C8 in GAP, Magma, Sage, TeX

(C_2\times {\rm Dic}_3)\rtimes C_8
% in TeX

G:=Group("(C2xDic3):C8");
// GroupNames label

G:=SmallGroup(192,28);
// by ID

G=gap.SmallGroup(192,28);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,141,36,758,100,570,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^6=d^8=1,c^2=b^3,a*b=b*a,a*c=c*a,d*a*d^-1=a*b^3,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=a*b^3*c>;
// generators/relations

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