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## G = C23×C3⋊S3order 144 = 24·32

### Direct product of C23 and C3⋊S3

Aliases: C23×C3⋊S3, C323C24, C6210C22, (C2×C6)⋊11D6, (C22×C6)⋊5S3, (C3×C6)⋊3C23, C32(S3×C23), C62(C22×S3), (C2×C62)⋊5C2, SmallGroup(144,196)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C23×C3⋊S3
 Chief series C1 — C3 — C32 — C3⋊S3 — C2×C3⋊S3 — C22×C3⋊S3 — C23×C3⋊S3
 Lower central C32 — C23×C3⋊S3
 Upper central C1 — C23

Generators and relations for C23×C3⋊S3
G = < a,b,c,d,e,f | a2=b2=c2=d3=e3=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, fdf=d-1, fef=e-1 >

Subgroups: 1218 in 402 conjugacy classes, 147 normal (5 characteristic)
C1, C2 [×7], C2 [×8], C3 [×4], C22 [×7], C22 [×28], S3 [×32], C6 [×28], C23, C23 [×14], C32, D6 [×112], C2×C6 [×28], C24, C3⋊S3 [×8], C3×C6 [×7], C22×S3 [×56], C22×C6 [×4], C2×C3⋊S3 [×28], C62 [×7], S3×C23 [×4], C22×C3⋊S3 [×14], C2×C62, C23×C3⋊S3
Quotients: C1, C2 [×15], C22 [×35], S3 [×4], C23 [×15], D6 [×28], C24, C3⋊S3, C22×S3 [×28], C2×C3⋊S3 [×7], S3×C23 [×4], C22×C3⋊S3 [×7], C23×C3⋊S3

Smallest permutation representation of C23×C3⋊S3
On 72 points
Generators in S72
(1 65)(2 66)(3 64)(4 67)(5 68)(6 69)(7 70)(8 71)(9 72)(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 29)(2 30)(3 28)(4 31)(5 32)(6 33)(7 34)(8 35)(9 36)(10 19)(11 20)(12 21)(13 22)(14 23)(15 24)(16 25)(17 26)(18 27)(37 64)(38 65)(39 66)(40 67)(41 68)(42 69)(43 70)(44 71)(45 72)(46 55)(47 56)(48 57)(49 58)(50 59)(51 60)(52 61)(53 62)(54 63)
(1 11)(2 12)(3 10)(4 13)(5 14)(6 15)(7 16)(8 17)(9 18)(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)
(1 5 8)(2 6 9)(3 4 7)(10 13 16)(11 14 17)(12 15 18)(19 22 25)(20 23 26)(21 24 27)(28 31 34)(29 32 35)(30 33 36)(37 40 43)(38 41 44)(39 42 45)(46 49 52)(47 50 53)(48 51 54)(55 58 61)(56 59 62)(57 60 63)(64 67 70)(65 68 71)(66 69 72)
(1 38)(2 37)(3 39)(4 45)(5 44)(6 43)(7 42)(8 41)(9 40)(10 48)(11 47)(12 46)(13 54)(14 53)(15 52)(16 51)(17 50)(18 49)(19 57)(20 56)(21 55)(22 63)(23 62)(24 61)(25 60)(26 59)(27 58)(28 66)(29 65)(30 64)(31 72)(32 71)(33 70)(34 69)(35 68)(36 67)

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

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

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

C23×C3⋊S3 is a maximal subgroup of   C62.116C23  C628D4  C6212D4  C6213D4  S32×C23
C23×C3⋊S3 is a maximal quotient of   C3282+ 1+4  C3272- 1+4  C62.154C23  C3292- 1+4

48 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2O 3A 3B 3C 3D 6A ··· 6AB order 1 2 ··· 2 2 ··· 2 3 3 3 3 6 ··· 6 size 1 1 ··· 1 9 ··· 9 2 2 2 2 2 ··· 2

48 irreducible representations

 dim 1 1 1 2 2 type + + + + + image C1 C2 C2 S3 D6 kernel C23×C3⋊S3 C22×C3⋊S3 C2×C62 C22×C6 C2×C6 # reps 1 14 1 4 28

Matrix representation of C23×C3⋊S3 in GL6(ℤ)

 1 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 0 0 0 0 0 0 1
,
 -1 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 0 0 0 0 0 0 -1
,
 1 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 0 0 0 0 0 0 -1
,
 1 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 -1 0
,
 1 -3 0 0 0 0 1 -2 0 0 0 0 0 0 -1 -1 0 0 0 0 1 0 0 0 0 0 0 0 -1 1 0 0 0 0 -1 0
,
 -1 0 0 0 0 0 -1 1 0 0 0 0 0 0 1 0 0 0 0 0 -1 -1 0 0 0 0 0 0 0 -1 0 0 0 0 -1 0

G:=sub<GL(6,Integers())| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,-1,0,0,0,0,1,0],[1,1,0,0,0,0,-3,-2,0,0,0,0,0,0,-1,1,0,0,0,0,-1,0,0,0,0,0,0,0,-1,-1,0,0,0,0,1,0],[-1,-1,0,0,0,0,0,1,0,0,0,0,0,0,1,-1,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,-1,0] >;

C23×C3⋊S3 in GAP, Magma, Sage, TeX

C_2^3\times C_3\rtimes S_3
% in TeX

G:=Group("C2^3xC3:S3");
// GroupNames label

G:=SmallGroup(144,196);
// by ID

G=gap.SmallGroup(144,196);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-3,964,3461]);
// Polycyclic

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

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