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## G = (C3×D4)⋊14D4order 192 = 26·3

### 2nd semidirect product of C3×D4 and D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — (C3×D4)⋊14D4
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — C2×D12 — C2×D4⋊S3 — (C3×D4)⋊14D4
 Lower central C3 — C6 — C2×C12 — (C3×D4)⋊14D4
 Upper central C1 — C22 — C22×C4 — C2×C4○D4

Generators and relations for (C3×D4)⋊14D4
G = < a,b,c,d,e | a3=b4=c2=d4=e2=1, ab=ba, ac=ca, dad-1=eae=a-1, cbc=dbd-1=ebe=b-1, dcd-1=b-1c, ece=bc, ede=d-1 >

Subgroups: 456 in 162 conjugacy classes, 47 normal (39 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, Dic3, C12, C12, D6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C2×C8, D8, SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C3⋊C8, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C3×Q8, C22×S3, C22×C6, C22×C6, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊D4, C2×D8, C2×SD16, C2×C4○D4, C2×C3⋊C8, C4⋊Dic3, D6⋊C4, D4⋊S3, Q82S3, C2×D12, C2×C3⋊D4, C22×C12, C22×C12, C6×D4, C6×D4, C6×Q8, C3×C4○D4, D4⋊D4, C12.55D4, D4⋊Dic3, Q82Dic3, C127D4, C2×D4⋊S3, C2×Q82S3, C6×C4○D4, (C3×D4)⋊14D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊D4, C22×S3, C22≀C2, C4○D8, C8⋊C22, C2×C3⋊D4, D4⋊D4, D4⋊D6, Q8.13D6, C244S3, (C3×D4)⋊14D4

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

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

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

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

39 conjugacy classes

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

39 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 S3 D4 D4 D4 D4 D6 D6 D6 C3⋊D4 C3⋊D4 C3⋊D4 C3⋊D4 C4○D8 C8⋊C22 D4⋊D6 Q8.13D6 kernel (C3×D4)⋊14D4 C12.55D4 D4⋊Dic3 Q8⋊2Dic3 C12⋊7D4 C2×D4⋊S3 C2×Q8⋊2S3 C6×C4○D4 C2×C4○D4 C2×C12 C3×D4 C3×Q8 C22×C6 C22×C4 C2×D4 C2×Q8 C2×C4 D4 Q8 C23 C6 C6 C2 C2 # reps 1 1 1 1 1 1 1 1 1 1 2 2 1 1 1 1 2 4 4 2 4 1 2 2

Matrix representation of (C3×D4)⋊14D4 in GL4(𝔽73) generated by

 72 72 0 0 1 0 0 0 0 0 1 0 0 0 0 1
,
 72 0 0 0 0 72 0 0 0 0 72 71 0 0 1 1
,
 43 13 0 0 60 30 0 0 0 0 41 41 0 0 16 32
,
 43 13 0 0 43 30 0 0 0 0 46 19 0 0 0 27
,
 1 0 0 0 72 72 0 0 0 0 1 0 0 0 72 72
G:=sub<GL(4,GF(73))| [72,1,0,0,72,0,0,0,0,0,1,0,0,0,0,1],[72,0,0,0,0,72,0,0,0,0,72,1,0,0,71,1],[43,60,0,0,13,30,0,0,0,0,41,16,0,0,41,32],[43,43,0,0,13,30,0,0,0,0,46,0,0,0,19,27],[1,72,0,0,0,72,0,0,0,0,1,72,0,0,0,72] >;

(C3×D4)⋊14D4 in GAP, Magma, Sage, TeX

(C_3\times D_4)\rtimes_{14}D_4
% in TeX

G:=Group("(C3xD4):14D4");
// GroupNames label

G:=SmallGroup(192,797);
// by ID

G=gap.SmallGroup(192,797);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,254,184,1684,851,102,6278]);
// Polycyclic

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

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