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## G = Q8×C3.A4order 288 = 25·32

### Direct product of Q8 and C3.A4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C23 — Q8×C3.A4
 Chief series C1 — C22 — C2×C6 — C22×C6 — C2×C3.A4 — C4×C3.A4 — Q8×C3.A4
 Lower central C22 — C23 — Q8×C3.A4
 Upper central C1 — C6 — C3×Q8

Generators and relations for Q8×C3.A4
G = < a,b,c,d,e,f | a4=c3=d2=e2=1, b2=a2, f3=c, bab-1=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, fdf-1=de=ed, fef-1=d >

Subgroups: 186 in 80 conjugacy classes, 30 normal (15 characteristic)
C1, C2, C2 [×2], C3, C4 [×3], C4 [×3], C22, C22 [×2], C6, C6 [×2], C2×C4 [×6], Q8, Q8 [×5], C23, C9, C12 [×3], C12 [×3], C2×C6, C2×C6 [×2], C22×C4 [×3], C2×Q8 [×4], C18, C2×C12 [×6], C3×Q8, C3×Q8 [×5], C22×C6, C22×Q8, C36 [×3], C3.A4, C22×C12 [×3], C6×Q8 [×4], Q8×C9, C2×C3.A4, Q8×C2×C6, C4×C3.A4 [×3], Q8×C3.A4
Quotients: C1, C2 [×3], C3, C22, C6 [×3], Q8, C9, A4, C2×C6, C18 [×3], C3×Q8, C2×A4 [×3], C3.A4, C2×C18, C22×A4, Q8×C9, C2×C3.A4 [×3], Q8×A4, C22×C3.A4, Q8×C3.A4

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

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

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

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

60 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4A 4B 4C 4D 4E 4F 6A 6B 6C 6D 6E 6F 9A ··· 9F 12A ··· 12F 12G ··· 12L 18A ··· 18F 36A ··· 36R order 1 2 2 2 3 3 4 4 4 4 4 4 6 6 6 6 6 6 9 ··· 9 12 ··· 12 12 ··· 12 18 ··· 18 36 ··· 36 size 1 1 3 3 1 1 2 2 2 6 6 6 1 1 3 3 3 3 4 ··· 4 2 ··· 2 6 ··· 6 4 ··· 4 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 C6 C9 C18 Q8 C3×Q8 Q8×C9 A4 C2×A4 C3.A4 C2×C3.A4 Q8×A4 Q8×C3.A4 kernel Q8×C3.A4 C4×C3.A4 Q8×C2×C6 C22×C12 C22×Q8 C22×C4 C3.A4 C2×C6 C22 C3×Q8 C12 Q8 C4 C3 C1 # reps 1 3 2 6 6 18 1 2 6 1 3 2 6 1 2

Matrix representation of Q8×C3.A4 in GL5(𝔽37)

 34 8 0 0 0 8 3 0 0 0 0 0 36 0 0 0 0 0 36 0 0 0 0 0 36
,
 0 36 0 0 0 1 0 0 0 0 0 0 36 0 0 0 0 0 36 0 0 0 0 0 36
,
 26 0 0 0 0 0 26 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 36 36 36 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 36 36 36 0 0 1 0 0
,
 16 0 0 0 0 0 16 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0

G:=sub<GL(5,GF(37))| [34,8,0,0,0,8,3,0,0,0,0,0,36,0,0,0,0,0,36,0,0,0,0,0,36],[0,1,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,36,0,0,0,0,0,36],[26,0,0,0,0,0,26,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,36,0,0,0,0,36,0,1,0,0,36,1,0],[1,0,0,0,0,0,1,0,0,0,0,0,0,36,1,0,0,0,36,0,0,0,1,36,0],[16,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0] >;

Q8×C3.A4 in GAP, Magma, Sage, TeX

Q_8\times C_3.A_4
% in TeX

G:=Group("Q8xC3.A4");
// GroupNames label

G:=SmallGroup(288,346);
// by ID

G=gap.SmallGroup(288,346);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-3,-2,2,84,197,92,142,1531,2666]);
// Polycyclic

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