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## G = C32×2+ 1+4order 288 = 25·32

### Direct product of C32 and 2+ 1+4

direct product, metabelian, nilpotent (class 2), monomial

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C32×2+ 1+4
 Chief series C1 — C2 — C6 — C3×C6 — C62 — D4×C32 — D4×C3×C6 — C32×2+ 1+4
 Lower central C1 — C2 — C32×2+ 1+4
 Upper central C1 — C3×C6 — C32×2+ 1+4

Generators and relations for C32×2+ 1+4
G = < a,b,c,d,e,f | a3=b3=c4=d2=f2=1, e2=c2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd=c-1, ce=ec, cf=fc, de=ed, df=fd, fef=c2e >

Subgroups: 660 in 498 conjugacy classes, 408 normal (6 characteristic)
C1, C2, C2, C3, C4, C22, C22, C6, C6, C2×C4, D4, Q8, C23, C32, C12, C2×C6, C2×C6, C2×D4, C4○D4, C3×C6, C3×C6, C2×C12, C3×D4, C3×Q8, C22×C6, 2+ 1+4, C3×C12, C62, C62, C6×D4, C3×C4○D4, C6×C12, D4×C32, Q8×C32, C2×C62, C3×2+ 1+4, D4×C3×C6, C32×C4○D4, C32×2+ 1+4
Quotients: C1, C2, C3, C22, C6, C23, C32, C2×C6, C24, C3×C6, C22×C6, 2+ 1+4, C62, C23×C6, C2×C62, C3×2+ 1+4, C22×C62, C32×2+ 1+4

Smallest permutation representation of C32×2+ 1+4
On 72 points
Generators in S72
(1 56 18)(2 53 19)(3 54 20)(4 55 17)(5 33 9)(6 34 10)(7 35 11)(8 36 12)(13 23 68)(14 24 65)(15 21 66)(16 22 67)(25 39 71)(26 40 72)(27 37 69)(28 38 70)(29 64 60)(30 61 57)(31 62 58)(32 63 59)(41 49 45)(42 50 46)(43 51 47)(44 52 48)
(1 65 37)(2 66 38)(3 67 39)(4 68 40)(5 43 29)(6 44 30)(7 41 31)(8 42 32)(9 47 60)(10 48 57)(11 45 58)(12 46 59)(13 72 55)(14 69 56)(15 70 53)(16 71 54)(17 23 26)(18 24 27)(19 21 28)(20 22 25)(33 51 64)(34 52 61)(35 49 62)(36 50 63)
(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 42)(2 41)(3 44)(4 43)(5 40)(6 39)(7 38)(8 37)(9 26)(10 25)(11 28)(12 27)(13 64)(14 63)(15 62)(16 61)(17 47)(18 46)(19 45)(20 48)(21 58)(22 57)(23 60)(24 59)(29 68)(30 67)(31 66)(32 65)(33 72)(34 71)(35 70)(36 69)(49 53)(50 56)(51 55)(52 54)
(1 4 3 2)(5 6 7 8)(9 10 11 12)(13 16 15 14)(17 20 19 18)(21 24 23 22)(25 28 27 26)(29 30 31 32)(33 34 35 36)(37 40 39 38)(41 42 43 44)(45 46 47 48)(49 50 51 52)(53 56 55 54)(57 58 59 60)(61 62 63 64)(65 68 67 66)(69 72 71 70)
(1 41)(2 42)(3 43)(4 44)(5 39)(6 40)(7 37)(8 38)(9 25)(10 26)(11 27)(12 28)(13 61)(14 62)(15 63)(16 64)(17 48)(18 45)(19 46)(20 47)(21 59)(22 60)(23 57)(24 58)(29 67)(30 68)(31 65)(32 66)(33 71)(34 72)(35 69)(36 70)(49 56)(50 53)(51 54)(52 55)

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

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

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

153 conjugacy classes

 class 1 2A 2B ··· 2J 3A ··· 3H 4A ··· 4F 6A ··· 6H 6I ··· 6CB 12A ··· 12AV order 1 2 2 ··· 2 3 ··· 3 4 ··· 4 6 ··· 6 6 ··· 6 12 ··· 12 size 1 1 2 ··· 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 2 ··· 2

153 irreducible representations

 dim 1 1 1 1 1 1 4 4 type + + + + image C1 C2 C2 C3 C6 C6 2+ 1+4 C3×2+ 1+4 kernel C32×2+ 1+4 D4×C3×C6 C32×C4○D4 C3×2+ 1+4 C6×D4 C3×C4○D4 C32 C3 # reps 1 9 6 8 72 48 1 8

Matrix representation of C32×2+ 1+4 in GL5(𝔽13)

 1 0 0 0 0 0 3 0 0 0 0 0 3 0 0 0 0 0 3 0 0 0 0 0 3
,
 9 0 0 0 0 0 3 0 0 0 0 0 3 0 0 0 0 0 3 0 0 0 0 0 3
,
 1 0 0 0 0 0 9 6 1 2 0 0 0 1 0 0 0 12 0 0 0 11 6 12 4
,
 12 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 4 7 12 11 0 8 5 0 1
,
 1 0 0 0 0 0 4 7 12 11 0 0 0 1 0 0 0 12 0 0 0 2 8 8 9
,
 1 0 0 0 0 0 0 0 1 0 0 4 7 12 11 0 1 0 0 0 0 7 11 5 6

G:=sub<GL(5,GF(13))| [1,0,0,0,0,0,3,0,0,0,0,0,3,0,0,0,0,0,3,0,0,0,0,0,3],[9,0,0,0,0,0,3,0,0,0,0,0,3,0,0,0,0,0,3,0,0,0,0,0,3],[1,0,0,0,0,0,9,0,0,11,0,6,0,12,6,0,1,1,0,12,0,2,0,0,4],[12,0,0,0,0,0,0,1,4,8,0,1,0,7,5,0,0,0,12,0,0,0,0,11,1],[1,0,0,0,0,0,4,0,0,2,0,7,0,12,8,0,12,1,0,8,0,11,0,0,9],[1,0,0,0,0,0,0,4,1,7,0,0,7,0,11,0,1,12,0,5,0,0,11,0,6] >;

C32×2+ 1+4 in GAP, Magma, Sage, TeX

C_3^2\times 2_+^{1+4}
% in TeX

G:=Group("C3^2xES+(2,2)");
// GroupNames label

G:=SmallGroup(288,1022);
// by ID

G=gap.SmallGroup(288,1022);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-2,2045,1563,4259]);
// Polycyclic

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

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