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## G = C5×Q8.D6order 480 = 25·3·5

### Direct product of C5 and Q8.D6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — SL2(𝔽3) — C5×Q8.D6
 Chief series C1 — C2 — Q8 — SL2(𝔽3) — C5×SL2(𝔽3) — C5×GL2(𝔽3) — C5×Q8.D6
 Lower central SL2(𝔽3) — C5×Q8.D6
 Upper central C1 — C10 — C2×C10

Generators and relations for C5×Q8.D6
G = < a,b,c,d,e | a5=b4=d6=1, c2=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe-1=b-1, dbd-1=bc, dcd-1=b, ece-1=b-1c, ede-1=b2d-1 >

Subgroups: 266 in 78 conjugacy classes, 20 normal (all characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C8, C2×C4, D4, Q8, Q8, C10, C10, Dic3, D6, C2×C6, C15, M4(2), SD16, Q16, C2×Q8, C4○D4, C20, C2×C10, C2×C10, SL2(𝔽3), C3⋊D4, C5×S3, C30, C8.C22, C40, C2×C20, C5×D4, C5×Q8, C5×Q8, CSU2(𝔽3), GL2(𝔽3), C2×SL2(𝔽3), C5×Dic3, S3×C10, C2×C30, C5×M4(2), C5×SD16, C5×Q16, Q8×C10, C5×C4○D4, Q8.D6, C5×SL2(𝔽3), C5×C3⋊D4, C5×C8.C22, C5×CSU2(𝔽3), C5×GL2(𝔽3), C10×SL2(𝔽3), C5×Q8.D6
Quotients: C1, C2, C22, C5, S3, C10, D6, C2×C10, S4, C5×S3, C2×S4, S3×C10, Q8.D6, C5×S4, C10×S4, C5×Q8.D6

Smallest permutation representation of C5×Q8.D6
On 80 points
Generators in S80
(1 12 18 10 4)(2 11 17 9 3)(5 13 20 16 8)(6 14 19 15 7)(21 38 66 72 39)(22 33 67 73 40)(23 34 68 74 41)(24 35 63 69 42)(25 36 64 70 43)(26 37 65 71 44)(27 50 79 56 60)(28 45 80 51 61)(29 46 75 52 62)(30 47 76 53 57)(31 48 77 54 58)(32 49 78 55 59)
(1 74 5 50)(2 71 6 47)(3 65 7 30)(4 68 8 27)(9 37 15 57)(10 34 16 60)(11 44 14 76)(12 41 13 79)(17 26 19 53)(18 23 20 56)(21 52 54 25)(22 24 55 51)(28 67 63 32)(29 31 64 66)(33 35 59 61)(36 38 62 58)(39 75 77 43)(40 42 78 80)(45 73 69 49)(46 48 70 72)
(1 72 5 48)(2 69 6 45)(3 63 7 28)(4 66 8 31)(9 35 15 61)(10 38 16 58)(11 42 14 80)(12 39 13 77)(17 24 19 51)(18 21 20 54)(22 53 55 26)(23 25 56 52)(27 29 68 64)(30 32 65 67)(33 57 59 37)(34 36 60 62)(40 76 78 44)(41 43 79 75)(46 74 70 50)(47 49 71 73)
(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 73 74)(75 76 77 78 79 80)
(1 6 5 2)(3 4 7 8)(9 10 15 16)(11 12 14 13)(17 18 19 20)(21 22 54 55)(23 26 56 53)(24 52 51 25)(27 30 68 65)(28 64 63 29)(31 32 66 67)(33 58 59 38)(34 37 60 57)(35 62 61 36)(39 40 77 78)(41 44 79 76)(42 75 80 43)(45 70 69 46)(47 74 71 50)(48 49 72 73)

G:=sub<Sym(80)| (1,12,18,10,4)(2,11,17,9,3)(5,13,20,16,8)(6,14,19,15,7)(21,38,66,72,39)(22,33,67,73,40)(23,34,68,74,41)(24,35,63,69,42)(25,36,64,70,43)(26,37,65,71,44)(27,50,79,56,60)(28,45,80,51,61)(29,46,75,52,62)(30,47,76,53,57)(31,48,77,54,58)(32,49,78,55,59), (1,74,5,50)(2,71,6,47)(3,65,7,30)(4,68,8,27)(9,37,15,57)(10,34,16,60)(11,44,14,76)(12,41,13,79)(17,26,19,53)(18,23,20,56)(21,52,54,25)(22,24,55,51)(28,67,63,32)(29,31,64,66)(33,35,59,61)(36,38,62,58)(39,75,77,43)(40,42,78,80)(45,73,69,49)(46,48,70,72), (1,72,5,48)(2,69,6,45)(3,63,7,28)(4,66,8,31)(9,35,15,61)(10,38,16,58)(11,42,14,80)(12,39,13,77)(17,24,19,51)(18,21,20,54)(22,53,55,26)(23,25,56,52)(27,29,68,64)(30,32,65,67)(33,57,59,37)(34,36,60,62)(40,76,78,44)(41,43,79,75)(46,74,70,50)(47,49,71,73), (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)(69,70,71,72,73,74)(75,76,77,78,79,80), (1,6,5,2)(3,4,7,8)(9,10,15,16)(11,12,14,13)(17,18,19,20)(21,22,54,55)(23,26,56,53)(24,52,51,25)(27,30,68,65)(28,64,63,29)(31,32,66,67)(33,58,59,38)(34,37,60,57)(35,62,61,36)(39,40,77,78)(41,44,79,76)(42,75,80,43)(45,70,69,46)(47,74,71,50)(48,49,72,73)>;

G:=Group( (1,12,18,10,4)(2,11,17,9,3)(5,13,20,16,8)(6,14,19,15,7)(21,38,66,72,39)(22,33,67,73,40)(23,34,68,74,41)(24,35,63,69,42)(25,36,64,70,43)(26,37,65,71,44)(27,50,79,56,60)(28,45,80,51,61)(29,46,75,52,62)(30,47,76,53,57)(31,48,77,54,58)(32,49,78,55,59), (1,74,5,50)(2,71,6,47)(3,65,7,30)(4,68,8,27)(9,37,15,57)(10,34,16,60)(11,44,14,76)(12,41,13,79)(17,26,19,53)(18,23,20,56)(21,52,54,25)(22,24,55,51)(28,67,63,32)(29,31,64,66)(33,35,59,61)(36,38,62,58)(39,75,77,43)(40,42,78,80)(45,73,69,49)(46,48,70,72), (1,72,5,48)(2,69,6,45)(3,63,7,28)(4,66,8,31)(9,35,15,61)(10,38,16,58)(11,42,14,80)(12,39,13,77)(17,24,19,51)(18,21,20,54)(22,53,55,26)(23,25,56,52)(27,29,68,64)(30,32,65,67)(33,57,59,37)(34,36,60,62)(40,76,78,44)(41,43,79,75)(46,74,70,50)(47,49,71,73), (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)(69,70,71,72,73,74)(75,76,77,78,79,80), (1,6,5,2)(3,4,7,8)(9,10,15,16)(11,12,14,13)(17,18,19,20)(21,22,54,55)(23,26,56,53)(24,52,51,25)(27,30,68,65)(28,64,63,29)(31,32,66,67)(33,58,59,38)(34,37,60,57)(35,62,61,36)(39,40,77,78)(41,44,79,76)(42,75,80,43)(45,70,69,46)(47,74,71,50)(48,49,72,73) );

G=PermutationGroup([[(1,12,18,10,4),(2,11,17,9,3),(5,13,20,16,8),(6,14,19,15,7),(21,38,66,72,39),(22,33,67,73,40),(23,34,68,74,41),(24,35,63,69,42),(25,36,64,70,43),(26,37,65,71,44),(27,50,79,56,60),(28,45,80,51,61),(29,46,75,52,62),(30,47,76,53,57),(31,48,77,54,58),(32,49,78,55,59)], [(1,74,5,50),(2,71,6,47),(3,65,7,30),(4,68,8,27),(9,37,15,57),(10,34,16,60),(11,44,14,76),(12,41,13,79),(17,26,19,53),(18,23,20,56),(21,52,54,25),(22,24,55,51),(28,67,63,32),(29,31,64,66),(33,35,59,61),(36,38,62,58),(39,75,77,43),(40,42,78,80),(45,73,69,49),(46,48,70,72)], [(1,72,5,48),(2,69,6,45),(3,63,7,28),(4,66,8,31),(9,35,15,61),(10,38,16,58),(11,42,14,80),(12,39,13,77),(17,24,19,51),(18,21,20,54),(22,53,55,26),(23,25,56,52),(27,29,68,64),(30,32,65,67),(33,57,59,37),(34,36,60,62),(40,76,78,44),(41,43,79,75),(46,74,70,50),(47,49,71,73)], [(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),(69,70,71,72,73,74),(75,76,77,78,79,80)], [(1,6,5,2),(3,4,7,8),(9,10,15,16),(11,12,14,13),(17,18,19,20),(21,22,54,55),(23,26,56,53),(24,52,51,25),(27,30,68,65),(28,64,63,29),(31,32,66,67),(33,58,59,38),(34,37,60,57),(35,62,61,36),(39,40,77,78),(41,44,79,76),(42,75,80,43),(45,70,69,46),(47,74,71,50),(48,49,72,73)]])

65 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 5A 5B 5C 5D 6A 6B 6C 8A 8B 10A 10B 10C 10D 10E 10F 10G 10H 10I 10J 10K 10L 15A 15B 15C 15D 20A ··· 20H 20I 20J 20K 20L 30A ··· 30L 40A ··· 40H order 1 2 2 2 3 4 4 4 5 5 5 5 6 6 6 8 8 10 10 10 10 10 10 10 10 10 10 10 10 15 15 15 15 20 ··· 20 20 20 20 20 30 ··· 30 40 ··· 40 size 1 1 2 12 8 6 6 12 1 1 1 1 8 8 8 12 12 1 1 1 1 2 2 2 2 12 12 12 12 8 8 8 8 6 ··· 6 12 12 12 12 8 ··· 8 12 ··· 12

65 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 3 3 3 3 4 4 4 type + + + + + + + + - image C1 C2 C2 C2 C5 C10 C10 C10 S3 D6 C5×S3 S3×C10 S4 C2×S4 C5×S4 C10×S4 Q8.D6 Q8.D6 C5×Q8.D6 kernel C5×Q8.D6 C5×CSU2(𝔽3) C5×GL2(𝔽3) C10×SL2(𝔽3) Q8.D6 CSU2(𝔽3) GL2(𝔽3) C2×SL2(𝔽3) Q8×C10 C5×Q8 C2×Q8 Q8 C2×C10 C10 C22 C2 C5 C5 C1 # reps 1 1 1 1 4 4 4 4 1 1 4 4 2 2 8 8 1 2 12

Matrix representation of C5×Q8.D6 in GL4(𝔽241) generated by

 98 0 0 0 0 98 0 0 0 0 98 0 0 0 0 98
,
 0 0 1 0 0 0 0 1 240 0 0 0 0 240 0 0
,
 0 240 0 0 1 0 0 0 0 0 0 1 0 0 240 0
,
 0 171 70 171 70 171 0 70 171 171 171 0 70 0 171 171
,
 0 70 171 70 70 70 70 0 171 70 0 171 70 0 171 171
G:=sub<GL(4,GF(241))| [98,0,0,0,0,98,0,0,0,0,98,0,0,0,0,98],[0,0,240,0,0,0,0,240,1,0,0,0,0,1,0,0],[0,1,0,0,240,0,0,0,0,0,0,240,0,0,1,0],[0,70,171,70,171,171,171,0,70,0,171,171,171,70,0,171],[0,70,171,70,70,70,70,0,171,70,0,171,70,0,171,171] >;

C5×Q8.D6 in GAP, Magma, Sage, TeX

C_5\times Q_8.D_6
% in TeX

G:=Group("C5xQ8.D6");
// GroupNames label

G:=SmallGroup(480,1018);
// by ID

G=gap.SmallGroup(480,1018);
# by ID

G:=PCGroup([7,-2,-2,-5,-3,-2,2,-2,3389,1123,4204,655,172,2525,404,285,124]);
// Polycyclic

G:=Group<a,b,c,d,e|a^5=b^4=d^6=1,c^2=e^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=e*b*e^-1=b^-1,d*b*d^-1=b*c,d*c*d^-1=b,e*c*e^-1=b^-1*c,e*d*e^-1=b^2*d^-1>;
// generators/relations

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