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## G = C3×Q8×A4order 288 = 25·32

### Direct product of C3, Q8 and A4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C23 — C3×Q8×A4
 Chief series C1 — C22 — C23 — C22×C6 — C6×A4 — C12×A4 — C3×Q8×A4
 Lower central C22 — C23 — 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=d2=e2=f3=1, c2=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, fdf-1=de=ed, fef-1=d >

Subgroups: 276 in 116 conjugacy classes, 48 normal (15 characteristic)
C1, C2, C2 [×2], C3, C3 [×3], C4 [×3], C4 [×3], C22, C22 [×2], C6, C6 [×5], C2×C4 [×6], Q8, Q8 [×5], C23, C32, C12 [×3], C12 [×12], A4 [×3], C2×C6, C2×C6 [×2], C22×C4 [×3], C2×Q8 [×4], C3×C6, C2×C12 [×6], C3×Q8, C3×Q8 [×8], C2×A4 [×3], C22×C6, C22×Q8, C3×C12 [×3], C3×A4, C4×A4 [×9], C22×C12 [×3], C6×Q8 [×4], Q8×C32, C6×A4, Q8×A4 [×3], Q8×C2×C6, C12×A4 [×3], C3×Q8×A4
Quotients: C1, C2 [×3], C3 [×4], C22, C6 [×12], Q8, C32, A4, C2×C6 [×4], C3×C6 [×3], C3×Q8 [×4], C2×A4 [×3], C3×A4, C62, C22×A4, Q8×C32, C6×A4 [×3], Q8×A4, A4×C2×C6, C3×Q8×A4

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

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

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

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

60 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C ··· 3H 4A 4B 4C 4D 4E 4F 6A 6B 6C 6D 6E 6F 6G ··· 6L 12A ··· 12F 12G ··· 12L 12M ··· 12AD order 1 2 2 2 3 3 3 ··· 3 4 4 4 4 4 4 6 6 6 6 6 6 6 ··· 6 12 ··· 12 12 ··· 12 12 ··· 12 size 1 1 3 3 1 1 4 ··· 4 2 2 2 6 6 6 1 1 3 3 3 3 4 ··· 4 2 ··· 2 6 ··· 6 8 ··· 8

60 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 3 3 3 3 6 6 type + + - + + - image C1 C2 C3 C3 C6 C6 Q8 C3×Q8 C3×Q8 A4 C2×A4 C3×A4 C6×A4 Q8×A4 C3×Q8×A4 kernel C3×Q8×A4 C12×A4 Q8×A4 Q8×C2×C6 C4×A4 C22×C12 C3×A4 A4 C2×C6 C3×Q8 C12 Q8 C4 C3 C1 # reps 1 3 6 2 18 6 1 6 2 1 3 2 6 1 2

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

 3 0 0 0 0 0 3 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 5 0 0 0 0 7 8 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 6 0 0 0 4 12 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 1 12 0 0 1 0 12 0 0 0 0 12
,
 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
,
 3 0 0 0 0 0 3 0 0 0 0 0 3 0 10 0 0 0 0 10 0 0 0 3 10

G:=sub<GL(5,GF(13))| [3,0,0,0,0,0,3,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[5,7,0,0,0,0,8,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,4,0,0,0,6,12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,12,12,12],[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],[3,0,0,0,0,0,3,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,10,10,10] >;

C3×Q8×A4 in GAP, Magma, Sage, TeX

C_3\times Q_8\times A_4
% in TeX

G:=Group("C3xQ8xA4");
// GroupNames label

G:=SmallGroup(288,982);
// by ID

G=gap.SmallGroup(288,982);
# by ID

G:=PCGroup([7,-2,-2,-3,-3,-2,-2,2,252,533,260,1531,2666]);
// 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,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,f*d*f^-1=d*e=e*d,f*e*f^-1=d>;
// generators/relations

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