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## G = C3×C24.C22order 192 = 26·3

### Direct product of C3 and C24.C22

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C3×C24.C22
 Chief series C1 — C2 — C22 — C23 — C22×C6 — C22×C12 — C6×C22⋊C4 — C3×C24.C22
 Lower central C1 — C22 — C3×C24.C22
 Upper central C1 — C22×C6 — C3×C24.C22

Generators and relations for C3×C24.C22
G = < a,b,c,d,e,f,g | a3=b2=c2=d2=e2=1, f2=e, g2=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, fbf-1=bc=cb, gbg-1=bd=db, be=eb, cd=dc, ce=ec, cf=fc, cg=gc, de=ed, gfg-1=df=fd, dg=gd, ef=fe, eg=ge >

Subgroups: 322 in 190 conjugacy classes, 90 normal (62 characteristic)
C1, C2, C2, C3, C4, C22, C22, C6, C6, C2×C4, C2×C4, C23, C23, C23, C12, C2×C6, C2×C6, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C24, C2×C12, C2×C12, C22×C6, C22×C6, C22×C6, C2.C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C4×C12, C3×C22⋊C4, C3×C22⋊C4, C3×C4⋊C4, C22×C12, C23×C6, C24.C22, C3×C2.C42, C2×C4×C12, C6×C22⋊C4, C6×C4⋊C4, C3×C24.C22
Quotients:

Smallest permutation representation of C3×C24.C22
On 96 points
Generators in S96
(1 16 12)(2 13 9)(3 14 10)(4 15 11)(5 58 54)(6 59 55)(7 60 56)(8 57 53)(17 27 21)(18 28 22)(19 25 23)(20 26 24)(29 37 33)(30 38 34)(31 39 35)(32 40 36)(41 49 45)(42 50 46)(43 51 47)(44 52 48)(61 69 65)(62 70 66)(63 71 67)(64 72 68)(73 81 77)(74 82 78)(75 83 79)(76 84 80)(85 93 89)(86 94 90)(87 95 91)(88 96 92)
(2 54)(4 56)(5 13)(7 15)(9 58)(11 60)(17 42)(18 86)(19 44)(20 88)(21 46)(22 90)(23 48)(24 92)(25 52)(26 96)(27 50)(28 94)(29 76)(31 74)(33 80)(35 78)(37 84)(39 82)(41 61)(43 63)(45 65)(47 67)(49 69)(51 71)(62 85)(64 87)(66 89)(68 91)(70 93)(72 95)
(1 53)(2 54)(3 55)(4 56)(5 13)(6 14)(7 15)(8 16)(9 58)(10 59)(11 60)(12 57)(17 62)(18 63)(19 64)(20 61)(21 66)(22 67)(23 68)(24 65)(25 72)(26 69)(27 70)(28 71)(29 76)(30 73)(31 74)(32 75)(33 80)(34 77)(35 78)(36 79)(37 84)(38 81)(39 82)(40 83)(41 88)(42 85)(43 86)(44 87)(45 92)(46 89)(47 90)(48 91)(49 96)(50 93)(51 94)(52 95)
(1 73)(2 74)(3 75)(4 76)(5 39)(6 40)(7 37)(8 38)(9 78)(10 79)(11 80)(12 77)(13 82)(14 83)(15 84)(16 81)(17 42)(18 43)(19 44)(20 41)(21 46)(22 47)(23 48)(24 45)(25 52)(26 49)(27 50)(28 51)(29 56)(30 53)(31 54)(32 55)(33 60)(34 57)(35 58)(36 59)(61 88)(62 85)(63 86)(64 87)(65 92)(66 89)(67 90)(68 91)(69 96)(70 93)(71 94)(72 95)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)(25 27)(26 28)(29 31)(30 32)(33 35)(34 36)(37 39)(38 40)(41 43)(42 44)(45 47)(46 48)(49 51)(50 52)(53 55)(54 56)(57 59)(58 60)(61 63)(62 64)(65 67)(66 68)(69 71)(70 72)(73 75)(74 76)(77 79)(78 80)(81 83)(82 84)(85 87)(86 88)(89 91)(90 92)(93 95)(94 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 42 53 85)(2 18 54 63)(3 44 55 87)(4 20 56 61)(5 71 13 28)(6 95 14 52)(7 69 15 26)(8 93 16 50)(9 22 58 67)(10 48 59 91)(11 24 60 65)(12 46 57 89)(17 30 62 73)(19 32 64 75)(21 34 66 77)(23 36 68 79)(25 40 72 83)(27 38 70 81)(29 88 76 41)(31 86 74 43)(33 92 80 45)(35 90 78 47)(37 96 84 49)(39 94 82 51)

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

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

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

84 conjugacy classes

 class 1 2A ··· 2G 2H 2I 3A 3B 4A ··· 4L 4M ··· 4R 6A ··· 6N 6O 6P 6Q 6R 12A ··· 12X 12Y ··· 12AJ order 1 2 ··· 2 2 2 3 3 4 ··· 4 4 ··· 4 6 ··· 6 6 6 6 6 12 ··· 12 12 ··· 12 size 1 1 ··· 1 4 4 1 1 2 ··· 2 4 ··· 4 1 ··· 1 4 4 4 4 2 ··· 2 4 ··· 4

84 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 type + + + + + + image C1 C2 C2 C2 C2 C3 C4 C6 C6 C6 C6 C12 D4 C4○D4 C3×D4 C3×C4○D4 kernel C3×C24.C22 C3×C2.C42 C2×C4×C12 C6×C22⋊C4 C6×C4⋊C4 C24.C22 C3×C22⋊C4 C2.C42 C2×C42 C2×C22⋊C4 C2×C4⋊C4 C22⋊C4 C2×C12 C2×C6 C2×C4 C22 # reps 1 2 1 3 1 2 8 4 2 6 2 16 4 8 8 16

Matrix representation of C3×C24.C22 in GL5(𝔽13)

 1 0 0 0 0 0 3 0 0 0 0 0 3 0 0 0 0 0 1 0 0 0 0 0 1
,
 12 0 0 0 0 0 1 0 0 0 0 0 12 0 0 0 0 0 1 0 0 0 0 0 12
,
 1 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12
,
 1 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 1 0 0 0 0 0 1
,
 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
,
 5 0 0 0 0 0 0 5 0 0 0 5 0 0 0 0 0 0 0 1 0 0 0 1 0
,
 1 0 0 0 0 0 0 1 0 0 0 12 0 0 0 0 0 0 8 0 0 0 0 0 8

G:=sub<GL(5,GF(13))| [1,0,0,0,0,0,3,0,0,0,0,0,3,0,0,0,0,0,1,0,0,0,0,0,1],[12,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,12],[1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,1],[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],[5,0,0,0,0,0,0,5,0,0,0,5,0,0,0,0,0,0,0,1,0,0,0,1,0],[1,0,0,0,0,0,0,12,0,0,0,1,0,0,0,0,0,0,8,0,0,0,0,0,8] >;

C3×C24.C22 in GAP, Magma, Sage, TeX

C_3\times C_2^4.C_2^2
% in TeX

G:=Group("C3xC2^4.C2^2");
// GroupNames label

G:=SmallGroup(192,821);
// by ID

G=gap.SmallGroup(192,821);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,672,365,848,1094,142]);
// Polycyclic

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

׿
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