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## G = C32×C4.D4order 288 = 25·32

### Direct product of C32 and C4.D4

direct product, metabelian, nilpotent (class 3), monomial

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C32×C4.D4
 Chief series C1 — C2 — C4 — C2×C4 — C2×C12 — C6×C12 — C32×M4(2) — C32×C4.D4
 Lower central C1 — C2 — C22 — C32×C4.D4
 Upper central C1 — C3×C6 — C6×C12 — C32×C4.D4

Generators and relations for C32×C4.D4
G = < a,b,c,d,e | a3=b3=c4=1, d4=c2, e2=c, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=c-1, ce=ec, ede-1=c-1d3 >

Subgroups: 252 in 138 conjugacy classes, 72 normal (12 characteristic)
C1, C2, C2 [×3], C3 [×4], C4 [×2], C22, C22 [×4], C6 [×4], C6 [×12], C8 [×2], C2×C4, D4 [×2], C23 [×2], C32, C12 [×8], C2×C6 [×4], C2×C6 [×16], M4(2) [×2], C2×D4, C3×C6, C3×C6 [×3], C24 [×8], C2×C12 [×4], C3×D4 [×8], C22×C6 [×8], C4.D4, C3×C12 [×2], C62, C62 [×4], C3×M4(2) [×8], C6×D4 [×4], C3×C24 [×2], C6×C12, D4×C32 [×2], C2×C62 [×2], C3×C4.D4 [×4], C32×M4(2) [×2], D4×C3×C6, C32×C4.D4
Quotients: C1, C2 [×3], C3 [×4], C4 [×2], C22, C6 [×12], C2×C4, D4 [×2], C32, C12 [×8], C2×C6 [×4], C22⋊C4, C3×C6 [×3], C2×C12 [×4], C3×D4 [×8], C4.D4, C3×C12 [×2], C62, C3×C22⋊C4 [×4], C6×C12, D4×C32 [×2], C3×C4.D4 [×4], C32×C22⋊C4, C32×C4.D4

Smallest permutation representation of C32×C4.D4
On 72 points
Generators in S72
(1 35 61)(2 36 62)(3 37 63)(4 38 64)(5 39 57)(6 40 58)(7 33 59)(8 34 60)(9 65 42)(10 66 43)(11 67 44)(12 68 45)(13 69 46)(14 70 47)(15 71 48)(16 72 41)(17 56 32)(18 49 25)(19 50 26)(20 51 27)(21 52 28)(22 53 29)(23 54 30)(24 55 31)
(1 31 13)(2 32 14)(3 25 15)(4 26 16)(5 27 9)(6 28 10)(7 29 11)(8 30 12)(17 70 36)(18 71 37)(19 72 38)(20 65 39)(21 66 40)(22 67 33)(23 68 34)(24 69 35)(41 64 50)(42 57 51)(43 58 52)(44 59 53)(45 60 54)(46 61 55)(47 62 56)(48 63 49)
(1 3 5 7)(2 8 6 4)(9 11 13 15)(10 16 14 12)(17 23 21 19)(18 20 22 24)(25 27 29 31)(26 32 30 28)(33 35 37 39)(34 40 38 36)(41 47 45 43)(42 44 46 48)(49 51 53 55)(50 56 54 52)(57 59 61 63)(58 64 62 60)(65 67 69 71)(66 72 70 68)
(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 4 3 2 5 8 7 6)(9 12 11 10 13 16 15 14)(17 20 23 22 21 24 19 18)(25 32 27 30 29 28 31 26)(33 40 35 38 37 36 39 34)(41 48 47 42 45 44 43 46)(49 56 51 54 53 52 55 50)(57 60 59 58 61 64 63 62)(65 68 67 66 69 72 71 70)

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

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

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

99 conjugacy classes

 class 1 2A 2B 2C 2D 3A ··· 3H 4A 4B 6A ··· 6H 6I ··· 6P 6Q ··· 6AF 8A 8B 8C 8D 12A ··· 12P 24A ··· 24AF order 1 2 2 2 2 3 ··· 3 4 4 6 ··· 6 6 ··· 6 6 ··· 6 8 8 8 8 12 ··· 12 24 ··· 24 size 1 1 2 4 4 1 ··· 1 2 2 1 ··· 1 2 ··· 2 4 ··· 4 4 4 4 4 2 ··· 2 4 ··· 4

99 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 4 4 type + + + + + image C1 C2 C2 C3 C4 C6 C6 C12 D4 C3×D4 C4.D4 C3×C4.D4 kernel C32×C4.D4 C32×M4(2) D4×C3×C6 C3×C4.D4 C2×C62 C3×M4(2) C6×D4 C22×C6 C3×C12 C12 C32 C3 # reps 1 2 1 8 4 16 8 32 2 16 1 8

Matrix representation of C32×C4.D4 in GL6(𝔽73)

 64 0 0 0 0 0 0 64 0 0 0 0 0 0 8 0 0 0 0 0 0 8 0 0 0 0 0 0 8 0 0 0 0 0 0 8
,
 8 0 0 0 0 0 0 8 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 72 0 0 0 0 0 0 72 0 0 0 0 0 0 0 1 0 0 0 0 72 0 0 0 0 0 21 52 1 71 0 0 21 0 1 72
,
 72 0 0 0 0 0 2 1 0 0 0 0 0 0 21 52 1 71 0 0 0 0 1 0 0 0 72 0 0 0 0 0 1 71 0 52
,
 72 72 0 0 0 0 2 1 0 0 0 0 0 0 0 0 1 0 0 0 21 52 1 71 0 0 0 1 0 0 0 0 72 2 0 21

G:=sub<GL(6,GF(73))| [64,0,0,0,0,0,0,64,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,8],[8,0,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,21,21,0,0,1,0,52,0,0,0,0,0,1,1,0,0,0,0,71,72],[72,2,0,0,0,0,0,1,0,0,0,0,0,0,21,0,72,1,0,0,52,0,0,71,0,0,1,1,0,0,0,0,71,0,0,52],[72,2,0,0,0,0,72,1,0,0,0,0,0,0,0,21,0,72,0,0,0,52,1,2,0,0,1,1,0,0,0,0,0,71,0,21] >;

C32×C4.D4 in GAP, Magma, Sage, TeX

C_3^2\times C_4.D_4
% in TeX

G:=Group("C3^2xC4.D4");
// GroupNames label

G:=SmallGroup(288,318);
// by ID

G=gap.SmallGroup(288,318);
# by ID

G:=PCGroup([7,-2,-2,-3,-3,-2,-2,-2,504,533,6304,4548,124]);
// Polycyclic

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

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