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G = C2×Dic3.D4  order 192 = 26·3

Direct product of C2 and Dic3.D4

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2×Dic3.D4, C23⋊4Dic6, C24.62D6, (C22×C6)⋊4Q8, C6⋊1(C22⋊Q8), C6.5(C22×Q8), (C2×C6).25C24, C22⋊C4.84D6, C22⋊3(C2×Dic6), C6.32(C22×D4), C4⋊Dic3⋊49C22, Dic3.41(C2×D4), (C22×Dic6)⋊5C2, (C22×C4).183D6, C22.122(S3×D4), C2.7(C22×Dic6), (C2×C12).125C23, Dic3⋊C4⋊46C22, (C2×Dic3).189D4, (C2×Dic6)⋊47C22, (C23×C6).51C22, C22.67(S3×C23), (C23×Dic3).8C2, (C22×C6).117C23, C23.328(C22×S3), (C22×C12).70C22, C22.64(D4⋊2S3), (C2×Dic3).175C23, C6.D4.83C22, (C22×Dic3).202C22, C2.7(C2×S3×D4), (C2×C6)⋊4(C2×Q8), C3⋊1(C2×C22⋊Q8), C6.66(C2×C4○D4), (C2×C4⋊Dic3)⋊17C2, (C2×C6).378(C2×D4), C2.7(C2×D4⋊2S3), (C2×Dic3⋊C4)⋊21C2, (C6×C22⋊C4).17C2, (C2×C22⋊C4).16S3, (C2×C6).166(C4○D4), (C2×C4).132(C22×S3), (C2×C6.D4).20C2, (C3×C22⋊C4).96C22, SmallGroup(192,1040)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C6 — C2×Dic3.D4
Chief series
C1 — C3 — C6 — C2×C6 — C2×Dic3 — C22×Dic3 — C23×Dic3 — C2×Dic3.D4
Lower central
C3 — C2×C6 — C2×Dic3.D4

Subgroups: 712 in 322 conjugacy classes, 135 normal (31 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C3, C4 [×14], C22, C22 [×10], C22 [×12], C6 [×3], C6 [×4], C6 [×4], C2×C4 [×4], C2×C4 [×30], Q8 [×8], C23, C23 [×6], C23 [×4], Dic3 [×4], Dic3 [×6], C12 [×4], C2×C6, C2×C6 [×10], C2×C6 [×12], C22⋊C4 [×4], C22⋊C4 [×4], C4⋊C4 [×12], C22×C4 [×2], C22×C4 [×12], C2×Q8 [×8], C24, Dic6 [×8], C2×Dic3 [×12], C2×Dic3 [×14], C2×C12 [×4], C2×C12 [×4], C22×C6, C22×C6 [×6], C22×C6 [×4], C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4 [×3], C22⋊Q8 [×8], C23×C4, C22×Q8, Dic3⋊C4 [×8], C4⋊Dic3 [×4], C6.D4 [×4], C3×C22⋊C4 [×4], C2×Dic6 [×4], C2×Dic6 [×4], C22×Dic3 [×4], C22×Dic3 [×4], C22×Dic3 [×4], C22×C12 [×2], C23×C6, C2×C22⋊Q8, Dic3.D4 [×8], C2×Dic3⋊C4 [×2], C2×C4⋊Dic3, C2×C6.D4, C6×C22⋊C4, C22×Dic6, C23×Dic3, C2×Dic3.D4

Quotients:
C1, C2 [×15], C22 [×35], S3, D4 [×4], Q8 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C2×Q8 [×6], C4○D4 [×2], C24, Dic6 [×4], C22×S3 [×7], C22⋊Q8 [×4], C22×D4, C22×Q8, C2×C4○D4, C2×Dic6 [×6], S3×D4 [×2], D4⋊2S3 [×2], S3×C23, C2×C22⋊Q8, Dic3.D4 [×4], C22×Dic6, C2×S3×D4, C2×D4⋊2S3, C2×Dic3.D4

Generators and relations
 G = < a,b,c,d,e | a2=b6=d4=e2=1, c2=b3, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, dcd-1=b3c, ce=ec, ede=b3d-1 >

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

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

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

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

Matrix representation ►G ⊆ GL7(𝔽13)

12000000
0100000
0010000
0001000
0000100
0000010
0000001
,
1000000
00120000
01120000
00012000
00001200
0000010
0000001
,
1000000
0730000
01060000
0005000
0000800
0000010
0000001
,
12000000
01200000
00120000
00001200
00012000
00000012
0000010
,
12000000
0100000
0010000
0001000
00001200
0000010
00000012

G:=sub<GL(7,GF(13))| [12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,12,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,7,10,0,0,0,0,0,3,6,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,12,0],[12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12] >;
 

48 conjugacy classes

class 1 2A···2G2H2I2J2K 3 4A4B4C4D4E···4L4M4N4O4P6A···6G6H6I6J6K12A···12H
order12···22222344444···444446···6666612···12
size11···12222244446···6121212122···244444···4

48 irreducible representations

dim111111112222222244
type++++++++++-+++-+-
imageC1C2C2C2C2C2C2C2S3D4Q8D6D6D6C4○D4Dic6S3×D4D4⋊2S3
kernelC2×Dic3.D4Dic3.D4C2×Dic3⋊C4C2×C4⋊Dic3C2×C6.D4C6×C22⋊C4C22×Dic6C23×Dic3C2×C22⋊C4C2×Dic3C22×C6C22⋊C4C22×C4C24C2×C6C23C22C22
# reps182111111444214822

In GAP, Magma, Sage, TeX 

C_2\times Dic_3.D_4
 
% in TeX
 
G:=Group("C2xDic3.D4");
 
// GroupNames label
 
G:=SmallGroup(192,1040);
 
// by ID
 
G=gap.SmallGroup(192,1040);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,675,297,80,6278]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^6=d^4=e^2=1,c^2=b^3,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,d*c*d^-1=b^3*c,c*e=e*c,e*d*e=b^3*d^-1>;
 
// generators/relations
 

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