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## G = (C3×D4).D4order 192 = 26·3

### 8th non-split extension by C3×D4 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — (C3×D4).D4
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C4×Dic3 — D4×Dic3 — (C3×D4).D4
 Lower central C3 — C6 — C2×C12 — (C3×D4).D4
 Upper central C1 — C22 — C2×C4 — C2×SD16

Generators and relations for (C3×D4).D4
G = < a,b,c,d,e | a4=b2=c6=e2=1, d2=c3, bab=eae=a-1, ac=ca, ad=da, bc=cb, bd=db, ebe=ab, dcd-1=ece=c-1, ede=a2c3d >

Subgroups: 392 in 124 conjugacy classes, 41 normal (37 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, C23, Dic3, C12, C12, D6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, D8, SD16, C22×C4, C2×D4, C2×D4, C2×Q8, C3⋊C8, C24, D12, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C22×S3, C22×C6, D4⋊C4, Q8⋊C4, C4⋊C8, C4×D4, C4.4D4, C2×D8, C2×SD16, C2×C3⋊C8, C4×Dic3, C4⋊Dic3, D6⋊C4, D4⋊S3, C6.D4, C2×C24, C3×SD16, C2×D12, C22×Dic3, C6×D4, C6×Q8, D4.2D4, Dic3⋊C8, C2.D24, Q82Dic3, C2×D4⋊S3, D4×Dic3, C12.23D4, C6×SD16, (C3×D4).D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C3⋊D4, C22×S3, C4⋊D4, C4○D8, C8⋊C22, S3×D4, D42S3, C2×C3⋊D4, D4.2D4, Q83D6, Q8.7D6, C23.14D6, (C3×D4).D4

Smallest permutation representation of (C3×D4).D4
On 96 points
Generators in S96
(1 29 17 22)(2 30 18 23)(3 25 13 24)(4 26 14 19)(5 27 15 20)(6 28 16 21)(7 87 92 80)(8 88 93 81)(9 89 94 82)(10 90 95 83)(11 85 96 84)(12 86 91 79)(31 43 38 50)(32 44 39 51)(33 45 40 52)(34 46 41 53)(35 47 42 54)(36 48 37 49)(55 67 62 74)(56 68 63 75)(57 69 64 76)(58 70 65 77)(59 71 66 78)(60 72 61 73)
(1 58)(2 59)(3 60)(4 55)(5 56)(6 57)(7 51)(8 52)(9 53)(10 54)(11 49)(12 50)(13 61)(14 62)(15 63)(16 64)(17 65)(18 66)(19 67)(20 68)(21 69)(22 70)(23 71)(24 72)(25 73)(26 74)(27 75)(28 76)(29 77)(30 78)(31 79)(32 80)(33 81)(34 82)(35 83)(36 84)(37 85)(38 86)(39 87)(40 88)(41 89)(42 90)(43 91)(44 92)(45 93)(46 94)(47 95)(48 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 34 4 31)(2 33 5 36)(3 32 6 35)(7 69 10 72)(8 68 11 71)(9 67 12 70)(13 39 16 42)(14 38 17 41)(15 37 18 40)(19 50 22 53)(20 49 23 52)(21 54 24 51)(25 44 28 47)(26 43 29 46)(27 48 30 45)(55 79 58 82)(56 84 59 81)(57 83 60 80)(61 87 64 90)(62 86 65 89)(63 85 66 88)(73 92 76 95)(74 91 77 94)(75 96 78 93)
(2 6)(3 5)(7 81)(8 80)(9 79)(10 84)(11 83)(12 82)(13 15)(16 18)(19 26)(20 25)(21 30)(22 29)(23 28)(24 27)(31 41)(32 40)(33 39)(34 38)(35 37)(36 42)(43 46)(44 45)(47 48)(49 54)(50 53)(51 52)(55 74)(56 73)(57 78)(58 77)(59 76)(60 75)(61 68)(62 67)(63 72)(64 71)(65 70)(66 69)(85 95)(86 94)(87 93)(88 92)(89 91)(90 96)

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

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

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

33 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 3 4A 4B 4C 4D 4E 4F 4G 4H 6A 6B 6C 6D 6E 8A 8B 8C 8D 12A 12B 12C 12D 24A 24B 24C 24D order 1 2 2 2 2 2 2 3 4 4 4 4 4 4 4 4 6 6 6 6 6 8 8 8 8 12 12 12 12 24 24 24 24 size 1 1 1 1 4 4 24 2 2 2 6 6 8 12 12 12 2 2 2 8 8 4 4 12 12 4 4 8 8 4 4 4 4

33 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + + + + + + + - + + image C1 C2 C2 C2 C2 C2 C2 C2 S3 D4 D4 D6 D6 D6 C4○D4 C3⋊D4 C4○D8 C8⋊C22 D4⋊2S3 S3×D4 Q8⋊3D6 Q8.7D6 kernel (C3×D4).D4 Dic3⋊C8 C2.D24 Q8⋊2Dic3 C2×D4⋊S3 D4×Dic3 C12.23D4 C6×SD16 C2×SD16 C2×Dic3 C3×D4 C2×C8 C2×D4 C2×Q8 C12 D4 C6 C6 C4 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 1 2 2 1 1 1 2 4 4 1 1 1 2 2

Matrix representation of (C3×D4).D4 in GL4(𝔽73) generated by

 72 71 0 0 1 1 0 0 0 0 1 0 0 0 0 1
,
 41 41 0 0 16 32 0 0 0 0 72 0 0 0 0 72
,
 72 0 0 0 0 72 0 0 0 0 0 1 0 0 72 1
,
 27 0 0 0 0 27 0 0 0 0 60 43 0 0 30 13
,
 1 0 0 0 72 72 0 0 0 0 0 1 0 0 1 0
G:=sub<GL(4,GF(73))| [72,1,0,0,71,1,0,0,0,0,1,0,0,0,0,1],[41,16,0,0,41,32,0,0,0,0,72,0,0,0,0,72],[72,0,0,0,0,72,0,0,0,0,0,72,0,0,1,1],[27,0,0,0,0,27,0,0,0,0,60,30,0,0,43,13],[1,72,0,0,0,72,0,0,0,0,0,1,0,0,1,0] >;

(C3×D4).D4 in GAP, Magma, Sage, TeX

(C_3\times D_4).D_4
% in TeX

G:=Group("(C3xD4).D4");
// GroupNames label

G:=SmallGroup(192,724);
// by ID

G=gap.SmallGroup(192,724);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,1094,135,184,570,297,136,6278]);
// Polycyclic

G:=Group<a,b,c,d,e|a^4=b^2=c^6=e^2=1,d^2=c^3,b*a*b=e*a*e=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,e*b*e=a*b,d*c*d^-1=e*c*e=c^-1,e*d*e=a^2*c^3*d>;
// generators/relations

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