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## G = C4⋊(He3⋊C4)  order 432 = 24·33

### The semidirect product of C4 and He3⋊C4 acting via He3⋊C4/He3⋊C2=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C2×He3 — C4⋊(He3⋊C4)
 Chief series C1 — C3 — He3 — He3⋊C2 — C2×He3⋊C2 — C2×He3⋊C4 — C4⋊(He3⋊C4)
 Lower central He3 — C2×He3 — C4⋊(He3⋊C4)
 Upper central C1 — C6 — 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, eae-1=a-1, bc=cb, dbd-1=bc-1, ebe-1=bcd, cd=dc, ce=ec, ede-1=bd-1 >

Subgroups: 405 in 71 conjugacy classes, 17 normal (15 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C2×C4, C32, Dic3, C12, C12, D6, C2×C6, C4⋊C4, C3×S3, C3×C6, C4×S3, C2×C12, He3, C3×Dic3, C3×C12, S3×C6, C3×C4⋊C4, 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, D4, Q8, C4⋊C4, 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 29 21)(2 30 22)(3 31 23)(4 32 24)(5 35 51)(6 36 52)(7 33 49)(8 34 50)(9 18 63)(10 19 64)(11 20 61)(12 17 62)(25 41 56)(26 42 53)(27 43 54)(28 44 55)(45 65 72)(46 66 69)(47 67 70)(48 68 71)
(1 21 29)(2 22 30)(3 23 31)(4 24 32)(5 35 51)(6 36 52)(7 33 49)(8 34 50)(9 43 69)(10 44 70)(11 41 71)(12 42 72)(13 38 60)(14 39 57)(15 40 58)(16 37 59)(17 53 45)(18 54 46)(19 55 47)(20 56 48)(25 68 61)(26 65 62)(27 66 63)(28 67 64)
(1 7 40)(2 8 37)(3 5 38)(4 6 39)(9 66 18)(10 67 19)(11 68 20)(12 65 17)(13 31 51)(14 32 52)(15 29 49)(16 30 50)(21 33 58)(22 34 59)(23 35 60)(24 36 57)(25 48 71)(26 45 72)(27 46 69)(28 47 70)(41 61 56)(42 62 53)(43 63 54)(44 64 55)
(1 20)(2 19)(3 18)(4 17)(5 63 38 43)(6 62 39 42)(7 61 40 41)(8 64 37 44)(9 51 66 13)(10 50 67 16)(11 49 68 15)(12 52 65 14)(21 56)(22 55)(23 54)(24 53)(25 58 71 33)(26 57 72 36)(27 60 69 35)(28 59 70 34)(29 48)(30 47)(31 46)(32 45)

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

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

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

38 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 4A 4B ··· 4F 6A 6B 6C 6D 6E 6F 6G 6H 12A 12B 12C 12D 12E 12F 12G ··· 12P order 1 2 2 2 3 3 3 3 4 4 ··· 4 6 6 6 6 6 6 6 6 12 12 12 12 12 12 12 ··· 12 size 1 1 9 9 1 1 12 12 2 18 ··· 18 1 1 9 9 9 9 12 12 2 2 12 12 12 12 18 ··· 18

38 irreducible representations

 dim 1 1 1 1 1 2 2 3 3 4 4 4 6 type + + + + - + + image C1 C2 C2 C4 C4 D4 Q8 He3⋊C4 C2×He3⋊C4 C32⋊C4 C2×C32⋊C4 C4⋊(C32⋊C4) C4⋊(He3⋊C4) kernel C4⋊(He3⋊C4) C4×He3⋊C2 C2×He3⋊C4 He3⋊3C4 C4×He3 He3⋊C2 He3⋊C2 C4 C2 C12 C6 C3 C1 # reps 1 1 2 2 2 1 1 8 8 2 2 4 4

Matrix representation of C4⋊(He3⋊C4) in GL5(𝔽13)

 12 12 0 0 0 2 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 3 0 0 0 0 0 9 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 9 0 0 0 0 0 9 0 0 0 0 0 9
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0
,
 12 3 0 0 0 8 1 0 0 0 0 0 3 9 9 0 0 1 9 1 0 0 1 1 9

G:=sub<GL(5,GF(13))| [12,2,0,0,0,12,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,3,0,0,0,0,0,9,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,9],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0],[12,8,0,0,0,3,1,0,0,0,0,0,3,1,1,0,0,9,9,1,0,0,9,1,9] >;

C4⋊(He3⋊C4) in GAP, Magma, Sage, TeX

C_4\rtimes ({\rm He}_3\rtimes C_4)
% in TeX

G:=Group("C4:(He3:C4)");
// GroupNames label

G:=SmallGroup(432,276);
// by ID

G=gap.SmallGroup(432,276);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,28,141,64,3924,298,5381,2539,537]);
// 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,e*a*e^-1=a^-1,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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