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## G = C2×Q8⋊D15order 480 = 25·3·5

### Direct product of C2 and Q8⋊D15

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — C5×SL2(𝔽3) — C2×Q8⋊D15
 Chief series C1 — C2 — Q8 — C5×Q8 — C5×SL2(𝔽3) — Q8⋊D15 — C2×Q8⋊D15
 Lower central C5×SL2(𝔽3) — C2×Q8⋊D15
 Upper central C1 — C22

Generators and relations for C2×Q8⋊D15
G = < a,b,c,d,e | a2=b4=d15=e2=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, dbd-1=c, ebe=b-1c, dcd-1=bc, ece=b2c, ede=d-1 >

Subgroups: 994 in 102 conjugacy classes, 21 normal (15 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C5, S3, C6, C8, C2×C4, D4, Q8, Q8, C23, D5, C10, C10, D6, C2×C6, C15, C2×C8, SD16, C2×D4, C2×Q8, C20, D10, C2×C10, SL2(𝔽3), C22×S3, D15, C30, C2×SD16, C52C8, D20, C2×C20, C5×Q8, C5×Q8, C22×D5, GL2(𝔽3), C2×SL2(𝔽3), D30, C2×C30, C2×C52C8, Q8⋊D5, C2×D20, Q8×C10, C2×GL2(𝔽3), C5×SL2(𝔽3), C22×D15, C2×Q8⋊D5, Q8⋊D15, C10×SL2(𝔽3), C2×Q8⋊D15
Quotients: C1, C2, C22, S3, D5, D6, D10, S4, D15, GL2(𝔽3), C2×S4, D30, C2×GL2(𝔽3), C5⋊S4, Q8⋊D15, C2×C5⋊S4, C2×Q8⋊D15

Smallest permutation representation of C2×Q8⋊D15
On 80 points
Generators in S80
(1 17)(2 18)(3 19)(4 20)(5 16)(6 12)(7 13)(8 14)(9 15)(10 11)(21 74)(22 75)(23 76)(24 77)(25 78)(26 79)(27 80)(28 66)(29 67)(30 68)(31 69)(32 70)(33 71)(34 72)(35 73)(36 65)(37 51)(38 52)(39 53)(40 54)(41 55)(42 56)(43 57)(44 58)(45 59)(46 60)(47 61)(48 62)(49 63)(50 64)
(1 66 9 40)(2 72 10 46)(3 78 6 37)(4 69 7 43)(5 75 8 49)(11 60 18 34)(12 51 19 25)(13 57 20 31)(14 63 16 22)(15 54 17 28)(21 52 62 26)(23 33 64 59)(24 55 65 29)(27 58 53 32)(30 61 56 35)(36 67 77 41)(38 48 79 74)(39 70 80 44)(42 73 68 47)(45 76 71 50)
(1 71 9 45)(2 77 10 36)(3 68 6 42)(4 74 7 48)(5 80 8 39)(11 65 18 24)(12 56 19 30)(13 62 20 21)(14 53 16 27)(15 59 17 33)(22 32 63 58)(23 54 64 28)(25 35 51 61)(26 57 52 31)(29 60 55 34)(37 47 78 73)(38 69 79 43)(40 50 66 76)(41 72 67 46)(44 75 70 49)
(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 16)(2 20)(3 19)(4 18)(5 17)(6 12)(7 11)(8 15)(9 14)(10 13)(21 36)(22 50)(23 49)(24 48)(25 47)(26 46)(27 45)(28 44)(29 43)(30 42)(31 41)(32 40)(33 39)(34 38)(35 37)(51 73)(52 72)(53 71)(54 70)(55 69)(56 68)(57 67)(58 66)(59 80)(60 79)(61 78)(62 77)(63 76)(64 75)(65 74)

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

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

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

44 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 5A 5B 6A 6B 6C 8A 8B 8C 8D 10A ··· 10F 15A 15B 15C 15D 20A 20B 20C 20D 30A ··· 30L order 1 2 2 2 2 2 3 4 4 5 5 6 6 6 8 8 8 8 10 ··· 10 15 15 15 15 20 20 20 20 30 ··· 30 size 1 1 1 1 60 60 8 6 6 2 2 8 8 8 30 30 30 30 2 ··· 2 8 8 8 8 12 12 12 12 8 ··· 8

44 irreducible representations

 dim 1 1 1 2 2 2 2 2 2 2 3 3 4 4 6 6 type + + + + + + + + + + + + + + + image C1 C2 C2 S3 D5 D6 D10 D15 GL2(𝔽3) D30 S4 C2×S4 GL2(𝔽3) Q8⋊D15 C5⋊S4 C2×C5⋊S4 kernel C2×Q8⋊D15 Q8⋊D15 C10×SL2(𝔽3) Q8×C10 C2×SL2(𝔽3) C5×Q8 SL2(𝔽3) C2×Q8 C10 Q8 C2×C10 C10 C10 C2 C22 C2 # reps 1 2 1 1 2 1 2 4 4 4 2 2 2 12 2 2

Matrix representation of C2×Q8⋊D15 in GL4(𝔽241) generated by

 240 0 0 0 0 240 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 106 67 0 0 174 135
,
 1 0 0 0 0 1 0 0 0 0 68 105 0 0 174 173
,
 8 11 0 0 197 30 0 0 0 0 67 68 0 0 135 173
,
 30 237 0 0 44 211 0 0 0 0 68 105 0 0 135 173
G:=sub<GL(4,GF(241))| [240,0,0,0,0,240,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,106,174,0,0,67,135],[1,0,0,0,0,1,0,0,0,0,68,174,0,0,105,173],[8,197,0,0,11,30,0,0,0,0,67,135,0,0,68,173],[30,44,0,0,237,211,0,0,0,0,68,135,0,0,105,173] >;

C2×Q8⋊D15 in GAP, Magma, Sage, TeX

C_2\times Q_8\rtimes D_{15}
% in TeX

G:=Group("C2xQ8:D15");
// GroupNames label

G:=SmallGroup(480,1028);
// by ID

G=gap.SmallGroup(480,1028);
# by ID

G:=PCGroup([7,-2,-2,-3,-5,-2,2,-2,170,1347,4204,3168,172,2525,1909,285,124]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^4=d^15=e^2=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,d*b*d^-1=c,e*b*e=b^-1*c,d*c*d^-1=b*c,e*c*e=b^2*c,e*d*e=d^-1>;
// generators/relations

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