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## G = C4○D4⋊He3order 432 = 24·33

### The semidirect product of C4○D4 and He3 acting via He3/C32=C3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C3×Q8 — C4○D4⋊He3
 Chief series C1 — C2 — Q8 — C3×Q8 — Q8×C32 — Q8⋊He3 — C4○D4⋊He3
 Lower central Q8 — C3×Q8 — C4○D4⋊He3
 Upper central C1 — C12 — C3×C12

Generators and relations for C4○D4⋊He3
G = < a,b,c,d,e,f | a4=c2=d3=e3=f3=1, b2=a2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, cbc=a2b, bd=db, be=eb, fbf-1=a-1bc, cd=dc, ce=ec, fcf-1=a-1b, de=ed, fdf-1=de-1, ef=fe >

Subgroups: 320 in 85 conjugacy classes, 23 normal (17 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C6, C6, C2×C4, D4, Q8, C32, C32, C12, C12, C2×C6, C4○D4, C3×C6, C3×C6, SL2(𝔽3), C2×C12, C3×D4, C3×Q8, C3×Q8, He3, C3×C12, C3×C12, C62, C4.A4, C3×C4○D4, C3×C4○D4, C2×He3, C3×SL2(𝔽3), C6×C12, D4×C32, Q8×C32, C4×He3, C3×C4.A4, C32×C4○D4, Q8⋊He3, C4○D4⋊He3
Quotients: C1, C2, C3, C6, C32, A4, C3×C6, C2×A4, He3, C3×A4, C4.A4, C2×He3, C6×A4, C32⋊A4, C3×C4.A4, C2×C32⋊A4, C4○D4⋊He3

Smallest permutation representation of C4○D4⋊He3
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 4 3 2)(5 6 7 8)(9 12 11 10)(13 14 15 16)(17 20 19 18)(21 22 23 24)(25 32 27 30)(26 29 28 31)(33 40 35 38)(34 37 36 39)(41 48 43 46)(42 45 44 47)(49 53 51 55)(50 54 52 56)(57 61 59 63)(58 62 60 64)(65 69 67 71)(66 70 68 72)
(1 5)(2 6)(3 7)(4 8)(9 15)(10 16)(11 13)(12 14)(17 23)(18 24)(19 21)(20 22)(25 32)(26 29)(27 30)(28 31)(33 40)(34 37)(35 38)(36 39)(41 48)(42 45)(43 46)(44 47)(49 51)(50 52)(57 59)(58 60)(65 67)(66 68)
(25 41 33)(26 42 34)(27 43 35)(28 44 36)(29 45 37)(30 46 38)(31 47 39)(32 48 40)(49 57 65)(50 58 66)(51 59 67)(52 60 68)(53 61 69)(54 62 70)(55 63 71)(56 64 72)
(1 19 11)(2 20 12)(3 17 9)(4 18 10)(5 21 13)(6 22 14)(7 23 15)(8 24 16)(25 41 33)(26 42 34)(27 43 35)(28 44 36)(29 45 37)(30 46 38)(31 47 39)(32 48 40)(49 65 57)(50 66 58)(51 67 59)(52 68 60)(53 69 61)(54 70 62)(55 71 63)(56 72 64)
(1 51 27)(2 52 28)(3 49 25)(4 50 26)(5 53 29)(6 54 30)(7 55 31)(8 56 32)(9 57 33)(10 58 34)(11 59 35)(12 60 36)(13 61 37)(14 62 38)(15 63 39)(16 64 40)(17 65 41)(18 66 42)(19 67 43)(20 68 44)(21 69 45)(22 70 46)(23 71 47)(24 72 48)

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

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

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

62 conjugacy classes

 class 1 2A 2B 3A 3B 3C 3D 3E ··· 3J 4A 4B 4C 6A 6B 6C 6D 6E ··· 6L 6M ··· 6R 12A 12B 12C 12D 12E 12F 12G 12H 12I ··· 12P 12Q ··· 12AB order 1 2 2 3 3 3 3 3 ··· 3 4 4 4 6 6 6 6 6 ··· 6 6 ··· 6 12 12 12 12 12 12 12 12 12 ··· 12 12 ··· 12 size 1 1 6 1 1 3 3 12 ··· 12 1 1 6 1 1 3 3 6 ··· 6 12 ··· 12 1 1 1 1 3 3 3 3 6 ··· 6 12 ··· 12

62 irreducible representations

 dim 1 1 1 1 1 1 2 2 3 3 3 3 3 3 3 3 6 type + + + + image C1 C2 C3 C3 C6 C6 C4.A4 C3×C4.A4 A4 C2×A4 He3 C3×A4 C2×He3 C6×A4 C32⋊A4 C2×C32⋊A4 C4○D4⋊He3 kernel C4○D4⋊He3 Q8⋊He3 C3×C4.A4 C32×C4○D4 C3×SL2(𝔽3) Q8×C32 C32 C3 C3×C12 C3×C6 C4○D4 C12 Q8 C6 C4 C2 C1 # reps 1 1 6 2 6 2 6 12 1 1 2 2 2 2 6 6 4

Matrix representation of C4○D4⋊He3 in GL5(𝔽13)

 5 0 0 0 0 0 5 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 8 0 0 0 0 0 5 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 0 12 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 3 0 0 0 0 0 3 0 0 0 0 0 1 0 0 0 0 0 9 0 0 0 0 0 3
,
 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
,
 2 3 0 0 0 2 10 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0

G:=sub<GL(5,GF(13))| [5,0,0,0,0,0,5,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[8,0,0,0,0,0,5,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[0,12,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[3,0,0,0,0,0,3,0,0,0,0,0,1,0,0,0,0,0,9,0,0,0,0,0,3],[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],[2,2,0,0,0,3,10,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0] >;

C4○D4⋊He3 in GAP, Magma, Sage, TeX

C_4\circ D_4\rtimes {\rm He}_3
% in TeX

G:=Group("C4oD4:He3");
// GroupNames label

G:=SmallGroup(432,339);
// by ID

G=gap.SmallGroup(432,339);
# by ID

G:=PCGroup([7,-2,-3,-3,-3,-2,2,-2,1512,261,1901,172,3414,285,124]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^4=c^2=d^3=e^3=f^3=1,b^2=a^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,c*b*c=a^2*b,b*d=d*b,b*e=e*b,f*b*f^-1=a^-1*b*c,c*d=d*c,c*e=e*c,f*c*f^-1=a^-1*b,d*e=e*d,f*d*f^-1=d*e^-1,e*f=f*e>;
// generators/relations

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