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## G = C4×D15order 120 = 23·3·5

### Direct product of C4 and D15

Aliases: C4×D15, C602C2, C202S3, C122D5, C4Dic15, C2.1D30, C10.9D6, C6.9D10, D30.2C2, Dic155C2, C30.9C22, C53(C4×S3), C32(C4×D5), C157(C2×C4), SmallGroup(120,27)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C15 — C4×D15
 Chief series C1 — C5 — C15 — C30 — D30 — C4×D15
 Lower central C15 — C4×D15
 Upper central C1 — C4

Generators and relations for C4×D15
G = < a,b,c | a4=b15=c2=1, ab=ba, ac=ca, cbc=b-1 >

Smallest permutation representation of C4×D15
On 60 points
Generators in S60
(1 54 17 31)(2 55 18 32)(3 56 19 33)(4 57 20 34)(5 58 21 35)(6 59 22 36)(7 60 23 37)(8 46 24 38)(9 47 25 39)(10 48 26 40)(11 49 27 41)(12 50 28 42)(13 51 29 43)(14 52 30 44)(15 53 16 45)
(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 16)(2 30)(3 29)(4 28)(5 27)(6 26)(7 25)(8 24)(9 23)(10 22)(11 21)(12 20)(13 19)(14 18)(15 17)(31 53)(32 52)(33 51)(34 50)(35 49)(36 48)(37 47)(38 46)(39 60)(40 59)(41 58)(42 57)(43 56)(44 55)(45 54)

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

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

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

C4×D15 is a maximal subgroup of
D152C8  D30.5C4  C40⋊S3  D20⋊S3  D12⋊D5  D15⋊Q8  D6.D10  C4×S3×D5  C20⋊D6  D6011C2  D42D15  Q83D15  C6.D30  C20.6S4
C4×D15 is a maximal quotient of
C40⋊S3  C30.4Q8  D303C4  C6.D30

36 conjugacy classes

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

36 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + image C1 C2 C2 C2 C4 S3 D5 D6 D10 C4×S3 D15 C4×D5 D30 C4×D15 kernel C4×D15 Dic15 C60 D30 D15 C20 C12 C10 C6 C5 C4 C3 C2 C1 # reps 1 1 1 1 4 1 2 1 2 2 4 4 4 8

Matrix representation of C4×D15 in GL2(𝔽29) generated by

 17 0 0 17
,
 28 21 21 22
,
 22 23 8 7
G:=sub<GL(2,GF(29))| [17,0,0,17],[28,21,21,22],[22,8,23,7] >;

C4×D15 in GAP, Magma, Sage, TeX

C_4\times D_{15}
% in TeX

G:=Group("C4xD15");
// GroupNames label

G:=SmallGroup(120,27);
// by ID

G=gap.SmallGroup(120,27);
# by ID

G:=PCGroup([5,-2,-2,-2,-3,-5,26,323,2404]);
// Polycyclic

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

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