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## G = C32×C22≀C2order 288 = 25·32

### Direct product of C32 and C22≀C2

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C32×C22≀C2
 Chief series C1 — C2 — C22 — C2×C6 — C62 — C2×C62 — D4×C3×C6 — C32×C22≀C2
 Lower central C1 — C22 — C32×C22≀C2
 Upper central C1 — C62 — C32×C22≀C2

Generators and relations for C32×C22≀C2
G = < a,b,c,d,e,f,g | a3=b3=c2=d2=e2=f2=g2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, bd=db, be=eb, bf=fb, bg=gb, cd=dc, gcg=ce=ec, cf=fc, de=ed, gdg=df=fd, ef=fe, eg=ge, fg=gf >

Subgroups: 636 in 390 conjugacy classes, 156 normal (10 characteristic)
C1, C2 [×3], C2 [×7], C3 [×4], C4 [×3], C22, C22 [×6], C22 [×17], C6 [×12], C6 [×28], C2×C4 [×3], D4 [×6], C23, C23 [×3], C23 [×6], C32, C12 [×12], C2×C6 [×28], C2×C6 [×68], C22⋊C4 [×3], C2×D4 [×3], C24, C3×C6 [×3], C3×C6 [×7], C2×C12 [×12], C3×D4 [×24], C22×C6 [×16], C22×C6 [×24], C22≀C2, C3×C12 [×3], C62, C62 [×6], C62 [×17], C3×C22⋊C4 [×12], C6×D4 [×12], C23×C6 [×4], C6×C12 [×3], D4×C32 [×6], C2×C62, C2×C62 [×3], C2×C62 [×6], C3×C22≀C2 [×4], C32×C22⋊C4 [×3], D4×C3×C6 [×3], C22×C62, C32×C22≀C2
Quotients: C1, C2 [×7], C3 [×4], C22 [×7], C6 [×28], D4 [×6], C23, C32, C2×C6 [×28], C2×D4 [×3], C3×C6 [×7], C3×D4 [×24], C22×C6 [×4], C22≀C2, C62 [×7], C6×D4 [×12], D4×C32 [×6], C2×C62, C3×C22≀C2 [×4], D4×C3×C6 [×3], C32×C22≀C2

Smallest permutation representation of C32×C22≀C2
On 72 points
Generators in S72
(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 23 28)(2 24 29)(3 22 30)(4 60 26)(5 58 27)(6 59 25)(7 15 35)(8 13 36)(9 14 34)(10 21 44)(11 19 45)(12 20 43)(16 50 67)(17 51 68)(18 49 69)(31 65 40)(32 66 41)(33 64 42)(37 57 62)(38 55 63)(39 56 61)(46 54 71)(47 52 72)(48 53 70)
(1 15)(2 13)(3 14)(4 21)(5 19)(6 20)(7 28)(8 29)(9 30)(10 26)(11 27)(12 25)(16 32)(17 33)(18 31)(22 34)(23 35)(24 36)(37 70)(38 71)(39 72)(40 69)(41 67)(42 68)(43 59)(44 60)(45 58)(46 55)(47 56)(48 57)(49 65)(50 66)(51 64)(52 61)(53 62)(54 63)
(16 54)(17 52)(18 53)(31 62)(32 63)(33 61)(37 65)(38 66)(39 64)(40 57)(41 55)(42 56)(46 67)(47 68)(48 69)(49 70)(50 71)(51 72)
(1 45)(2 43)(3 44)(4 9)(5 7)(6 8)(10 22)(11 23)(12 24)(13 59)(14 60)(15 58)(16 63)(17 61)(18 62)(19 28)(20 29)(21 30)(25 36)(26 34)(27 35)(31 53)(32 54)(33 52)(37 49)(38 50)(39 51)(40 48)(41 46)(42 47)(55 67)(56 68)(57 69)(64 72)(65 70)(66 71)
(1 15)(2 13)(3 14)(4 21)(5 19)(6 20)(7 28)(8 29)(9 30)(10 26)(11 27)(12 25)(16 54)(17 52)(18 53)(22 34)(23 35)(24 36)(31 62)(32 63)(33 61)(37 65)(38 66)(39 64)(40 57)(41 55)(42 56)(43 59)(44 60)(45 58)(46 67)(47 68)(48 69)(49 70)(50 71)(51 72)
(1 16)(2 17)(3 18)(4 40)(5 41)(6 42)(7 46)(8 47)(9 48)(10 37)(11 38)(12 39)(13 52)(14 53)(15 54)(19 55)(20 56)(21 57)(22 49)(23 50)(24 51)(25 64)(26 65)(27 66)(28 67)(29 68)(30 69)(31 60)(32 58)(33 59)(34 70)(35 71)(36 72)(43 61)(44 62)(45 63)

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

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

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

126 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 2J 3A ··· 3H 4A 4B 4C 6A ··· 6X 6Y ··· 6BT 6BU ··· 6CB 12A ··· 12X order 1 2 2 2 2 ··· 2 2 3 ··· 3 4 4 4 6 ··· 6 6 ··· 6 6 ··· 6 12 ··· 12 size 1 1 1 1 2 ··· 2 4 1 ··· 1 4 4 4 1 ··· 1 2 ··· 2 4 ··· 4 4 ··· 4

126 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 type + + + + + image C1 C2 C2 C2 C3 C6 C6 C6 D4 C3×D4 kernel C32×C22≀C2 C32×C22⋊C4 D4×C3×C6 C22×C62 C3×C22≀C2 C3×C22⋊C4 C6×D4 C23×C6 C62 C2×C6 # reps 1 3 3 1 8 24 24 8 6 48

Matrix representation of C32×C22≀C2 in GL4(𝔽13) generated by

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

C32×C22≀C2 in GAP, Magma, Sage, TeX

C_3^2\times C_2^2\wr C_2
% in TeX

G:=Group("C3^2xC2^2wrC2");
// GroupNames label

G:=SmallGroup(288,817);
// by ID

G=gap.SmallGroup(288,817);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-3,-2,-2,1037,3110]);
// Polycyclic

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

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