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## G = C3×Q32⋊C2order 192 = 26·3

### Direct product of C3 and Q32⋊C2

direct product, metabelian, nilpotent (class 4), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C8 — C3×Q32⋊C2
 Chief series C1 — C2 — C4 — C8 — C24 — C3×D8 — C3×SD32 — C3×Q32⋊C2
 Lower central C1 — C2 — C4 — C8 — C3×Q32⋊C2
 Upper central C1 — C6 — C2×C12 — C2×C24 — C3×Q32⋊C2

Generators and relations for C3×Q32⋊C2
G = < a,b,c,d | a3=b16=d2=1, c2=b8, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd=b9, cd=dc >

Subgroups: 162 in 82 conjugacy classes, 46 normal (30 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C12, C12, C2×C6, C2×C6, C16, C2×C8, D8, SD16, Q16, Q16, Q16, C2×Q8, C4○D4, C24, C2×C12, C2×C12, C3×D4, C3×Q8, M5(2), SD32, Q32, C2×Q16, C4○D8, C48, C2×C24, C3×D8, C3×SD16, C3×Q16, C3×Q16, C3×Q16, C6×Q8, C3×C4○D4, Q32⋊C2, C3×M5(2), C3×SD32, C3×Q32, C6×Q16, C3×C4○D8, C3×Q32⋊C2
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, D8, C2×D4, C3×D4, C22×C6, C2×D8, C3×D8, C6×D4, Q32⋊C2, C6×D8, C3×Q32⋊C2

Smallest permutation representation of C3×Q32⋊C2
On 96 points
Generators in S96
(1 34 72)(2 35 73)(3 36 74)(4 37 75)(5 38 76)(6 39 77)(7 40 78)(8 41 79)(9 42 80)(10 43 65)(11 44 66)(12 45 67)(13 46 68)(14 47 69)(15 48 70)(16 33 71)(17 56 83)(18 57 84)(19 58 85)(20 59 86)(21 60 87)(22 61 88)(23 62 89)(24 63 90)(25 64 91)(26 49 92)(27 50 93)(28 51 94)(29 52 95)(30 53 96)(31 54 81)(32 55 82)
(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 30 9 22)(2 29 10 21)(3 28 11 20)(4 27 12 19)(5 26 13 18)(6 25 14 17)(7 24 15 32)(8 23 16 31)(33 54 41 62)(34 53 42 61)(35 52 43 60)(36 51 44 59)(37 50 45 58)(38 49 46 57)(39 64 47 56)(40 63 48 55)(65 87 73 95)(66 86 74 94)(67 85 75 93)(68 84 76 92)(69 83 77 91)(70 82 78 90)(71 81 79 89)(72 96 80 88)
(1 21)(2 30)(3 23)(4 32)(5 25)(6 18)(7 27)(8 20)(9 29)(10 22)(11 31)(12 24)(13 17)(14 26)(15 19)(16 28)(33 51)(34 60)(35 53)(36 62)(37 55)(38 64)(39 57)(40 50)(41 59)(42 52)(43 61)(44 54)(45 63)(46 56)(47 49)(48 58)(65 88)(66 81)(67 90)(68 83)(69 92)(70 85)(71 94)(72 87)(73 96)(74 89)(75 82)(76 91)(77 84)(78 93)(79 86)(80 95)

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

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

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

48 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4A 4B 4C 4D 4E 6A 6B 6C 6D 6E 6F 8A 8B 8C 12A 12B 12C 12D 12E ··· 12J 16A 16B 16C 16D 24A 24B 24C 24D 24E 24F 48A ··· 48H order 1 2 2 2 3 3 4 4 4 4 4 6 6 6 6 6 6 8 8 8 12 12 12 12 12 ··· 12 16 16 16 16 24 24 24 24 24 24 48 ··· 48 size 1 1 2 8 1 1 2 2 8 8 8 1 1 2 2 8 8 2 2 4 2 2 2 2 8 ··· 8 4 4 4 4 2 2 2 2 4 4 4 ··· 4

48 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 D4 D4 D8 D8 C3×D4 C3×D4 C3×D8 C3×D8 Q32⋊C2 C3×Q32⋊C2 kernel C3×Q32⋊C2 C3×M5(2) C3×SD32 C3×Q32 C6×Q16 C3×C4○D8 Q32⋊C2 M5(2) SD32 Q32 C2×Q16 C4○D8 C24 C2×C12 C12 C2×C6 C8 C2×C4 C4 C22 C3 C1 # reps 1 1 2 2 1 1 2 2 4 4 2 2 1 1 2 2 2 2 4 4 2 4

Matrix representation of C3×Q32⋊C2 in GL4(𝔽7) generated by

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

C3×Q32⋊C2 in GAP, Magma, Sage, TeX

C_3\times Q_{32}\rtimes C_2
% in TeX

G:=Group("C3xQ32:C2");
// GroupNames label

G:=SmallGroup(192,943);
// by ID

G=gap.SmallGroup(192,943);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,365,680,2102,2524,1271,242,6053,3036,124]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^16=d^2=1,c^2=b^8,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,d*b*d=b^9,c*d=d*c>;
// generators/relations

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