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## G = C23×C3⋊D4order 192 = 26·3

### Direct product of C23 and C3⋊D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C23×C3⋊D4
 Chief series C1 — C3 — C6 — D6 — C22×S3 — S3×C23 — S3×C24 — C23×C3⋊D4
 Lower central C3 — C6 — C23×C3⋊D4
 Upper central C1 — C24 — C25

Generators and relations for C23×C3⋊D4
G = < a,b,c,d,e,f | a2=b2=c2=d3=e4=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, ede-1=fdf=d-1, fef=e-1 >

Subgroups: 3000 in 1362 conjugacy classes, 543 normal (11 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, S3, C6, C6, C6, C2×C4, D4, C23, C23, Dic3, D6, D6, C2×C6, C2×C6, C22×C4, C2×D4, C24, C24, C24, C2×Dic3, C3⋊D4, C22×S3, C22×S3, C22×C6, C22×C6, C23×C4, C22×D4, C25, C25, C22×Dic3, C2×C3⋊D4, S3×C23, S3×C23, C23×C6, C23×C6, C23×C6, D4×C23, C23×Dic3, C22×C3⋊D4, S3×C24, C24×C6, C23×C3⋊D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C24, C3⋊D4, C22×S3, C22×D4, C25, C2×C3⋊D4, S3×C23, D4×C23, C22×C3⋊D4, S3×C24, C23×C3⋊D4

Smallest permutation representation of C23×C3⋊D4
On 96 points
Generators in S96
(1 72)(2 69)(3 70)(4 71)(5 54)(6 55)(7 56)(8 53)(9 35)(10 36)(11 33)(12 34)(13 61)(14 62)(15 63)(16 64)(17 48)(18 45)(19 46)(20 47)(21 30)(22 31)(23 32)(24 29)(25 85)(26 86)(27 87)(28 88)(37 80)(38 77)(39 78)(40 79)(41 84)(42 81)(43 82)(44 83)(49 75)(50 76)(51 73)(52 74)(57 91)(58 92)(59 89)(60 90)(65 95)(66 96)(67 93)(68 94)
(1 66)(2 67)(3 68)(4 65)(5 92)(6 89)(7 90)(8 91)(9 81)(10 82)(11 83)(12 84)(13 87)(14 88)(15 85)(16 86)(17 22)(18 23)(19 24)(20 21)(25 63)(26 64)(27 61)(28 62)(29 46)(30 47)(31 48)(32 45)(33 44)(34 41)(35 42)(36 43)(37 51)(38 52)(39 49)(40 50)(53 57)(54 58)(55 59)(56 60)(69 93)(70 94)(71 95)(72 96)(73 80)(74 77)(75 78)(76 79)
(1 54)(2 55)(3 56)(4 53)(5 72)(6 69)(7 70)(8 71)(9 21)(10 22)(11 23)(12 24)(13 78)(14 79)(15 80)(16 77)(17 82)(18 83)(19 84)(20 81)(25 51)(26 52)(27 49)(28 50)(29 34)(30 35)(31 36)(32 33)(37 63)(38 64)(39 61)(40 62)(41 46)(42 47)(43 48)(44 45)(57 65)(58 66)(59 67)(60 68)(73 85)(74 86)(75 87)(76 88)(89 93)(90 94)(91 95)(92 96)
(1 15 20)(2 17 16)(3 13 18)(4 19 14)(5 37 42)(6 43 38)(7 39 44)(8 41 40)(9 58 73)(10 74 59)(11 60 75)(12 76 57)(21 66 85)(22 86 67)(23 68 87)(24 88 65)(25 30 96)(26 93 31)(27 32 94)(28 95 29)(33 90 49)(34 50 91)(35 92 51)(36 52 89)(45 70 61)(46 62 71)(47 72 63)(48 64 69)(53 84 79)(54 80 81)(55 82 77)(56 78 83)
(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)
(2 4)(6 8)(9 73)(10 76)(11 75)(12 74)(13 18)(14 17)(15 20)(16 19)(21 85)(22 88)(23 87)(24 86)(25 30)(26 29)(27 32)(28 31)(33 49)(34 52)(35 51)(36 50)(37 42)(38 41)(39 44)(40 43)(45 61)(46 64)(47 63)(48 62)(53 55)(57 59)(65 67)(69 71)(77 84)(78 83)(79 82)(80 81)(89 91)(93 95)

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

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

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

72 conjugacy classes

 class 1 2A ··· 2O 2P ··· 2W 2X ··· 2AE 3 4A ··· 4H 6A ··· 6AE order 1 2 ··· 2 2 ··· 2 2 ··· 2 3 4 ··· 4 6 ··· 6 size 1 1 ··· 1 2 ··· 2 6 ··· 6 2 6 ··· 6 2 ··· 2

72 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C2 S3 D4 D6 C3⋊D4 kernel C23×C3⋊D4 C23×Dic3 C22×C3⋊D4 S3×C24 C24×C6 C25 C22×C6 C24 C23 # reps 1 1 28 1 1 1 8 15 16

Matrix representation of C23×C3⋊D4 in GL5(𝔽13)

 12 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 12 0 0 0 0 0 12
,
 12 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 12 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 12 12 0 0 0 1 0
,
 1 0 0 0 0 0 12 0 0 0 0 0 1 0 0 0 0 0 11 9 0 0 0 11 2
,
 12 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 1 0 0 0 0 12 12

G:=sub<GL(5,GF(13))| [12,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,12],[12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,12,1,0,0,0,12,0],[1,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,11,11,0,0,0,9,2],[12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,1,12,0,0,0,0,12] >;

C23×C3⋊D4 in GAP, Magma, Sage, TeX

C_2^3\times C_3\rtimes D_4
% in TeX

G:=Group("C2^3xC3:D4");
// GroupNames label

G:=SmallGroup(192,1529);
// by ID

G=gap.SmallGroup(192,1529);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,1684,6278]);
// Polycyclic

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

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