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G = D304C4order 240 = 24·3·5

2nd semidirect product of D30 and C4 acting via C4/C2=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D304C4, C30.15D4, C6.11D20, C10.11D12, C52(D6⋊C4), C6.3(C4×D5), (C2×C6).8D10, (C2×C10).8D6, C10.10(C4×S3), C156(C22⋊C4), C30.31(C2×C4), (C2×Dic3)⋊2D5, (C2×Dic5)⋊2S3, (C6×Dic5)⋊2C2, C6.5(C5⋊D4), C31(D10⋊C4), C22.7(S3×D5), (C10×Dic3)⋊2C2, C10.5(C3⋊D4), C2.2(C5⋊D12), C2.2(C3⋊D20), (C2×C30).5C22, C2.4(D30.C2), (C22×D15).2C2, SmallGroup(240,28)

Series: Derived Chief Lower central Upper central

C1C30 — D304C4
C1C5C15C30C2×C30C6×Dic5 — D304C4
C15C30 — D304C4
C1C22

Generators and relations for D304C4
 G = < a,b,c | a30=b2=c4=1, bab=a-1, cac-1=a11, cbc-1=a25b >

Subgroups: 368 in 68 conjugacy classes, 28 normal (26 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C22, C22 [×4], C5, S3 [×2], C6 [×3], C2×C4 [×2], C23, D5 [×2], C10 [×3], Dic3, C12, D6 [×4], C2×C6, C15, C22⋊C4, Dic5, C20, D10 [×4], C2×C10, C2×Dic3, C2×C12, C22×S3, D15 [×2], C30 [×3], C2×Dic5, C2×C20, C22×D5, D6⋊C4, C5×Dic3, C3×Dic5, D30 [×2], D30 [×2], C2×C30, D10⋊C4, C6×Dic5, C10×Dic3, C22×D15, D304C4
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], D5, D6, C22⋊C4, D10, C4×S3, D12, C3⋊D4, C4×D5, D20, C5⋊D4, D6⋊C4, S3×D5, D10⋊C4, D30.C2, C3⋊D20, C5⋊D12, D304C4

Smallest permutation representation of D304C4
On 120 points
Generators in S120
(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 81 82 83 84 85 86 87 88 89 90)(91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120)
(1 30)(2 29)(3 28)(4 27)(5 26)(6 25)(7 24)(8 23)(9 22)(10 21)(11 20)(12 19)(13 18)(14 17)(15 16)(31 40)(32 39)(33 38)(34 37)(35 36)(41 60)(42 59)(43 58)(44 57)(45 56)(46 55)(47 54)(48 53)(49 52)(50 51)(61 65)(62 64)(66 90)(67 89)(68 88)(69 87)(70 86)(71 85)(72 84)(73 83)(74 82)(75 81)(76 80)(77 79)(91 105)(92 104)(93 103)(94 102)(95 101)(96 100)(97 99)(106 120)(107 119)(108 118)(109 117)(110 116)(111 115)(112 114)
(1 106 36 71)(2 117 37 82)(3 98 38 63)(4 109 39 74)(5 120 40 85)(6 101 41 66)(7 112 42 77)(8 93 43 88)(9 104 44 69)(10 115 45 80)(11 96 46 61)(12 107 47 72)(13 118 48 83)(14 99 49 64)(15 110 50 75)(16 91 51 86)(17 102 52 67)(18 113 53 78)(19 94 54 89)(20 105 55 70)(21 116 56 81)(22 97 57 62)(23 108 58 73)(24 119 59 84)(25 100 60 65)(26 111 31 76)(27 92 32 87)(28 103 33 68)(29 114 34 79)(30 95 35 90)

G:=sub<Sym(120)| (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)(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90)(91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120), (1,30)(2,29)(3,28)(4,27)(5,26)(6,25)(7,24)(8,23)(9,22)(10,21)(11,20)(12,19)(13,18)(14,17)(15,16)(31,40)(32,39)(33,38)(34,37)(35,36)(41,60)(42,59)(43,58)(44,57)(45,56)(46,55)(47,54)(48,53)(49,52)(50,51)(61,65)(62,64)(66,90)(67,89)(68,88)(69,87)(70,86)(71,85)(72,84)(73,83)(74,82)(75,81)(76,80)(77,79)(91,105)(92,104)(93,103)(94,102)(95,101)(96,100)(97,99)(106,120)(107,119)(108,118)(109,117)(110,116)(111,115)(112,114), (1,106,36,71)(2,117,37,82)(3,98,38,63)(4,109,39,74)(5,120,40,85)(6,101,41,66)(7,112,42,77)(8,93,43,88)(9,104,44,69)(10,115,45,80)(11,96,46,61)(12,107,47,72)(13,118,48,83)(14,99,49,64)(15,110,50,75)(16,91,51,86)(17,102,52,67)(18,113,53,78)(19,94,54,89)(20,105,55,70)(21,116,56,81)(22,97,57,62)(23,108,58,73)(24,119,59,84)(25,100,60,65)(26,111,31,76)(27,92,32,87)(28,103,33,68)(29,114,34,79)(30,95,35,90)>;

G:=Group( (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)(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90)(91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120), (1,30)(2,29)(3,28)(4,27)(5,26)(6,25)(7,24)(8,23)(9,22)(10,21)(11,20)(12,19)(13,18)(14,17)(15,16)(31,40)(32,39)(33,38)(34,37)(35,36)(41,60)(42,59)(43,58)(44,57)(45,56)(46,55)(47,54)(48,53)(49,52)(50,51)(61,65)(62,64)(66,90)(67,89)(68,88)(69,87)(70,86)(71,85)(72,84)(73,83)(74,82)(75,81)(76,80)(77,79)(91,105)(92,104)(93,103)(94,102)(95,101)(96,100)(97,99)(106,120)(107,119)(108,118)(109,117)(110,116)(111,115)(112,114), (1,106,36,71)(2,117,37,82)(3,98,38,63)(4,109,39,74)(5,120,40,85)(6,101,41,66)(7,112,42,77)(8,93,43,88)(9,104,44,69)(10,115,45,80)(11,96,46,61)(12,107,47,72)(13,118,48,83)(14,99,49,64)(15,110,50,75)(16,91,51,86)(17,102,52,67)(18,113,53,78)(19,94,54,89)(20,105,55,70)(21,116,56,81)(22,97,57,62)(23,108,58,73)(24,119,59,84)(25,100,60,65)(26,111,31,76)(27,92,32,87)(28,103,33,68)(29,114,34,79)(30,95,35,90) );

G=PermutationGroup([(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),(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90),(91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120)], [(1,30),(2,29),(3,28),(4,27),(5,26),(6,25),(7,24),(8,23),(9,22),(10,21),(11,20),(12,19),(13,18),(14,17),(15,16),(31,40),(32,39),(33,38),(34,37),(35,36),(41,60),(42,59),(43,58),(44,57),(45,56),(46,55),(47,54),(48,53),(49,52),(50,51),(61,65),(62,64),(66,90),(67,89),(68,88),(69,87),(70,86),(71,85),(72,84),(73,83),(74,82),(75,81),(76,80),(77,79),(91,105),(92,104),(93,103),(94,102),(95,101),(96,100),(97,99),(106,120),(107,119),(108,118),(109,117),(110,116),(111,115),(112,114)], [(1,106,36,71),(2,117,37,82),(3,98,38,63),(4,109,39,74),(5,120,40,85),(6,101,41,66),(7,112,42,77),(8,93,43,88),(9,104,44,69),(10,115,45,80),(11,96,46,61),(12,107,47,72),(13,118,48,83),(14,99,49,64),(15,110,50,75),(16,91,51,86),(17,102,52,67),(18,113,53,78),(19,94,54,89),(20,105,55,70),(21,116,56,81),(22,97,57,62),(23,108,58,73),(24,119,59,84),(25,100,60,65),(26,111,31,76),(27,92,32,87),(28,103,33,68),(29,114,34,79),(30,95,35,90)])

D304C4 is a maximal subgroup of
(C2×C20).D6  C4⋊Dic3⋊D5  C4⋊Dic5⋊S3  (C4×D15)⋊8C4  Dic3⋊C4⋊D5  Dic5.8D12  D6⋊Dic5⋊C2  Dic3.D20  D30.34D4  D30.35D4  C60.47D4  C60.70D4  D308Q8  C10.D4⋊S3  D309Q8  (C4×D15)⋊10C4  (C4×Dic5)⋊S3  D3010Q8  Dic1513D4  D30.C2⋊C4  D30.23(C2×C4)  Dic1514D4  D30⋊Q8  D10.16D12  D6017C4  D302Q8  D30⋊D4  D303Q8  D6.D20  D6014C4  D304Q8  D30.6D4  C4×C3⋊D20  C4×C5⋊D12  C127D20  (C2×Dic6)⋊D5  D302D4  C606D4  D5×D6⋊C4  S3×D10⋊C4  D305D4  Dic15.19D4  C10.(C2×D12)  C6.D4⋊D5  Dic153D4  C1526(C4×D4)  C1528(C4×D4)  Dic154D4  D30.45D4  D30.16D4  (C2×C10)⋊4D12  (C2×C6)⋊D20  D3018D4  D3019D4  D308D4
D304C4 is a maximal quotient of
D304C8  C60.29D4  C60.31D4  D6012C4  D6015C4  Dic3012C4  Dic3015C4  D6013C4  D6016C4  C30.24C42  C159(C23⋊C4)

42 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D5A5B6A6B6C10A···10F12A12B12C12D15A15B20A···20H30A···30F
order122222344445566610···1012121212151520···2030···30
size111130302661010222222···210101010446···64···4

42 irreducible representations

dim11111222222222224444
type+++++++++++++++
imageC1C2C2C2C4S3D4D5D6D10C4×S3D12C3⋊D4C4×D5D20C5⋊D4S3×D5D30.C2C3⋊D20C5⋊D12
kernelD304C4C6×Dic5C10×Dic3C22×D15D30C2×Dic5C30C2×Dic3C2×C10C2×C6C10C10C10C6C6C6C22C2C2C2
# reps11114122122224442222

Matrix representation of D304C4 in GL4(𝔽61) generated by

0100
606000
00431
00421
,
0100
1000
00044
00430
,
50000
111100
00257
005036
G:=sub<GL(4,GF(61))| [0,60,0,0,1,60,0,0,0,0,43,42,0,0,1,1],[0,1,0,0,1,0,0,0,0,0,0,43,0,0,44,0],[50,11,0,0,0,11,0,0,0,0,25,50,0,0,7,36] >;

D304C4 in GAP, Magma, Sage, TeX

D_{30}\rtimes_4C_4
% in TeX

G:=Group("D30:4C4");
// GroupNames label

G:=SmallGroup(240,28);
// by ID

G=gap.SmallGroup(240,28);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-5,24,121,79,490,6917]);
// Polycyclic

G:=Group<a,b,c|a^30=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^11,c*b*c^-1=a^25*b>;
// generators/relations

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