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G = C3xQ8xA4order 288 = 25·32

Direct product of C3, Q8 and A4

direct product, metabelian, soluble, monomial

Aliases: C3xQ8xA4, C23.8C62, C4.1(C6xA4), (C4xA4).3C6, (C12xA4).7C2, C12.11(C2xA4), C22:(Q8xC32), (C22xC12).5C6, C6.23(C22xA4), (C22xQ8):3C32, (C6xA4).25C22, C2.4(A4xC2xC6), (Q8xC2xC6):2C3, (C2xC6):3(C3xQ8), (C22xC4).(C3xC6), (C2xA4).8(C2xC6), (C22xC6).41(C2xC6), SmallGroup(288,982)

Series: Derived Chief Lower central Upper central

C1C23 — C3xQ8xA4
C1C22C23C22xC6C6xA4C12xA4 — C3xQ8xA4
C22C23 — C3xQ8xA4
C1C6C3xQ8

Generators and relations for C3xQ8xA4
 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, C3, C3, C4, C4, C22, C22, C6, C6, C2xC4, Q8, Q8, C23, C32, C12, C12, A4, C2xC6, C2xC6, C22xC4, C2xQ8, C3xC6, C2xC12, C3xQ8, C3xQ8, C2xA4, C22xC6, C22xQ8, C3xC12, C3xA4, C4xA4, C22xC12, C6xQ8, Q8xC32, C6xA4, Q8xA4, Q8xC2xC6, C12xA4, C3xQ8xA4
Quotients: C1, C2, C3, C22, C6, Q8, C32, A4, C2xC6, C3xC6, C3xQ8, C2xA4, C3xA4, C62, C22xA4, Q8xC32, C6xA4, Q8xA4, A4xC2xC6, C3xQ8xA4

Smallest permutation representation of C3xQ8xA4
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 2A2B2C3A3B3C···3H4A4B4C4D4E4F6A6B6C6D6E6F6G···6L12A···12F12G···12L12M···12AD
order1222333···34444446666666···612···1212···1212···12
size1133114···42226661133334···42···26···68···8

60 irreducible representations

dim111111222333366
type++-++-
imageC1C2C3C3C6C6Q8C3xQ8C3xQ8A4C2xA4C3xA4C6xA4Q8xA4C3xQ8xA4
kernelC3xQ8xA4C12xA4Q8xA4Q8xC2xC6C4xA4C22xC12C3xA4A4C2xC6C3xQ8C12Q8C4C3C1
# reps1362186162132612

Matrix representation of C3xQ8xA4 in GL5(F13)

30000
03000
00100
00010
00001
,
50000
78000
00100
00010
00001
,
16000
412000
001200
000120
000012
,
10000
01000
000112
001012
000012
,
10000
01000
000121
000120
001120
,
30000
03000
003010
000010
000310

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] >;

C3xQ8xA4 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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