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## G = C3×Q8⋊A4order 288 = 25·32

### Direct product of C3 and Q8⋊A4

Aliases: C3×Q8⋊A4, (C3×Q8)⋊1A4, Q82(C3×A4), (C2×C6)⋊SL2(𝔽3), C23.6(C3×A4), C6.4(C22⋊A4), (C22×C6).13A4, (C22×Q8)⋊4C32, C222(C3×SL2(𝔽3)), (Q8×C2×C6)⋊3C3, C2.1(C3×C22⋊A4), SmallGroup(288,986)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C22×Q8 — C3×Q8⋊A4
 Chief series C1 — C2 — Q8 — C22×Q8 — Q8⋊A4 — C3×Q8⋊A4
 Lower central C22×Q8 — C3×Q8⋊A4
 Upper central C1 — C6

Generators and relations for C3×Q8⋊A4
G = < a,b,c,d,e,f | a3=b4=d2=e2=f3=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, cbc-1=b-1, bd=db, be=eb, fbf-1=c, cd=dc, ce=ec, fcf-1=bc, fdf-1=de=ed, fef-1=d >

Subgroups: 384 in 100 conjugacy classes, 22 normal (11 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, C6, C6, C2×C4, Q8, Q8, C23, C32, C12, A4, C2×C6, C2×C6, C22×C4, C2×Q8, C3×C6, SL2(𝔽3), C2×C12, C3×Q8, C3×Q8, C2×A4, C22×C6, C22×Q8, C3×A4, C22×C12, C6×Q8, C3×SL2(𝔽3), C6×A4, Q8⋊A4, Q8×C2×C6, C3×Q8⋊A4
Quotients: C1, C3, C32, A4, SL2(𝔽3), C3×A4, C22⋊A4, C3×SL2(𝔽3), Q8⋊A4, C3×C22⋊A4, C3×Q8⋊A4

Smallest permutation representation of C3×Q8⋊A4
On 72 points
Generators in S72
(1 24 16)(2 21 13)(3 22 14)(4 23 15)(5 65 57)(6 66 58)(7 67 59)(8 68 60)(9 26 17)(10 27 18)(11 28 19)(12 25 20)(29 45 37)(30 46 38)(31 47 39)(32 48 40)(33 50 41)(34 51 42)(35 52 43)(36 49 44)(53 69 61)(54 70 62)(55 71 63)(56 72 64)
(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 9 3 11)(2 12 4 10)(5 71 7 69)(6 70 8 72)(13 20 15 18)(14 19 16 17)(21 25 23 27)(22 28 24 26)(29 35 31 33)(30 34 32 36)(37 43 39 41)(38 42 40 44)(45 52 47 50)(46 51 48 49)(53 57 55 59)(54 60 56 58)(61 65 63 67)(62 68 64 66)
(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)(53 55)(54 56)(57 59)(58 60)(61 63)(62 64)(65 67)(66 68)(69 71)(70 72)
(5 7)(6 8)(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)
(1 58 29)(2 53 35)(3 60 31)(4 55 33)(5 49 28)(6 45 24)(7 51 26)(8 47 22)(9 59 34)(10 56 32)(11 57 36)(12 54 30)(13 61 43)(14 68 39)(15 63 41)(16 66 37)(17 67 42)(18 64 40)(19 65 44)(20 62 38)(21 69 52)(23 71 50)(25 70 46)(27 72 48)

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

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

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

36 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C ··· 3H 4A 4B 4C 4D 6A 6B 6C 6D 6E 6F 6G ··· 6L 12A ··· 12H order 1 2 2 2 3 3 3 ··· 3 4 4 4 4 6 6 6 6 6 6 6 ··· 6 12 ··· 12 size 1 1 3 3 1 1 16 ··· 16 6 6 6 6 1 1 3 3 3 3 16 ··· 16 6 ··· 6

36 irreducible representations

 dim 1 1 1 2 2 2 3 3 3 3 6 6 type + - + + - image C1 C3 C3 SL2(𝔽3) SL2(𝔽3) C3×SL2(𝔽3) A4 A4 C3×A4 C3×A4 Q8⋊A4 C3×Q8⋊A4 kernel C3×Q8⋊A4 Q8⋊A4 Q8×C2×C6 C2×C6 C2×C6 C22 C3×Q8 C22×C6 Q8 C23 C3 C1 # reps 1 6 2 1 2 6 4 1 8 2 1 2

Matrix representation of C3×Q8⋊A4 in GL5(𝔽13)

 9 0 0 0 0 0 9 0 0 0 0 0 9 0 0 0 0 0 9 0 0 0 0 0 9
,
 0 1 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 9 3 0 0 0 3 4 0 0 0 0 0 1 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 12 0 1 0 0 12 1 0
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 12 1 0 0 0 12 0 0 0 1 12 0
,
 9 0 0 0 0 3 1 0 0 0 0 0 0 0 9 0 0 9 0 0 0 0 0 9 0

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

C3×Q8⋊A4 in GAP, Magma, Sage, TeX

C_3\times Q_8\rtimes A_4
% in TeX

G:=Group("C3xQ8:A4");
// GroupNames label

G:=SmallGroup(288,986);
// by ID

G=gap.SmallGroup(288,986);
# by ID

G:=PCGroup([7,-3,-3,-2,2,-2,2,-2,380,759,2524,172,4541,285,124]);
// Polycyclic

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

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