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## G = Q8×He3⋊C2order 432 = 24·33

### Direct product of Q8 and He3⋊C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C2×He3 — Q8×He3⋊C2
 Chief series C1 — C3 — C32 — He3 — C2×He3 — C2×He3⋊C2 — C4×He3⋊C2 — Q8×He3⋊C2
 Lower central He3 — C2×He3 — Q8×He3⋊C2
 Upper central C1 — C6 — C3×Q8

Generators and relations for Q8×He3⋊C2
G = < a,b,c,d,e,f | a4=c3=d3=e3=f2=1, b2=a2, bab-1=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ece-1=cd-1, fcf=c-1, de=ed, df=fd, fef=e-1 >

Subgroups: 673 in 209 conjugacy classes, 55 normal (11 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C2×C4, Q8, Q8, C32, Dic3, C12, C12, D6, C2×C6, C2×Q8, C3×S3, C3×C6, Dic6, C4×S3, C2×C12, C3×Q8, C3×Q8, He3, C3×Dic3, C3×C12, S3×C6, S3×Q8, C6×Q8, He3⋊C2, C2×He3, C3×Dic6, S3×C12, Q8×C32, He33C4, C4×He3, C2×He3⋊C2, C3×S3×Q8, He34Q8, C4×He3⋊C2, Q8×He3, Q8×He3⋊C2
Quotients: C1, C2, C22, S3, Q8, C23, D6, C2×Q8, C3⋊S3, C22×S3, C2×C3⋊S3, S3×Q8, He3⋊C2, C22×C3⋊S3, C2×He3⋊C2, Q8×C3⋊S3, C22×He3⋊C2, Q8×He3⋊C2

Smallest permutation representation of Q8×He3⋊C2
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 18 3 20)(2 17 4 19)(5 41 7 43)(6 44 8 42)(9 51 11 49)(10 50 12 52)(13 68 15 66)(14 67 16 65)(21 54 23 56)(22 53 24 55)(25 58 27 60)(26 57 28 59)(29 46 31 48)(30 45 32 47)(33 69 35 71)(34 72 36 70)(37 62 39 64)(38 61 40 63)
(1 40 7)(2 37 8)(3 38 5)(4 39 6)(9 46 66)(10 47 67)(11 48 68)(12 45 65)(13 51 31)(14 52 32)(15 49 29)(16 50 30)(17 62 42)(18 63 43)(19 64 44)(20 61 41)(21 58 33)(22 59 34)(23 60 35)(24 57 36)(25 71 56)(26 72 53)(27 69 54)(28 70 55)
(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 33 58)(2 34 59)(3 35 60)(4 36 57)(5 38 31)(6 39 32)(7 40 29)(8 37 30)(9 66 54)(10 67 55)(11 68 56)(12 65 53)(13 23 51)(14 24 52)(15 21 49)(16 22 50)(17 72 26)(18 69 27)(19 70 28)(20 71 25)(41 61 48)(42 62 45)(43 63 46)(44 64 47)
(1 3)(2 4)(5 40)(6 37)(7 38)(8 39)(9 68)(10 65)(11 66)(12 67)(13 49)(14 50)(15 51)(16 52)(17 19)(18 20)(21 23)(22 24)(25 69)(26 70)(27 71)(28 72)(29 31)(30 32)(33 60)(34 57)(35 58)(36 59)(41 63)(42 64)(43 61)(44 62)(45 47)(46 48)(53 55)(54 56)

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

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

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

50 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 3F 4A 4B 4C 4D 4E 4F 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 12A ··· 12F 12G ··· 12R 12S ··· 12X order 1 2 2 2 3 3 3 3 3 3 4 4 4 4 4 4 6 6 6 6 6 6 6 6 6 6 12 ··· 12 12 ··· 12 12 ··· 12 size 1 1 9 9 1 1 6 6 6 6 2 2 2 18 18 18 1 1 6 6 6 6 9 9 9 9 2 ··· 2 12 ··· 12 18 ··· 18

50 irreducible representations

 dim 1 1 1 1 2 2 2 3 3 4 6 type + + + + + - + - image C1 C2 C2 C2 S3 Q8 D6 He3⋊C2 C2×He3⋊C2 S3×Q8 Q8×He3⋊C2 kernel Q8×He3⋊C2 He3⋊4Q8 C4×He3⋊C2 Q8×He3 Q8×C32 He3⋊C2 C3×C12 Q8 C4 C32 C1 # reps 1 3 3 1 4 2 12 4 12 4 4

Matrix representation of Q8×He3⋊C2 in GL5(𝔽13)

 1 5 0 0 0 10 12 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 12 2 0 0 0 12 1 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 12 0 0 0 0 0 12 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
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 4 0 0 3 0 0
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 12

G:=sub<GL(5,GF(13))| [1,10,0,0,0,5,12,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[12,12,0,0,0,2,1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,12,0,0,1,0,0],[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],[1,0,0,0,0,0,1,0,0,0,0,0,0,0,3,0,0,12,0,0,0,0,0,4,0],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,12] >;

Q8×He3⋊C2 in GAP, Magma, Sage, TeX

Q_8\times {\rm He}_3\rtimes C_2
% in TeX

G:=Group("Q8xHe3:C2");
// GroupNames label

G:=SmallGroup(432,394);
// by ID

G=gap.SmallGroup(432,394);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,64,135,58,1124,4037,537]);
// Polycyclic

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

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