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## G = C4×He3⋊C4order 432 = 24·33

### Direct product of C4 and He3⋊C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — He3 — C4×He3⋊C4
 Chief series C1 — C3 — He3 — He3⋊C2 — C2×He3⋊C2 — C2×He3⋊C4 — C4×He3⋊C4
 Lower central He3 — C4×He3⋊C4
 Upper central C1 — C12

Generators and relations for C4×He3⋊C4
G = < a,b,c,d,e | a4=b3=c3=d3=e4=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=bc-1, ebe-1=bcd, cd=dc, ce=ec, ede-1=bd-1 >

Subgroups: 405 in 77 conjugacy classes, 21 normal (13 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C2×C4, C32, Dic3, C12, C12, D6, C2×C6, C42, C3×S3, C3×C6, C4×S3, C2×C12, He3, C3×Dic3, C3×C12, S3×C6, C4×C12, He3⋊C2, C2×He3, S3×C12, He33C4, C4×He3, He3⋊C4, C2×He3⋊C2, C4×He3⋊C2, C2×He3⋊C4, C4×He3⋊C4
Quotients: C1, C2, C4, C22, C2×C4, C42, C32⋊C4, C2×C32⋊C4, He3⋊C4, C4×C32⋊C4, C2×He3⋊C4, C4×He3⋊C4

Smallest permutation representation of C4×He3⋊C4
On 72 points
Generators in S72
(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 33 40)(2 34 37)(3 35 38)(4 36 39)(5 60 14)(6 57 15)(7 58 16)(8 59 13)(9 68 18)(10 65 19)(11 66 20)(12 67 17)(21 41 30)(22 42 31)(23 43 32)(24 44 29)(25 71 53)(26 72 54)(27 69 55)(28 70 56)(45 52 63)(46 49 64)(47 50 61)(48 51 62)
(1 7 29)(2 8 30)(3 5 31)(4 6 32)(9 61 55)(10 62 56)(11 63 53)(12 64 54)(13 41 37)(14 42 38)(15 43 39)(16 44 40)(17 49 72)(18 50 69)(19 51 70)(20 52 71)(21 34 59)(22 35 60)(23 36 57)(24 33 58)(25 66 45)(26 67 46)(27 68 47)(28 65 48)
(9 68 50)(10 65 51)(11 66 52)(12 67 49)(13 37 41)(14 38 42)(15 39 43)(16 40 44)(17 54 26)(18 55 27)(19 56 28)(20 53 25)(21 34 59)(22 35 60)(23 36 57)(24 33 58)(45 71 63)(46 72 64)(47 69 61)(48 70 62)
(1 9 3 11)(2 10 4 12)(5 63 7 61)(6 64 8 62)(13 70 36 46)(14 71 33 47)(15 72 34 48)(16 69 35 45)(17 59 28 43)(18 60 25 44)(19 57 26 41)(20 58 27 42)(21 65 39 49)(22 66 40 50)(23 67 37 51)(24 68 38 52)(29 55 31 53)(30 56 32 54)

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

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

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

56 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 4A 4B 4C ··· 4L 6A 6B 6C 6D 6E 6F 6G 6H 12A 12B 12C 12D 12E ··· 12X 12Y 12Z 12AA 12AB order 1 2 2 2 3 3 3 3 4 4 4 ··· 4 6 6 6 6 6 6 6 6 12 12 12 12 12 ··· 12 12 12 12 12 size 1 1 9 9 1 1 12 12 1 1 9 ··· 9 1 1 9 9 9 9 12 12 1 1 1 1 9 ··· 9 12 12 12 12

56 irreducible representations

 dim 1 1 1 1 1 1 3 3 3 4 4 4 type + + + + + image C1 C2 C2 C4 C4 C4 He3⋊C4 C2×He3⋊C4 C4×He3⋊C4 C32⋊C4 C2×C32⋊C4 C4×C32⋊C4 kernel C4×He3⋊C4 C4×He3⋊C2 C2×He3⋊C4 He3⋊3C4 C4×He3 He3⋊C4 C4 C2 C1 C12 C6 C3 # reps 1 1 2 2 2 8 8 8 16 2 2 4

Matrix representation of C4×He3⋊C4 in GL3(𝔽13) generated by

 8 0 0 0 8 0 0 0 8
,
 4 3 9 4 0 12 7 0 9
,
 3 0 0 0 3 0 0 0 3
,
 4 1 12 4 0 9 7 0 9
,
 3 4 9 3 0 9 12 4 9
G:=sub<GL(3,GF(13))| [8,0,0,0,8,0,0,0,8],[4,4,7,3,0,0,9,12,9],[3,0,0,0,3,0,0,0,3],[4,4,7,1,0,0,12,9,9],[3,3,12,4,0,4,9,9,9] >;

C4×He3⋊C4 in GAP, Magma, Sage, TeX

C_4\times {\rm He}_3\rtimes C_4
% in TeX

G:=Group("C4xHe3:C4");
// GroupNames label

G:=SmallGroup(432,275);
// by ID

G=gap.SmallGroup(432,275);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,28,64,3924,298,5381,2539,537]);
// Polycyclic

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

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