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## G = C3×C22.35C24order 192 = 26·3

### Direct product of C3 and C22.35C24

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C3×C22.35C24
 Chief series C1 — C2 — C22 — C2×C6 — C2×C12 — C3×C4⋊C4 — C3×C42⋊2C2 — C3×C22.35C24
 Lower central C1 — C22 — C3×C22.35C24
 Upper central C1 — C2×C6 — C3×C22.35C24

Generators and relations for C3×C22.35C24
G = < a,b,c,d,e,f,g | a3=b2=c2=f2=1, d2=g2=b, e2=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, ede-1=gdg-1=bd=db, fef=be=eb, bf=fb, bg=gb, fdf=cd=dc, ce=ec, cf=fc, cg=gc, eg=ge, fg=gf >

Subgroups: 242 in 192 conjugacy classes, 146 normal (26 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C6, C6, C6, C2×C4, C2×C4, C2×C4, Q8, C23, C12, C12, C2×C6, C2×C6, C42, C42, C22⋊C4, C4⋊C4, C22×C4, C2×Q8, C2×C12, C2×C12, C2×C12, C3×Q8, C22×C6, C42⋊C2, C4×Q8, C22⋊Q8, C42.C2, C42.C2, C422C2, C4⋊Q8, C4×C12, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C22×C12, C6×Q8, C22.35C24, C3×C42⋊C2, Q8×C12, C3×C22⋊Q8, C3×C42.C2, C3×C42.C2, C3×C422C2, C3×C4⋊Q8, C3×C22.35C24
Quotients: C1, C2, C3, C22, C6, C23, C2×C6, C4○D4, C24, C22×C6, C2×C4○D4, 2- 1+4, C3×C4○D4, C23×C6, C22.35C24, C6×C4○D4, C3×2- 1+4, C3×C22.35C24

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

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

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

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

66 conjugacy classes

 class 1 2A 2B 2C 2D 3A 3B 4A ··· 4F 4G ··· 4Q 6A ··· 6F 6G 6H 12A ··· 12L 12M ··· 12AH order 1 2 2 2 2 3 3 4 ··· 4 4 ··· 4 6 ··· 6 6 6 12 ··· 12 12 ··· 12 size 1 1 1 1 4 1 1 2 ··· 2 4 ··· 4 1 ··· 1 4 4 2 ··· 2 4 ··· 4

66 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 4 4 type + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 C6 C4○D4 C3×C4○D4 2- 1+4 C3×2- 1+4 kernel C3×C22.35C24 C3×C42⋊C2 Q8×C12 C3×C22⋊Q8 C3×C42.C2 C3×C42⋊2C2 C3×C4⋊Q8 C22.35C24 C42⋊C2 C4×Q8 C22⋊Q8 C42.C2 C42⋊2C2 C4⋊Q8 C12 C4 C6 C2 # reps 1 1 2 2 5 4 1 2 2 4 4 10 8 2 4 8 2 4

Matrix representation of C3×C22.35C24 in GL6(𝔽13)

 3 0 0 0 0 0 0 3 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 7 7 12 5 0 0 11 11 7 6 0 0 3 4 0 0 0 0 9 12 2 8
,
 5 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 1 0 0 0 5 5 12 11 0 0 12 0 0 0 0 0 0 0 0 8
,
 1 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 5 5 0 12
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 12 0 0 0 0 0 5 5 12 11 0 0 8 0 1 1

G:=sub<GL(6,GF(13))| [3,0,0,0,0,0,0,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,7,11,3,9,0,0,7,11,4,12,0,0,12,7,0,2,0,0,5,6,0,8],[5,0,0,0,0,0,0,5,0,0,0,0,0,0,0,5,12,0,0,0,0,5,0,0,0,0,1,12,0,0,0,0,0,11,0,8],[1,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,5,0,0,0,1,0,5,0,0,0,0,12,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,5,8,0,0,1,0,5,0,0,0,0,0,12,1,0,0,0,0,11,1] >;

C3×C22.35C24 in GAP, Magma, Sage, TeX

C_3\times C_2^2._{35}C_2^4
% in TeX

G:=Group("C3xC2^2.35C2^4");
// GroupNames label

G:=SmallGroup(192,1430);
// by ID

G=gap.SmallGroup(192,1430);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-2,336,701,680,2102,555,1571,192]);
// Polycyclic

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

׿
×
𝔽