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## G = C23×He3order 216 = 23·33

### Direct product of C23 and He3

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

Aliases: C23×He3, C628C6, C6.10C62, (C2×C62)⋊2C3, C3.1(C2×C62), C323(C22×C6), (C22×C6).6C32, (C3×C6)⋊3(C2×C6), (C2×C6).12(C3×C6), SmallGroup(216,115)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C23×He3
 Chief series C1 — C3 — C32 — He3 — C2×He3 — C22×He3 — C23×He3
 Lower central C1 — C3 — C23×He3
 Upper central C1 — C22×C6 — C23×He3

Generators and relations for C23×He3
G = < a,b,c,d,e,f | a2=b2=c2=d3=e3=f3=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-1=de-1, ef=fe >

Subgroups: 304 in 176 conjugacy classes, 112 normal (6 characteristic)
C1, C2, C3, C3, C22, C6, C6, C23, C32, C2×C6, C2×C6, C3×C6, C22×C6, C22×C6, He3, C62, C2×He3, C2×C62, C22×He3, C23×He3
Quotients: C1, C2, C3, C22, C6, C23, C32, C2×C6, C3×C6, C22×C6, He3, C62, C2×He3, C2×C62, C22×He3, C23×He3

Smallest permutation representation of C23×He3
On 72 points
Generators in S72
(1 70)(2 71)(3 72)(4 12)(5 10)(6 11)(7 15)(8 13)(9 14)(16 61)(17 62)(18 63)(19 64)(20 65)(21 66)(22 67)(23 68)(24 69)(25 52)(26 53)(27 54)(28 55)(29 56)(30 57)(31 58)(32 59)(33 60)(34 43)(35 44)(36 45)(37 46)(38 47)(39 48)(40 49)(41 50)(42 51)
(1 25)(2 26)(3 27)(4 57)(5 55)(6 56)(7 60)(8 58)(9 59)(10 28)(11 29)(12 30)(13 31)(14 32)(15 33)(16 34)(17 35)(18 36)(19 37)(20 38)(21 39)(22 40)(23 41)(24 42)(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 16)(2 17)(3 18)(4 66)(5 64)(6 65)(7 69)(8 67)(9 68)(10 19)(11 20)(12 21)(13 22)(14 23)(15 24)(25 34)(26 35)(27 36)(28 37)(29 38)(30 39)(31 40)(32 41)(33 42)(43 52)(44 53)(45 54)(46 55)(47 56)(48 57)(49 58)(50 59)(51 60)(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 14 10)(2 15 11)(3 13 12)(4 72 8)(5 70 9)(6 71 7)(16 23 19)(17 24 20)(18 22 21)(25 32 28)(26 33 29)(27 31 30)(34 41 37)(35 42 38)(36 40 39)(43 50 46)(44 51 47)(45 49 48)(52 59 55)(53 60 56)(54 58 57)(61 68 64)(62 69 65)(63 67 66)
(1 3 11)(2 14 13)(4 7 5)(6 70 72)(8 71 9)(10 12 15)(16 18 20)(17 23 22)(19 21 24)(25 27 29)(26 32 31)(28 30 33)(34 36 38)(35 41 40)(37 39 42)(43 45 47)(44 50 49)(46 48 51)(52 54 56)(53 59 58)(55 57 60)(61 63 65)(62 68 67)(64 66 69)

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

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

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

C23×He3 is a maximal subgroup of   C623C12  C624Dic3

88 conjugacy classes

 class 1 2A ··· 2G 3A 3B 3C ··· 3J 6A ··· 6N 6O ··· 6BR order 1 2 ··· 2 3 3 3 ··· 3 6 ··· 6 6 ··· 6 size 1 1 ··· 1 1 1 3 ··· 3 1 ··· 1 3 ··· 3

88 irreducible representations

 dim 1 1 1 1 3 3 type + + image C1 C2 C3 C6 He3 C2×He3 kernel C23×He3 C22×He3 C2×C62 C62 C23 C22 # reps 1 7 8 56 2 14

Matrix representation of C23×He3 in GL6(𝔽7)

 6 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 6 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 6 0 0 0 0 0 0 6 0 0 0 0 0 0 6 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 2 0 0 0 0 0 0 2 0 0 0 0 0 0 2 0 0 0 0 0 0 0 1 0 0 0 0 5 3 3 0 0 0 0 0 4
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 4
,
 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 0 0 0 0 0 0 6 5 5 0 0 0 1 0 0 0 0 0 3 4 1

G:=sub<GL(6,GF(7))| [6,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,6,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[6,0,0,0,0,0,0,6,0,0,0,0,0,0,6,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[2,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,0,5,0,0,0,0,1,3,0,0,0,0,0,3,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,2,0,0,0,0,0,0,1,0,0,0,0,0,0,6,1,3,0,0,0,5,0,4,0,0,0,5,0,1] >;

C23×He3 in GAP, Magma, Sage, TeX

C_2^3\times {\rm He}_3
% in TeX

G:=Group("C2^3xHe3");
// GroupNames label

G:=SmallGroup(216,115);
// by ID

G=gap.SmallGroup(216,115);
# by ID

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

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

׿
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