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## G = C3×Q8.A4order 288 = 25·32

### Direct product of C3 and Q8.A4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — C3×Q8.A4
 Chief series C1 — C2 — Q8 — C3×Q8 — C3×SL2(𝔽3) — C3×C4.A4 — C3×Q8.A4
 Lower central Q8 — C3×Q8.A4
 Upper central C1 — C6 — C3×Q8

Generators and relations for C3×Q8.A4
G = < a,b,c,d,e,f | a3=b4=f3=1, c2=d2=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, cbc-1=b-1, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, ede-1=b2d, fdf-1=b2de, fef-1=d >

Subgroups: 336 in 114 conjugacy classes, 42 normal (12 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C6, C6, C2×C4, D4, Q8, C23, C32, C12, C12, C2×C6, C2×D4, C4○D4, C4○D4, C3×C6, SL2(𝔽3), C2×C12, C3×D4, C3×Q8, C3×Q8, C22×C6, 2+ 1+4, C3×C12, C4.A4, C6×D4, C3×C4○D4, C3×C4○D4, C3×SL2(𝔽3), Q8×C32, Q8.A4, C3×2+ 1+4, C3×C4.A4, C3×Q8.A4
Quotients: C1, C2, C3, C22, C6, C32, A4, C2×C6, C3×C6, C2×A4, C3×A4, C62, C22×A4, C6×A4, Q8.A4, A4×C2×C6, 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 66 58)(6 67 59)(7 68 60)(8 65 57)(9 25 17)(10 26 18)(11 27 19)(12 28 20)(29 45 37)(30 46 38)(31 47 39)(32 48 40)(33 49 41)(34 50 42)(35 51 43)(36 52 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 11 3 9)(2 10 4 12)(5 71 7 69)(6 70 8 72)(13 18 15 20)(14 17 16 19)(21 26 23 28)(22 25 24 27)(29 34 31 36)(30 33 32 35)(37 42 39 44)(38 41 40 43)(45 50 47 52)(46 49 48 51)(53 58 55 60)(54 57 56 59)(61 66 63 68)(62 65 64 67)
(1 11 3 9)(2 12 4 10)(5 70 7 72)(6 71 8 69)(13 20 15 18)(14 17 16 19)(21 28 23 26)(22 25 24 27)(29 30 31 32)(33 36 35 34)(37 38 39 40)(41 44 43 42)(45 46 47 48)(49 52 51 50)(53 59 55 57)(54 60 56 58)(61 67 63 65)(62 68 64 66)
(1 10 3 12)(2 11 4 9)(5 8 7 6)(13 19 15 17)(14 20 16 18)(21 27 23 25)(22 28 24 26)(29 36 31 34)(30 33 32 35)(37 44 39 42)(38 41 40 43)(45 52 47 50)(46 49 48 51)(53 54 55 56)(57 60 59 58)(61 62 63 64)(65 68 67 66)(69 70 71 72)
(1 32 56)(2 29 53)(3 30 54)(4 31 55)(5 26 50)(6 27 51)(7 28 52)(8 25 49)(9 33 57)(10 34 58)(11 35 59)(12 36 60)(13 37 61)(14 38 62)(15 39 63)(16 40 64)(17 41 65)(18 42 66)(19 43 67)(20 44 68)(21 45 69)(22 46 70)(23 47 71)(24 48 72)

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

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

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

57 conjugacy classes

 class 1 2A 2B 2C 2D 3A 3B 3C ··· 3H 4A 4B 4C 4D 6A 6B 6C ··· 6H 6I ··· 6N 12A ··· 12F 12G 12H 12I ··· 12Z order 1 2 2 2 2 3 3 3 ··· 3 4 4 4 4 6 6 6 ··· 6 6 ··· 6 12 ··· 12 12 12 12 ··· 12 size 1 1 6 6 6 1 1 4 ··· 4 2 2 2 6 1 1 4 ··· 4 6 ··· 6 2 ··· 2 6 6 8 ··· 8

57 irreducible representations

 dim 1 1 1 1 1 1 3 3 3 3 4 4 4 type + + + + + image C1 C2 C3 C3 C6 C6 A4 C2×A4 C3×A4 C6×A4 Q8.A4 Q8.A4 C3×Q8.A4 kernel C3×Q8.A4 C3×C4.A4 Q8.A4 C3×2+ 1+4 C4.A4 C3×C4○D4 C3×Q8 C12 Q8 C4 C3 C3 C1 # reps 1 3 6 2 18 6 1 3 2 6 1 2 6

Matrix representation of C3×Q8.A4 in GL4(𝔽7) generated by

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

C3×Q8.A4 in GAP, Magma, Sage, TeX

C_3\times Q_8.A_4
% in TeX

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

G:=SmallGroup(288,984);
// by ID

G=gap.SmallGroup(288,984);
# by ID

G:=PCGroup([7,-2,-2,-3,-3,-2,2,-2,1008,2045,1016,648,172,1153,285,124]);
// Polycyclic

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

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