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## G = C23×3- 1+2order 216 = 23·33

### Direct product of C23 and 3- 1+2

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

Aliases: C23×3- 1+2, C62.8C6, C6.11C62, C182(C2×C6), (C2×C18)⋊8C6, C92(C22×C6), (C22×C18)⋊3C3, (C2×C62).3C3, C3.2(C2×C62), C32.(C22×C6), (C22×C6).7C32, (C2×C6).13(C3×C6), (C3×C6).17(C2×C6), SmallGroup(216,116)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C23×3- 1+2
 Chief series C1 — C3 — C32 — 3- 1+2 — C2×3- 1+2 — C22×3- 1+2 — C23×3- 1+2
 Lower central C1 — C3 — C23×3- 1+2
 Upper central C1 — C22×C6 — C23×3- 1+2

Generators and relations for C23×3- 1+2
G = < a,b,c,d,e | a2=b2=c2=d9=e3=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=d4 >

Subgroups: 160 in 128 conjugacy classes, 112 normal (8 characteristic)
C1, C2, C3, C3, C22, C6, C6, C23, C9, C32, C2×C6, C2×C6, C18, C3×C6, C22×C6, C22×C6, 3- 1+2, C2×C18, C62, C2×3- 1+2, C22×C18, C2×C62, C22×3- 1+2, C23×3- 1+2
Quotients: C1, C2, C3, C22, C6, C23, C32, C2×C6, C3×C6, C22×C6, 3- 1+2, C62, C2×3- 1+2, C2×C62, C22×3- 1+2, C23×3- 1+2

Smallest permutation representation of C23×3- 1+2
On 72 points
Generators in S72
(1 68)(2 69)(3 70)(4 71)(5 72)(6 64)(7 65)(8 66)(9 67)(10 55)(11 56)(12 57)(13 58)(14 59)(15 60)(16 61)(17 62)(18 63)(19 46)(20 47)(21 48)(22 49)(23 50)(24 51)(25 52)(26 53)(27 54)(28 37)(29 38)(30 39)(31 40)(32 41)(33 42)(34 43)(35 44)(36 45)
(1 23)(2 24)(3 25)(4 26)(5 27)(6 19)(7 20)(8 21)(9 22)(10 28)(11 29)(12 30)(13 31)(14 32)(15 33)(16 34)(17 35)(18 36)(37 55)(38 56)(39 57)(40 58)(41 59)(42 60)(43 61)(44 62)(45 63)(46 64)(47 65)(48 66)(49 67)(50 68)(51 69)(52 70)(53 71)(54 72)
(1 14)(2 15)(3 16)(4 17)(5 18)(6 10)(7 11)(8 12)(9 13)(19 28)(20 29)(21 30)(22 31)(23 32)(24 33)(25 34)(26 35)(27 36)(37 46)(38 47)(39 48)(40 49)(41 50)(42 51)(43 52)(44 53)(45 54)(55 64)(56 65)(57 66)(58 67)(59 68)(60 69)(61 70)(62 71)(63 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)
(2 8 5)(3 6 9)(10 13 16)(12 18 15)(19 22 25)(21 27 24)(28 31 34)(30 36 33)(37 40 43)(39 45 42)(46 49 52)(48 54 51)(55 58 61)(57 63 60)(64 67 70)(66 72 69)

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

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

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

C23×3- 1+2 is a maximal subgroup of   C62.27D6

88 conjugacy classes

 class 1 2A ··· 2G 3A 3B 3C 3D 6A ··· 6N 6O ··· 6AB 9A ··· 9F 18A ··· 18AP order 1 2 ··· 2 3 3 3 3 6 ··· 6 6 ··· 6 9 ··· 9 18 ··· 18 size 1 1 ··· 1 1 1 3 3 1 ··· 1 3 ··· 3 3 ··· 3 3 ··· 3

88 irreducible representations

 dim 1 1 1 1 1 1 3 3 type + + image C1 C2 C3 C3 C6 C6 3- 1+2 C2×3- 1+2 kernel C23×3- 1+2 C22×3- 1+2 C22×C18 C2×C62 C2×C18 C62 C23 C22 # reps 1 7 6 2 42 14 2 14

Matrix representation of C23×3- 1+2 in GL6(𝔽19)

 18 0 0 0 0 0 0 1 0 0 0 0 0 0 18 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 18 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
,
 1 0 0 0 0 0 0 18 0 0 0 0 0 0 18 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 7 0 0 0 0 0 0 11 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 11 0 0 0 1 0 0
,
 11 0 0 0 0 0 0 7 0 0 0 0 0 0 11 0 0 0 0 0 0 1 0 0 0 0 0 0 11 0 0 0 0 0 0 7

G:=sub<GL(6,GF(19))| [18,0,0,0,0,0,0,1,0,0,0,0,0,0,18,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,18,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[7,0,0,0,0,0,0,11,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,11,0],[11,0,0,0,0,0,0,7,0,0,0,0,0,0,11,0,0,0,0,0,0,1,0,0,0,0,0,0,11,0,0,0,0,0,0,7] >;

C23×3- 1+2 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(216,116);
// by ID

G=gap.SmallGroup(216,116);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-3,-3,237,382]);
// Polycyclic

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

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