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## G = C22⋊C4×3- 1+2order 432 = 24·33

### Direct product of C22⋊C4 and 3- 1+2

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C22⋊C4×3- 1+2
 Chief series C1 — C3 — C6 — C2×C6 — C62 — C22×3- 1+2 — C2×C4×3- 1+2 — C22⋊C4×3- 1+2
 Lower central C1 — C6 — C22⋊C4×3- 1+2
 Upper central C1 — C2×C6 — C22⋊C4×3- 1+2

Generators and relations for C22⋊C4×3- 1+2
G = < a,b,c,d,e | a2=b2=c4=d9=e3=1, cac-1=ab=ba, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=d4 >

Subgroups: 230 in 136 conjugacy classes, 77 normal (20 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C3, C4 [×2], C22, C22 [×2], C22 [×2], C6, C6 [×2], C6 [×7], C2×C4 [×2], C23, C9 [×3], C32, C12 [×4], C2×C6, C2×C6 [×2], C2×C6 [×7], C22⋊C4, C18 [×9], C18 [×6], C3×C6, C3×C6 [×2], C3×C6 [×2], C2×C12 [×2], C2×C12 [×2], C22×C6, C22×C6, 3- 1+2, C36 [×6], C2×C18 [×9], C2×C18 [×6], C3×C12 [×2], C62, C62 [×2], C62 [×2], C3×C22⋊C4, C3×C22⋊C4, C2×3- 1+2, C2×3- 1+2 [×2], C2×3- 1+2 [×2], C2×C36 [×6], C22×C18 [×3], C6×C12 [×2], C2×C62, C4×3- 1+2 [×2], C22×3- 1+2, C22×3- 1+2 [×2], C22×3- 1+2 [×2], C9×C22⋊C4 [×3], C32×C22⋊C4, C2×C4×3- 1+2 [×2], C23×3- 1+2, C22⋊C4×3- 1+2
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], 3- 1+2, C3×C12 [×2], C62, C3×C22⋊C4 [×4], C2×3- 1+2 [×3], C6×C12, D4×C32 [×2], C4×3- 1+2 [×2], C22×3- 1+2, C32×C22⋊C4, C2×C4×3- 1+2, D4×3- 1+2 [×2], C22⋊C4×3- 1+2

Smallest permutation representation of C22⋊C4×3- 1+2
On 72 points
Generators in S72
(1 34)(2 35)(3 36)(4 28)(5 29)(6 30)(7 31)(8 32)(9 33)(10 20)(11 21)(12 22)(13 23)(14 24)(15 25)(16 26)(17 27)(18 19)(37 49)(38 50)(39 51)(40 52)(41 53)(42 54)(43 46)(44 47)(45 48)(55 69)(56 70)(57 71)(58 72)(59 64)(60 65)(61 66)(62 67)(63 68)
(1 40)(2 41)(3 42)(4 43)(5 44)(6 45)(7 37)(8 38)(9 39)(10 55)(11 56)(12 57)(13 58)(14 59)(15 60)(16 61)(17 62)(18 63)(19 68)(20 69)(21 70)(22 71)(23 72)(24 64)(25 65)(26 66)(27 67)(28 46)(29 47)(30 48)(31 49)(32 50)(33 51)(34 52)(35 53)(36 54)
(1 70 34 11)(2 71 35 12)(3 72 36 13)(4 64 28 14)(5 65 29 15)(6 66 30 16)(7 67 31 17)(8 68 32 18)(9 69 33 10)(19 50 63 38)(20 51 55 39)(21 52 56 40)(22 53 57 41)(23 54 58 42)(24 46 59 43)(25 47 60 44)(26 48 61 45)(27 49 62 37)
(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)
(2 8 5)(3 6 9)(10 13 16)(12 18 15)(19 25 22)(20 23 26)(29 35 32)(30 33 36)(38 44 41)(39 42 45)(47 53 50)(48 51 54)(55 58 61)(57 63 60)(65 71 68)(66 69 72)

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

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

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

110 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 3D 4A 4B 4C 4D 6A ··· 6F 6G 6H 6I 6J 6K ··· 6P 6Q 6R 6S 6T 9A ··· 9F 12A ··· 12H 12I ··· 12P 18A ··· 18R 18S ··· 18AD 36A ··· 36X order 1 2 2 2 2 2 3 3 3 3 4 4 4 4 6 ··· 6 6 6 6 6 6 ··· 6 6 6 6 6 9 ··· 9 12 ··· 12 12 ··· 12 18 ··· 18 18 ··· 18 36 ··· 36 size 1 1 1 1 2 2 1 1 3 3 2 2 2 2 1 ··· 1 2 2 2 2 3 ··· 3 6 6 6 6 3 ··· 3 2 ··· 2 6 ··· 6 3 ··· 3 6 ··· 6 6 ··· 6

110 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 3 3 3 3 6 type + + + + image C1 C2 C2 C3 C3 C4 C6 C6 C6 C6 C12 C12 D4 C3×D4 C3×D4 3- 1+2 C2×3- 1+2 C2×3- 1+2 C4×3- 1+2 D4×3- 1+2 kernel C22⋊C4×3- 1+2 C2×C4×3- 1+2 C23×3- 1+2 C9×C22⋊C4 C32×C22⋊C4 C22×3- 1+2 C2×C36 C22×C18 C6×C12 C2×C62 C2×C18 C62 C2×3- 1+2 C18 C3×C6 C22⋊C4 C2×C4 C23 C22 C2 # reps 1 2 1 6 2 4 12 6 4 2 24 8 2 12 4 2 4 2 8 4

Matrix representation of C22⋊C4×3- 1+2 in GL5(𝔽37)

 1 0 0 0 0 0 36 0 0 0 0 0 36 0 0 0 0 0 36 0 0 0 0 0 36
,
 36 0 0 0 0 0 36 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 0 6 0 0 0 31 0 0 0 0 0 0 31 0 0 0 0 0 31 0 0 0 0 0 31
,
 26 0 0 0 0 0 26 0 0 0 0 0 10 0 27 0 0 0 0 26 0 0 9 11 27
,
 26 0 0 0 0 0 26 0 0 0 0 0 1 27 10 0 0 0 26 0 0 0 0 0 10

G:=sub<GL(5,GF(37))| [1,0,0,0,0,0,36,0,0,0,0,0,36,0,0,0,0,0,36,0,0,0,0,0,36],[36,0,0,0,0,0,36,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[0,31,0,0,0,6,0,0,0,0,0,0,31,0,0,0,0,0,31,0,0,0,0,0,31],[26,0,0,0,0,0,26,0,0,0,0,0,10,0,9,0,0,0,0,11,0,0,27,26,27],[26,0,0,0,0,0,26,0,0,0,0,0,1,0,0,0,0,27,26,0,0,0,10,0,10] >;

C22⋊C4×3- 1+2 in GAP, Magma, Sage, TeX

C_2^2\rtimes C_4\times 3_-^{1+2}
% in TeX

G:=Group("C2^2:C4xES-(3,1)");
// GroupNames label

G:=SmallGroup(432,205);
// by ID

G=gap.SmallGroup(432,205);
# by ID

G:=PCGroup([7,-2,-2,-3,-3,-2,-2,-3,504,533,772,1109]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^4=d^9=e^3=1,c*a*c^-1=a*b=b*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^4>;
// generators/relations

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