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## G = C23×He3⋊C2order 432 = 24·33

### Direct product of C23 and He3⋊C2

Series: Derived Chief Lower central Upper central

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

Generators and relations for C23×He3⋊C2
G = < a,b,c,d,e,f,g | a2=b2=c2=d3=e3=f3=g2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, bd=db, be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, de=ed, fdf-1=de-1, gdg=d-1, ef=fe, eg=ge, gfg=f-1 >

Subgroups: 2497 in 737 conjugacy classes, 163 normal (7 characteristic)
C1, C2, C2, C3, C3, C22, C22, S3, C6, C6, C23, C23, C32, D6, C2×C6, C2×C6, C24, C3×S3, C3×C6, C22×S3, C22×C6, C22×C6, He3, S3×C6, C62, S3×C23, C23×C6, He3⋊C2, C2×He3, S3×C2×C6, C2×C62, C2×He3⋊C2, C22×He3, S3×C22×C6, C22×He3⋊C2, C23×He3, C23×He3⋊C2
Quotients: C1, C2, C22, S3, C23, D6, C24, C3⋊S3, C22×S3, C2×C3⋊S3, S3×C23, He3⋊C2, C22×C3⋊S3, C2×He3⋊C2, C23×C3⋊S3, C22×He3⋊C2, C23×He3⋊C2

Smallest permutation representation of C23×He3⋊C2
On 72 points
Generators in S72
(1 61)(2 62)(3 63)(4 21)(5 19)(6 20)(7 24)(8 22)(9 23)(10 64)(11 65)(12 66)(13 67)(14 68)(15 69)(16 70)(17 71)(18 72)(25 43)(26 44)(27 45)(28 46)(29 47)(30 48)(31 49)(32 50)(33 51)(34 52)(35 53)(36 54)(37 55)(38 56)(39 57)(40 58)(41 59)(42 60)
(1 34)(2 35)(3 36)(4 48)(5 46)(6 47)(7 51)(8 49)(9 50)(10 37)(11 38)(12 39)(13 40)(14 41)(15 42)(16 25)(17 26)(18 27)(19 28)(20 29)(21 30)(22 31)(23 32)(24 33)(43 70)(44 71)(45 72)(52 61)(53 62)(54 63)(55 64)(56 65)(57 66)(58 67)(59 68)(60 69)
(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 13 15)(2 10 3)(4 6 9)(5 72 71)(7 70 8)(11 14 12)(16 22 24)(17 19 18)(20 23 21)(25 31 33)(26 28 27)(29 32 30)(34 40 42)(35 37 36)(38 41 39)(43 49 51)(44 46 45)(47 50 48)(52 58 60)(53 55 54)(56 59 57)(61 67 69)(62 64 63)(65 68 66)
(1 61)(2 63)(3 62)(4 20)(5 19)(6 21)(7 22)(8 24)(9 23)(10 64)(11 66)(12 65)(13 69)(14 68)(15 67)(16 70)(17 72)(18 71)(25 43)(26 45)(27 44)(28 46)(29 48)(30 47)(31 51)(32 50)(33 49)(34 52)(35 54)(36 53)(37 55)(38 57)(39 56)(40 60)(41 59)(42 58)

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

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

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

80 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2O 3A 3B 3C 3D 3E 3F 6A ··· 6N 6O ··· 6AP 6AQ ··· 6BF order 1 2 ··· 2 2 ··· 2 3 3 3 3 3 3 6 ··· 6 6 ··· 6 6 ··· 6 size 1 1 ··· 1 9 ··· 9 1 1 6 6 6 6 1 ··· 1 6 ··· 6 9 ··· 9

80 irreducible representations

 dim 1 1 1 2 2 3 3 type + + + + + image C1 C2 C2 S3 D6 He3⋊C2 C2×He3⋊C2 kernel C23×He3⋊C2 C22×He3⋊C2 C23×He3 C2×C62 C62 C23 C22 # reps 1 14 1 4 28 4 28

Matrix representation of C23×He3⋊C2 in GL7(𝔽7)

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

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

C23×He3⋊C2 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(432,561);
// by ID

G=gap.SmallGroup(432,561);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,1124,4037,537]);
// Polycyclic

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

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