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## G = C3×A4⋊Q8order 288 = 25·32

### Direct product of C3 and A4⋊Q8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C2×A4 — C3×A4⋊Q8
 Chief series C1 — C22 — A4 — C2×A4 — C6×A4 — C3×A4⋊C4 — C3×A4⋊Q8
 Lower central A4 — C2×A4 — C3×A4⋊Q8
 Upper central C1 — C6 — C12

Generators and relations for C3×A4⋊Q8
G = < a,b,c,d,e,f | a3=b2=c2=d3=e4=1, f2=e2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, dbd-1=fbf-1=bc=cb, be=eb, dcd-1=b, ce=ec, cf=fc, de=ed, fdf-1=d-1, fef-1=e-1 >

Subgroups: 282 in 90 conjugacy classes, 24 normal (20 characteristic)
C1, C2, C2 [×2], C3, C3 [×2], C4, C4 [×5], C22, C22 [×2], C6, C6 [×4], C2×C4 [×6], Q8 [×2], C23, C32, Dic3 [×2], C12, C12 [×7], A4, A4, C2×C6, C2×C6 [×2], C22⋊C4 [×2], C4⋊C4 [×3], C22×C4, C2×Q8, C3×C6, Dic6, C2×C12 [×6], C3×Q8 [×2], C2×A4, C2×A4, C22×C6, C22⋊Q8, C3×Dic3 [×2], C3×C12, C3×A4, C3×C22⋊C4 [×2], C3×C4⋊C4 [×3], A4⋊C4 [×2], C4×A4, C4×A4, C22×C12, C6×Q8, C3×Dic6, C6×A4, C3×C22⋊Q8, A4⋊Q8, C3×A4⋊C4 [×2], C12×A4, C3×A4⋊Q8
Quotients: C1, C2 [×3], C3, C22, S3, C6 [×3], Q8, D6, C2×C6, C3×S3, Dic6, C3×Q8, S4, S3×C6, C2×S4, C3×Dic6, C3×S4, A4⋊Q8, C6×S4, C3×A4⋊Q8

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

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

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

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

42 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 4A 4B 4C 4D 4E 4F 6A 6B 6C 6D 6E 6F 6G 6H 6I 12A 12B 12C 12D 12E ··· 12J 12K ··· 12R order 1 2 2 2 3 3 3 3 3 4 4 4 4 4 4 6 6 6 6 6 6 6 6 6 12 12 12 12 12 ··· 12 12 ··· 12 size 1 1 3 3 1 1 8 8 8 2 6 12 12 12 12 1 1 3 3 3 3 8 8 8 2 2 6 6 8 ··· 8 12 ··· 12

42 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 3 3 3 3 6 6 type + + + + - + - + + - image C1 C2 C2 C3 C6 C6 S3 Q8 D6 C3×S3 C3×Q8 Dic6 S3×C6 C3×Dic6 S4 C2×S4 C3×S4 C6×S4 A4⋊Q8 C3×A4⋊Q8 kernel C3×A4⋊Q8 C3×A4⋊C4 C12×A4 A4⋊Q8 A4⋊C4 C4×A4 C22×C12 C3×A4 C22×C6 C22×C4 A4 C2×C6 C23 C22 C12 C6 C4 C2 C3 C1 # reps 1 2 1 2 4 2 1 1 1 2 2 2 2 4 2 2 4 4 1 2

Matrix representation of C3×A4⋊Q8 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
,
 1 0 0 0 0 0 1 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 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 0 1 0 0 1 0 0 0 0 0 1 0
,
 10 9 0 0 0 9 3 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12
,
 0 12 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0

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],[1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0],[10,9,0,0,0,9,3,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[0,1,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0] >;

C3×A4⋊Q8 in GAP, Magma, Sage, TeX

C_3\times A_4\rtimes Q_8
% in TeX

G:=Group("C3xA4:Q8");
// GroupNames label

G:=SmallGroup(288,896);
// by ID

G=gap.SmallGroup(288,896);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-3,-2,2,84,197,92,1684,6053,285,3534,475]);
// Polycyclic

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

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