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G = C2×D4.Dic3  order 192 = 26·3

Direct product of C2 and D4.Dic3

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

Aliases: C2×D4.Dic3, C12.75C24, C6⋊3(C8○D4), C4○D4.60D6, (C6×D4).12C4, C3⋊C8.37C23, (C6×Q8).12C4, C4.74(S3×C23), C6.48(C23×C4), C4○D4.7Dic3, D4.8(C2×Dic3), C12.97(C22×C4), (C2×Q8).13Dic3, (C2×D4).12Dic3, Q8.14(C2×Dic3), (C22×C4).402D6, (C2×C12).553C23, C4.Dic3⋊34C22, C4.40(C22×Dic3), C23.24(C2×Dic3), C2.10(C23×Dic3), (C22×C12).288C22, C22.33(C22×Dic3), C3⋊4(C2×C8○D4), (C2×C3⋊C8)⋊41C22, (C22×C3⋊C8)⋊14C2, (C3×C4○D4).3C4, (C6×C4○D4).9C2, (C2×C4○D4).19S3, (C3×D4).25(C2×C4), (C3×Q8).27(C2×C4), (C2×C12).135(C2×C4), (C2×C4.Dic3)⋊28C2, (C22×C6).80(C2×C4), (C2×C6).28(C22×C4), (C2×C4).56(C2×Dic3), (C2×C4).831(C22×S3), (C3×C4○D4).48C22, SmallGroup(192,1377)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C6 — C2×D4.Dic3
Chief series
C1 — C3 — C6 — C12 — C3⋊C8 — C2×C3⋊C8 — C22×C3⋊C8 — C2×D4.Dic3
Lower central
C3 — C6 — C2×D4.Dic3

Subgroups: 392 in 266 conjugacy classes, 191 normal (18 characteristic)
C1, C2, C2 [×2], C2 [×6], C3, C4 [×2], C4 [×6], C22, C22 [×6], C22 [×6], C6, C6 [×2], C6 [×6], C8 [×8], C2×C4, C2×C4 [×15], D4 [×12], Q8 [×4], C23 [×3], C12 [×2], C12 [×6], C2×C6, C2×C6 [×6], C2×C6 [×6], C2×C8 [×16], M4(2) [×12], C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4 [×8], C3⋊C8 [×8], C2×C12, C2×C12 [×15], C3×D4 [×12], C3×Q8 [×4], C22×C6 [×3], C22×C8 [×3], C2×M4(2) [×3], C8○D4 [×8], C2×C4○D4, C2×C3⋊C8, C2×C3⋊C8 [×15], C4.Dic3 [×12], C22×C12 [×3], C6×D4 [×3], C6×Q8, C3×C4○D4 [×8], C2×C8○D4, C22×C3⋊C8 [×3], C2×C4.Dic3 [×3], D4.Dic3 [×8], C6×C4○D4, C2×D4.Dic3

Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], S3, C2×C4 [×28], C23 [×15], Dic3 [×8], D6 [×7], C22×C4 [×14], C24, C2×Dic3 [×28], C22×S3 [×7], C8○D4 [×2], C23×C4, C22×Dic3 [×14], S3×C23, C2×C8○D4, D4.Dic3 [×2], C23×Dic3, C2×D4.Dic3

Generators and relations
 G = < a,b,c,d,e | a2=b4=c2=1, d6=b2, e2=b2d3, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d5 >

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

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

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

Matrix representation ►G ⊆ GL5(𝔽73)

720000
01000
00100
00010
00001
,
720000
072000
007200
000270
000046
,
10000
072000
007200
000046
000270
,
10000
00100
072100
000270
000027
,
720000
0301300
0434300
000630
000063

G:=sub<GL(5,GF(73))| [72,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[72,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,27,0,0,0,0,0,46],[1,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,0,27,0,0,0,46,0],[1,0,0,0,0,0,0,72,0,0,0,1,1,0,0,0,0,0,27,0,0,0,0,0,27],[72,0,0,0,0,0,30,43,0,0,0,13,43,0,0,0,0,0,63,0,0,0,0,0,63] >;
 

60 conjugacy classes

class 1 2A2B2C2D···2I 3 4A4B4C4D4E···4J6A6B6C6D···6I8A···8H8I···8T12A12B12C12D12E···12J
order12222···2344444···46666···68···88···81212121212···12
size11112···2211112···22224···43···36···622224···4

60 irreducible representations

dim1111111122222224
type+++++++---+
imageC1C2C2C2C2C4C4C4S3D6Dic3Dic3Dic3D6C8○D4D4.Dic3
kernelC2×D4.Dic3C22×C3⋊C8C2×C4.Dic3D4.Dic3C6×C4○D4C6×D4C6×Q8C3×C4○D4C2×C4○D4C22×C4C2×D4C2×Q8C4○D4C4○D4C6C2
# reps1338162813314484

In GAP, Magma, Sage, TeX 

C_2\times D_4.Dic_3
 
% in TeX
 
G:=Group("C2xD4.Dic3");
 
// GroupNames label
 
G:=SmallGroup(192,1377);
 
// by ID
 
G=gap.SmallGroup(192,1377);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,297,102,6278]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^4=c^2=1,d^6=b^2,e^2=b^2*d^3,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^5>;
 
// generators/relations
 

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