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## G = C3×C32⋊4Q8order 216 = 23·33

### Direct product of C3 and C32⋊4Q8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C3×C32⋊4Q8
 Chief series C1 — C3 — C32 — C3×C6 — C32×C6 — C3×C3⋊Dic3 — C3×C32⋊4Q8
 Lower central C32 — C3×C6 — C3×C32⋊4Q8
 Upper central C1 — C6 — C12

Generators and relations for C3×C324Q8
G = < a,b,c,d,e | a3=b3=c3=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe-1=b-1, cd=dc, ece-1=c-1, ede-1=d-1 >

Subgroups: 216 in 96 conjugacy classes, 42 normal (14 characteristic)
C1, C2, C3, C3, C3, C4, C4, C6, C6, C6, Q8, C32, C32, C32, Dic3, C12, C12, C12, C3×C6, C3×C6, C3×C6, Dic6, C3×Q8, C33, C3×Dic3, C3⋊Dic3, C3×C12, C3×C12, C3×C12, C32×C6, C3×Dic6, C324Q8, C3×C3⋊Dic3, C32×C12, C3×C324Q8
Quotients: C1, C2, C3, C22, S3, C6, Q8, D6, C2×C6, C3×S3, C3⋊S3, Dic6, C3×Q8, S3×C6, C2×C3⋊S3, C3×C3⋊S3, C3×Dic6, C324Q8, C6×C3⋊S3, C3×C324Q8

Smallest permutation representation of C3×C324Q8
On 72 points
Generators in S72
(1 49 58)(2 50 59)(3 51 60)(4 52 57)(5 13 23)(6 14 24)(7 15 21)(8 16 22)(9 27 18)(10 28 19)(11 25 20)(12 26 17)(29 33 40)(30 34 37)(31 35 38)(32 36 39)(41 68 56)(42 65 53)(43 66 54)(44 67 55)(45 72 62)(46 69 63)(47 70 64)(48 71 61)
(1 40 7)(2 37 8)(3 38 5)(4 39 6)(9 66 46)(10 67 47)(11 68 48)(12 65 45)(13 51 31)(14 52 32)(15 49 29)(16 50 30)(17 42 62)(18 43 63)(19 44 64)(20 41 61)(21 58 33)(22 59 34)(23 60 35)(24 57 36)(25 56 71)(26 53 72)(27 54 69)(28 55 70)
(1 21 29)(2 22 30)(3 23 31)(4 24 32)(5 35 51)(6 36 52)(7 33 49)(8 34 50)(9 69 43)(10 70 44)(11 71 41)(12 72 42)(13 38 60)(14 39 57)(15 40 58)(16 37 59)(17 45 53)(18 46 54)(19 47 55)(20 48 56)(25 61 68)(26 62 65)(27 63 66)(28 64 67)
(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)
(1 18 3 20)(2 17 4 19)(5 41 7 43)(6 44 8 42)(9 51 11 49)(10 50 12 52)(13 68 15 66)(14 67 16 65)(21 54 23 56)(22 53 24 55)(25 58 27 60)(26 57 28 59)(29 46 31 48)(30 45 32 47)(33 69 35 71)(34 72 36 70)(37 62 39 64)(38 61 40 63)

G:=sub<Sym(72)| (1,49,58)(2,50,59)(3,51,60)(4,52,57)(5,13,23)(6,14,24)(7,15,21)(8,16,22)(9,27,18)(10,28,19)(11,25,20)(12,26,17)(29,33,40)(30,34,37)(31,35,38)(32,36,39)(41,68,56)(42,65,53)(43,66,54)(44,67,55)(45,72,62)(46,69,63)(47,70,64)(48,71,61), (1,40,7)(2,37,8)(3,38,5)(4,39,6)(9,66,46)(10,67,47)(11,68,48)(12,65,45)(13,51,31)(14,52,32)(15,49,29)(16,50,30)(17,42,62)(18,43,63)(19,44,64)(20,41,61)(21,58,33)(22,59,34)(23,60,35)(24,57,36)(25,56,71)(26,53,72)(27,54,69)(28,55,70), (1,21,29)(2,22,30)(3,23,31)(4,24,32)(5,35,51)(6,36,52)(7,33,49)(8,34,50)(9,69,43)(10,70,44)(11,71,41)(12,72,42)(13,38,60)(14,39,57)(15,40,58)(16,37,59)(17,45,53)(18,46,54)(19,47,55)(20,48,56)(25,61,68)(26,62,65)(27,63,66)(28,64,67), (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), (1,18,3,20)(2,17,4,19)(5,41,7,43)(6,44,8,42)(9,51,11,49)(10,50,12,52)(13,68,15,66)(14,67,16,65)(21,54,23,56)(22,53,24,55)(25,58,27,60)(26,57,28,59)(29,46,31,48)(30,45,32,47)(33,69,35,71)(34,72,36,70)(37,62,39,64)(38,61,40,63)>;

G:=Group( (1,49,58)(2,50,59)(3,51,60)(4,52,57)(5,13,23)(6,14,24)(7,15,21)(8,16,22)(9,27,18)(10,28,19)(11,25,20)(12,26,17)(29,33,40)(30,34,37)(31,35,38)(32,36,39)(41,68,56)(42,65,53)(43,66,54)(44,67,55)(45,72,62)(46,69,63)(47,70,64)(48,71,61), (1,40,7)(2,37,8)(3,38,5)(4,39,6)(9,66,46)(10,67,47)(11,68,48)(12,65,45)(13,51,31)(14,52,32)(15,49,29)(16,50,30)(17,42,62)(18,43,63)(19,44,64)(20,41,61)(21,58,33)(22,59,34)(23,60,35)(24,57,36)(25,56,71)(26,53,72)(27,54,69)(28,55,70), (1,21,29)(2,22,30)(3,23,31)(4,24,32)(5,35,51)(6,36,52)(7,33,49)(8,34,50)(9,69,43)(10,70,44)(11,71,41)(12,72,42)(13,38,60)(14,39,57)(15,40,58)(16,37,59)(17,45,53)(18,46,54)(19,47,55)(20,48,56)(25,61,68)(26,62,65)(27,63,66)(28,64,67), (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), (1,18,3,20)(2,17,4,19)(5,41,7,43)(6,44,8,42)(9,51,11,49)(10,50,12,52)(13,68,15,66)(14,67,16,65)(21,54,23,56)(22,53,24,55)(25,58,27,60)(26,57,28,59)(29,46,31,48)(30,45,32,47)(33,69,35,71)(34,72,36,70)(37,62,39,64)(38,61,40,63) );

G=PermutationGroup([[(1,49,58),(2,50,59),(3,51,60),(4,52,57),(5,13,23),(6,14,24),(7,15,21),(8,16,22),(9,27,18),(10,28,19),(11,25,20),(12,26,17),(29,33,40),(30,34,37),(31,35,38),(32,36,39),(41,68,56),(42,65,53),(43,66,54),(44,67,55),(45,72,62),(46,69,63),(47,70,64),(48,71,61)], [(1,40,7),(2,37,8),(3,38,5),(4,39,6),(9,66,46),(10,67,47),(11,68,48),(12,65,45),(13,51,31),(14,52,32),(15,49,29),(16,50,30),(17,42,62),(18,43,63),(19,44,64),(20,41,61),(21,58,33),(22,59,34),(23,60,35),(24,57,36),(25,56,71),(26,53,72),(27,54,69),(28,55,70)], [(1,21,29),(2,22,30),(3,23,31),(4,24,32),(5,35,51),(6,36,52),(7,33,49),(8,34,50),(9,69,43),(10,70,44),(11,71,41),(12,72,42),(13,38,60),(14,39,57),(15,40,58),(16,37,59),(17,45,53),(18,46,54),(19,47,55),(20,48,56),(25,61,68),(26,62,65),(27,63,66),(28,64,67)], [(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)], [(1,18,3,20),(2,17,4,19),(5,41,7,43),(6,44,8,42),(9,51,11,49),(10,50,12,52),(13,68,15,66),(14,67,16,65),(21,54,23,56),(22,53,24,55),(25,58,27,60),(26,57,28,59),(29,46,31,48),(30,45,32,47),(33,69,35,71),(34,72,36,70),(37,62,39,64),(38,61,40,63)]])

C3×C324Q8 is a maximal subgroup of
C3312SD16  C3317SD16  C336Q16  C338Q16  C3318SD16  C339Q16  C3×S3×Dic6  D12⋊(C3⋊S3)  C329(S3×Q8)  C12.58S32  C3⋊S34Dic6  C12⋊S312S3  C3×Q8×C3⋊S3

63 conjugacy classes

 class 1 2 3A 3B 3C ··· 3N 4A 4B 4C 6A 6B 6C ··· 6N 12A ··· 12Z 12AA 12AB 12AC 12AD order 1 2 3 3 3 ··· 3 4 4 4 6 6 6 ··· 6 12 ··· 12 12 12 12 12 size 1 1 1 1 2 ··· 2 2 18 18 1 1 2 ··· 2 2 ··· 2 18 18 18 18

63 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + - + - image C1 C2 C2 C3 C6 C6 S3 Q8 D6 C3×S3 Dic6 C3×Q8 S3×C6 C3×Dic6 kernel C3×C32⋊4Q8 C3×C3⋊Dic3 C32×C12 C32⋊4Q8 C3⋊Dic3 C3×C12 C3×C12 C33 C3×C6 C12 C32 C32 C6 C3 # reps 1 2 1 2 4 2 4 1 4 8 8 2 8 16

Matrix representation of C3×C324Q8 in GL4(𝔽13) generated by

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

C3×C324Q8 in GAP, Magma, Sage, TeX

C_3\times C_3^2\rtimes_4Q_8
% in TeX

G:=Group("C3xC3^2:4Q8");
// GroupNames label

G:=SmallGroup(216,140);
// by ID

G=gap.SmallGroup(216,140);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-3,-3,72,169,79,1444,5189]);
// Polycyclic

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

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