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## G = C22×D15order 120 = 23·3·5

### Direct product of C22 and D15

Aliases: C22×D15, C102D6, C62D10, C152C23, C302C22, (C2×C6)⋊3D5, (C2×C10)⋊5S3, (C2×C30)⋊3C2, C52(C22×S3), C32(C22×D5), SmallGroup(120,46)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C15 — C22×D15
 Chief series C1 — C5 — C15 — D15 — D30 — C22×D15
 Lower central C15 — C22×D15
 Upper central C1 — C22

Generators and relations for C22×D15
G = < a,b,c,d | a2=b2=c15=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 284 in 64 conjugacy classes, 31 normal (9 characteristic)
C1, C2 [×3], C2 [×4], C3, C22, C22 [×6], C5, S3 [×4], C6 [×3], C23, D5 [×4], C10 [×3], D6 [×6], C2×C6, C15, D10 [×6], C2×C10, C22×S3, D15 [×4], C30 [×3], C22×D5, D30 [×6], C2×C30, C22×D15
Quotients: C1, C2 [×7], C22 [×7], S3, C23, D5, D6 [×3], D10 [×3], C22×S3, D15, C22×D5, D30 [×3], C22×D15

Smallest permutation representation of C22×D15
On 60 points
Generators in S60
(1 39)(2 40)(3 41)(4 42)(5 43)(6 44)(7 45)(8 31)(9 32)(10 33)(11 34)(12 35)(13 36)(14 37)(15 38)(16 60)(17 46)(18 47)(19 48)(20 49)(21 50)(22 51)(23 52)(24 53)(25 54)(26 55)(27 56)(28 57)(29 58)(30 59)
(1 17)(2 18)(3 19)(4 20)(5 21)(6 22)(7 23)(8 24)(9 25)(10 26)(11 27)(12 28)(13 29)(14 30)(15 16)(31 53)(32 54)(33 55)(34 56)(35 57)(36 58)(37 59)(38 60)(39 46)(40 47)(41 48)(42 49)(43 50)(44 51)(45 52)
(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)
(1 60)(2 59)(3 58)(4 57)(5 56)(6 55)(7 54)(8 53)(9 52)(10 51)(11 50)(12 49)(13 48)(14 47)(15 46)(16 39)(17 38)(18 37)(19 36)(20 35)(21 34)(22 33)(23 32)(24 31)(25 45)(26 44)(27 43)(28 42)(29 41)(30 40)

G:=sub<Sym(60)| (1,39)(2,40)(3,41)(4,42)(5,43)(6,44)(7,45)(8,31)(9,32)(10,33)(11,34)(12,35)(13,36)(14,37)(15,38)(16,60)(17,46)(18,47)(19,48)(20,49)(21,50)(22,51)(23,52)(24,53)(25,54)(26,55)(27,56)(28,57)(29,58)(30,59), (1,17)(2,18)(3,19)(4,20)(5,21)(6,22)(7,23)(8,24)(9,25)(10,26)(11,27)(12,28)(13,29)(14,30)(15,16)(31,53)(32,54)(33,55)(34,56)(35,57)(36,58)(37,59)(38,60)(39,46)(40,47)(41,48)(42,49)(43,50)(44,51)(45,52), (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), (1,60)(2,59)(3,58)(4,57)(5,56)(6,55)(7,54)(8,53)(9,52)(10,51)(11,50)(12,49)(13,48)(14,47)(15,46)(16,39)(17,38)(18,37)(19,36)(20,35)(21,34)(22,33)(23,32)(24,31)(25,45)(26,44)(27,43)(28,42)(29,41)(30,40)>;

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

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

C22×D15 is a maximal subgroup of   D304C4  D303C4  D10⋊D6  C22×S3×D5
C22×D15 is a maximal quotient of   D6011C2  D42D15  Q83D15

36 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 5A 5B 6A 6B 6C 10A ··· 10F 15A 15B 15C 15D 30A ··· 30L order 1 2 2 2 2 2 2 2 3 5 5 6 6 6 10 ··· 10 15 15 15 15 30 ··· 30 size 1 1 1 1 15 15 15 15 2 2 2 2 2 2 2 ··· 2 2 2 2 2 2 ··· 2

36 irreducible representations

 dim 1 1 1 2 2 2 2 2 2 type + + + + + + + + + image C1 C2 C2 S3 D5 D6 D10 D15 D30 kernel C22×D15 D30 C2×C30 C2×C10 C2×C6 C10 C6 C22 C2 # reps 1 6 1 1 2 3 6 4 12

Matrix representation of C22×D15 in GL3(𝔽31) generated by

 30 0 0 0 30 0 0 0 30
,
 30 0 0 0 1 0 0 0 1
,
 1 0 0 0 12 4 0 27 22
,
 30 0 0 0 19 27 0 28 12
G:=sub<GL(3,GF(31))| [30,0,0,0,30,0,0,0,30],[30,0,0,0,1,0,0,0,1],[1,0,0,0,12,27,0,4,22],[30,0,0,0,19,28,0,27,12] >;

C22×D15 in GAP, Magma, Sage, TeX

C_2^2\times D_{15}
% in TeX

G:=Group("C2^2xD15");
// GroupNames label

G:=SmallGroup(120,46);
// by ID

G=gap.SmallGroup(120,46);
# by ID

G:=PCGroup([5,-2,-2,-2,-3,-5,323,2404]);
// Polycyclic

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

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