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## G = (Q8×He3)⋊C2order 432 = 24·33

### 4th semidirect product of Q8×He3 and C2 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — (Q8×He3)⋊C2
 Chief series C1 — C3 — C32 — C3×C6 — C2×He3 — C2×C32⋊C6 — C4×C32⋊C6 — (Q8×He3)⋊C2
 Lower central C32 — C3×C6 — (Q8×He3)⋊C2
 Upper central C1 — C2 — 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=faf=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ece-1=cd-1, fcf=c-1, de=ed, fdf=d-1, ef=fe >

Subgroups: 709 in 152 conjugacy classes, 52 normal (19 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C2×C4, D4, Q8, C32, C32, Dic3, C12, C12, D6, C2×C6, C4○D4, C3×S3, C3⋊S3, C3×C6, C3×C6, C4×S3, D12, C2×C12, C3×D4, C3×Q8, C3×Q8, He3, C3×Dic3, C3⋊Dic3, C3×C12, C3×C12, S3×C6, C2×C3⋊S3, Q83S3, C3×C4○D4, C32⋊C6, C2×He3, S3×C12, C3×D12, C4×C3⋊S3, C12⋊S3, Q8×C32, Q8×C32, C32⋊C12, C4×He3, C2×C32⋊C6, C3×Q83S3, C12.26D6, C4×C32⋊C6, He34D4, Q8×He3, (Q8×He3)⋊C2
Quotients: C1, C2, C3, C22, S3, C6, C23, D6, C2×C6, C4○D4, C3×S3, C22×S3, C22×C6, S3×C6, Q83S3, C3×C4○D4, C32⋊C6, S3×C2×C6, C2×C32⋊C6, C3×Q83S3, C22×C32⋊C6, (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 9 3 11)(2 12 4 10)(5 42 7 44)(6 41 8 43)(13 53 15 55)(14 56 16 54)(17 57 19 59)(18 60 20 58)(21 68 23 66)(22 67 24 65)(25 49 27 51)(26 52 28 50)(29 46 31 48)(30 45 32 47)(33 63 35 61)(34 62 36 64)(37 72 39 70)(38 71 40 69)
(1 16 8)(2 13 5)(3 14 6)(4 15 7)(9 54 43)(10 55 44)(11 56 41)(12 53 42)(17 45 65)(18 46 66)(19 47 67)(20 48 68)(21 60 31)(22 57 32)(23 58 29)(24 59 30)(25 71 61)(26 72 62)(27 69 63)(28 70 64)(33 49 40)(34 50 37)(35 51 38)(36 52 39)
(1 33 29)(2 34 30)(3 35 31)(4 36 32)(5 37 59)(6 38 60)(7 39 57)(8 40 58)(9 63 46)(10 64 47)(11 61 48)(12 62 45)(13 50 24)(14 51 21)(15 52 22)(16 49 23)(17 42 72)(18 43 69)(19 44 70)(20 41 71)(25 68 56)(26 65 53)(27 66 54)(28 67 55)
(5 59 37)(6 60 38)(7 57 39)(8 58 40)(13 50 24)(14 51 21)(15 52 22)(16 49 23)(17 72 42)(18 69 43)(19 70 44)(20 71 41)(25 68 56)(26 65 53)(27 66 54)(28 67 55)
(1 12)(2 11)(3 10)(4 9)(5 56)(6 55)(7 54)(8 53)(13 41)(14 44)(15 43)(16 42)(17 49)(18 52)(19 51)(20 50)(21 70)(22 69)(23 72)(24 71)(25 59)(26 58)(27 57)(28 60)(29 62)(30 61)(31 64)(32 63)(33 45)(34 48)(35 47)(36 46)(37 68)(38 67)(39 66)(40 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), (1,9,3,11)(2,12,4,10)(5,42,7,44)(6,41,8,43)(13,53,15,55)(14,56,16,54)(17,57,19,59)(18,60,20,58)(21,68,23,66)(22,67,24,65)(25,49,27,51)(26,52,28,50)(29,46,31,48)(30,45,32,47)(33,63,35,61)(34,62,36,64)(37,72,39,70)(38,71,40,69), (1,16,8)(2,13,5)(3,14,6)(4,15,7)(9,54,43)(10,55,44)(11,56,41)(12,53,42)(17,45,65)(18,46,66)(19,47,67)(20,48,68)(21,60,31)(22,57,32)(23,58,29)(24,59,30)(25,71,61)(26,72,62)(27,69,63)(28,70,64)(33,49,40)(34,50,37)(35,51,38)(36,52,39), (1,33,29)(2,34,30)(3,35,31)(4,36,32)(5,37,59)(6,38,60)(7,39,57)(8,40,58)(9,63,46)(10,64,47)(11,61,48)(12,62,45)(13,50,24)(14,51,21)(15,52,22)(16,49,23)(17,42,72)(18,43,69)(19,44,70)(20,41,71)(25,68,56)(26,65,53)(27,66,54)(28,67,55), (5,59,37)(6,60,38)(7,57,39)(8,58,40)(13,50,24)(14,51,21)(15,52,22)(16,49,23)(17,72,42)(18,69,43)(19,70,44)(20,71,41)(25,68,56)(26,65,53)(27,66,54)(28,67,55), (1,12)(2,11)(3,10)(4,9)(5,56)(6,55)(7,54)(8,53)(13,41)(14,44)(15,43)(16,42)(17,49)(18,52)(19,51)(20,50)(21,70)(22,69)(23,72)(24,71)(25,59)(26,58)(27,57)(28,60)(29,62)(30,61)(31,64)(32,63)(33,45)(34,48)(35,47)(36,46)(37,68)(38,67)(39,66)(40,65)>;

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

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

50 conjugacy classes

 class 1 2A 2B 2C 2D 3A 3B 3C 3D 3E 3F 4A 4B 4C 4D 4E 6A 6B 6C 6D 6E 6F 6G ··· 6L 12A 12B 12C 12D ··· 12I 12J 12K 12L 12M 12N ··· 12V order 1 2 2 2 2 3 3 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 ··· 12 size 1 1 18 18 18 2 3 3 6 6 6 2 2 2 9 9 2 3 3 6 6 6 18 ··· 18 4 4 4 6 ··· 6 9 9 9 9 12 ··· 12

50 irreducible representations

 dim 1 1 1 1 1 1 1 1 12 2 2 2 2 2 2 4 4 6 6 type + + + + + + + + + + image C1 C2 C2 C2 C3 C6 C6 C6 (Q8×He3)⋊C2 S3 D6 C4○D4 C3×S3 S3×C6 C3×C4○D4 Q8⋊3S3 C3×Q8⋊3S3 C32⋊C6 C2×C32⋊C6 kernel (Q8×He3)⋊C2 C4×C32⋊C6 He3⋊4D4 Q8×He3 C12.26D6 C4×C3⋊S3 C12⋊S3 Q8×C32 C1 Q8×C32 C3×C12 He3 C3×Q8 C12 C32 C32 C3 Q8 C4 # reps 1 3 3 1 2 6 6 2 1 1 3 2 2 6 4 1 2 1 3

Matrix representation of (Q8×He3)⋊C2 in GL10(𝔽13)

 8 0 0 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 12
,
 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 0 0 0 0 12 12 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 12 12 0 0 0 0 0 0 0 0 0 0 0 0 12 1 0 0 0 0 0 0 1 1 11 12 0 0 0 0 0 0 0 0 12 0 1 0 0 0 0 0 0 0 12 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 0 1 0 12 0 0 0
,
 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 12 1 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 12 0 0 1 0 0 0 0 0 0 0 1 12 12 0 0 0 0 0 0 12 0 0 0 0 1 0 0 0 0 0 1 0 0 12 12
,
 9 0 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 12 12 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 0 12 12
,
 0 0 0 8 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 12 12 0 0 1 1 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12 12 1 1 0 0

G:=sub<GL(10,GF(13))| [8,0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12],[0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[0,12,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,11,12,12,12,12,0,0,0,0,1,12,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,12,12,12,0,12,0,0,0,0,0,1,0,0,1,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,12],[9,0,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,0,1,0,1,0,0,1,0,0,0,0,0,1,1,0,0,1,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,12],[0,0,0,5,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,12,0,12,0,0,0,0,12,0,0,12,0,12,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,0,0,1,0,0] >;

(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(432,369);
// by ID

G=gap.SmallGroup(432,369);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,176,590,303,142,4037,1034,14118]);
// 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=f*a*f=a^-1,a*c=c*a,a*d=d*a,a*e=e*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,f*d*f=d^-1,e*f=f*e>;
// generators/relations

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