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## G = (C3×Q8).D4order 192 = 26·3

### 6th non-split extension by C3×Q8 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — (C3×Q8).D4
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C4×Dic3 — Q8×Dic3 — (C3×Q8).D4
 Lower central C3 — C6 — C2×C12 — (C3×Q8).D4
 Upper central C1 — C22 — C2×C4 — C2×SD16

Generators and relations for (C3×Q8).D4
G = < a,b,c,d,e | a3=b4=d4=1, c2=e2=b2, ab=ba, ac=ca, dad-1=eae-1=a-1, cbc-1=ebe-1=b-1, bd=db, cd=dc, ece-1=bc, ede-1=b2d-1 >

Subgroups: 296 in 112 conjugacy classes, 41 normal (37 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, Dic3, C12, C12, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, Q16, C2×D4, C2×Q8, C2×Q8, C3⋊C8, C24, Dic6, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×D4, C3×Q8, C3×Q8, C22×C6, D4⋊C4, Q8⋊C4, C4⋊C8, C4×Q8, C4.4D4, C2×SD16, C2×Q16, C2×C3⋊C8, C4×Dic3, C4×Dic3, C4⋊Dic3, C4⋊Dic3, C3⋊Q16, C6.D4, C2×C24, C3×SD16, C2×Dic6, C6×D4, C6×Q8, Q8.D4, Dic3⋊C8, C2.Dic12, D4⋊Dic3, C23.12D6, C2×C3⋊Q16, Q8×Dic3, C6×SD16, (C3×Q8).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, Q8.D4, D4.D6, Q8.7D6, C23.14D6, (C3×Q8).D4

Smallest permutation representation of (C3×Q8).D4
On 96 points
Generators in S96
(1 9 5)(2 10 6)(3 11 7)(4 12 8)(13 95 78)(14 96 79)(15 93 80)(16 94 77)(17 21 28)(18 22 25)(19 23 26)(20 24 27)(29 46 52)(30 47 49)(31 48 50)(32 45 51)(33 44 38)(34 41 39)(35 42 40)(36 43 37)(53 62 59)(54 63 60)(55 64 57)(56 61 58)(65 76 69)(66 73 70)(67 74 71)(68 75 72)(81 86 92)(82 87 89)(83 88 90)(84 85 91)
(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 36 3 34)(2 35 4 33)(5 37 7 39)(6 40 8 38)(9 43 11 41)(10 42 12 44)(13 76 15 74)(14 75 16 73)(17 32 19 30)(18 31 20 29)(21 45 23 47)(22 48 24 46)(25 50 27 52)(26 49 28 51)(53 85 55 87)(54 88 56 86)(57 82 59 84)(58 81 60 83)(61 92 63 90)(62 91 64 89)(65 80 67 78)(66 79 68 77)(69 93 71 95)(70 96 72 94)
(1 65 17 59)(2 66 18 60)(3 67 19 57)(4 68 20 58)(5 76 28 53)(6 73 25 54)(7 74 26 55)(8 75 27 56)(9 69 21 62)(10 70 22 63)(11 71 23 64)(12 72 24 61)(13 49 87 39)(14 50 88 40)(15 51 85 37)(16 52 86 38)(29 81 33 77)(30 82 34 78)(31 83 35 79)(32 84 36 80)(41 95 47 89)(42 96 48 90)(43 93 45 91)(44 94 46 92)
(1 57 3 59)(2 60 4 58)(5 55 7 53)(6 54 8 56)(9 64 11 62)(10 63 12 61)(13 52 15 50)(14 51 16 49)(17 67 19 65)(18 66 20 68)(21 71 23 69)(22 70 24 72)(25 73 27 75)(26 76 28 74)(29 80 31 78)(30 79 32 77)(33 84 35 82)(34 83 36 81)(37 86 39 88)(38 85 40 87)(41 90 43 92)(42 89 44 91)(45 94 47 96)(46 93 48 95)

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

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

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

33 conjugacy classes

 class 1 2A 2B 2C 2D 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 6A 6B 6C 6D 6E 8A 8B 8C 8D 12A 12B 12C 12D 24A 24B 24C 24D order 1 2 2 2 2 3 4 4 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 8 2 2 2 4 4 6 6 12 12 12 24 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 D4.D6 Q8.7D6 kernel (C3×Q8).D4 Dic3⋊C8 C2.Dic12 D4⋊Dic3 C23.12D6 C2×C3⋊Q16 Q8×Dic3 C6×SD16 C2×SD16 C2×Dic3 C3×Q8 C2×C8 C2×D4 C2×Q8 C12 Q8 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×Q8).D4 in GL4(𝔽73) generated by

 1 0 0 0 0 1 0 0 0 0 8 0 0 0 20 64
,
 0 1 0 0 72 0 0 0 0 0 1 0 0 0 0 1
,
 6 67 0 0 67 67 0 0 0 0 72 0 0 0 0 72
,
 46 0 0 0 0 46 0 0 0 0 32 2 0 0 35 41
,
 46 0 0 0 0 27 0 0 0 0 32 2 0 0 36 41
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,8,20,0,0,0,64],[0,72,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[6,67,0,0,67,67,0,0,0,0,72,0,0,0,0,72],[46,0,0,0,0,46,0,0,0,0,32,35,0,0,2,41],[46,0,0,0,0,27,0,0,0,0,32,36,0,0,2,41] >;

(C3×Q8).D4 in GAP, Magma, Sage, TeX

(C_3\times Q_8).D_4
% in TeX

G:=Group("(C3xQ8).D4");
// GroupNames label

G:=SmallGroup(192,725);
// by ID

G=gap.SmallGroup(192,725);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,232,1094,135,570,297,136,6278]);
// Polycyclic

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

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