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G = C22×D15order 120 = 23·3·5

Direct product of C22 and D15

direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary

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

C1C15 — C22×D15
C1C5C15D15D30 — C22×D15
C15 — C22×D15
C1C22

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 2A2B2C2D2E2F2G 3 5A5B6A6B6C10A···10F15A15B15C15D30A···30L
order1222222235566610···101515151530···30
size1111151515152222222···222222···2

36 irreducible representations

dim111222222
type+++++++++
imageC1C2C2S3D5D6D10D15D30
kernelC22×D15D30C2×C30C2×C10C2×C6C10C6C22C2
# reps1611236412

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

3000
0300
0030
,
3000
010
001
,
100
0124
02722
,
3000
01927
02812
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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