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## G = C4○D4×He3order 432 = 24·33

### Direct product of C4○D4 and He3

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C4○D4×He3
 Chief series C1 — C3 — C6 — C3×C6 — C2×He3 — C22×He3 — D4×He3 — C4○D4×He3
 Lower central C1 — C6 — C4○D4×He3
 Upper central C1 — C12 — C4○D4×He3

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, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, fdf-1=de-1, ef=fe >

Subgroups: 437 in 220 conjugacy classes, 119 normal (15 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C6, C6, C2×C4, D4, Q8, C32, C12, C12, C12, C2×C6, C2×C6, C4○D4, C3×C6, C3×C6, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C3×Q8, He3, C3×C12, C62, C3×C4○D4, C3×C4○D4, C2×He3, C2×He3, C6×C12, D4×C32, Q8×C32, C4×He3, C4×He3, C22×He3, C32×C4○D4, C2×C4×He3, D4×He3, Q8×He3, C4○D4×He3
Quotients: C1, C2, C3, C22, C6, C23, C32, C2×C6, C4○D4, C3×C6, C22×C6, He3, C62, C3×C4○D4, C2×He3, C2×C62, C22×He3, C32×C4○D4, C23×He3, 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 41 3 43)(2 42 4 44)(5 48 7 46)(6 45 8 47)(9 63 11 61)(10 64 12 62)(13 56 15 54)(14 53 16 55)(17 39 19 37)(18 40 20 38)(21 57 23 59)(22 58 24 60)(25 65 27 67)(26 66 28 68)(29 52 31 50)(30 49 32 51)(33 69 35 71)(34 70 36 72)
(17 19)(18 20)(21 23)(22 24)(41 43)(42 44)(45 47)(46 48)(49 51)(50 52)(53 55)(54 56)(61 63)(62 64)(65 67)(66 68)(69 71)(70 72)
(1 26 35)(2 27 36)(3 28 33)(4 25 34)(5 31 38)(6 32 39)(7 29 40)(8 30 37)(9 15 60)(10 16 57)(11 13 58)(12 14 59)(17 47 49)(18 48 50)(19 45 51)(20 46 52)(21 62 53)(22 63 54)(23 64 55)(24 61 56)(41 66 71)(42 67 72)(43 68 69)(44 65 70)
(1 14 29)(2 15 30)(3 16 31)(4 13 32)(5 33 10)(6 34 11)(7 35 12)(8 36 9)(17 67 22)(18 68 23)(19 65 24)(20 66 21)(25 58 39)(26 59 40)(27 60 37)(28 57 38)(41 53 52)(42 54 49)(43 55 50)(44 56 51)(45 70 61)(46 71 62)(47 72 63)(48 69 64)
(1 35 40)(2 36 37)(3 33 38)(4 34 39)(5 57 31)(6 58 32)(7 59 29)(8 60 30)(9 27 15)(10 28 16)(11 25 13)(12 26 14)(17 42 72)(18 43 69)(19 44 70)(20 41 71)(21 52 46)(22 49 47)(23 50 48)(24 51 45)(53 62 66)(54 63 67)(55 64 68)(56 61 65)

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

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

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

110 conjugacy classes

 class 1 2A 2B 2C 2D 3A 3B 3C ··· 3J 4A 4B 4C 4D 4E 6A 6B 6C ··· 6H 6I ··· 6P 6Q ··· 6AN 12A 12B 12C 12D 12E ··· 12J 12K ··· 12Z 12AA ··· 12AX order 1 2 2 2 2 3 3 3 ··· 3 4 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 2 2 2 1 1 3 ··· 3 1 1 2 2 2 1 1 2 ··· 2 3 ··· 3 6 ··· 6 1 1 1 1 2 ··· 2 3 ··· 3 6 ··· 6

110 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 3 3 3 3 6 type + + + + image C1 C2 C2 C2 C3 C6 C6 C6 C4○D4 C3×C4○D4 He3 C2×He3 C2×He3 C2×He3 C4○D4×He3 kernel C4○D4×He3 C2×C4×He3 D4×He3 Q8×He3 C32×C4○D4 C6×C12 D4×C32 Q8×C32 He3 C32 C4○D4 C2×C4 D4 Q8 C1 # reps 1 3 3 1 8 24 24 8 2 16 2 6 6 2 4

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

 8 0 0 0 0 0 8 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12
,
 12 1 0 0 0 11 1 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12
,
 1 12 0 0 0 0 12 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 0 1 0 0 0 0 0 1 0 0 1 0 0
,
 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
,
 9 0 0 0 0 0 9 0 0 0 0 0 0 0 9 0 0 1 0 0 0 0 0 3 0

G:=sub<GL(5,GF(13))| [8,0,0,0,0,0,8,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[12,11,0,0,0,1,1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,12,12,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0],[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],[9,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,9,0,0] >;

C4○D4×He3 in GAP, Magma, Sage, TeX

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

G:=Group("C4oD4xHe3");
// GroupNames label

G:=SmallGroup(432,410);
// by ID

G=gap.SmallGroup(432,410);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-3,-2,-3,1037,394,760]);
// 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,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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