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G = C2×C157D4order 240 = 24·3·5

Direct product of C2 and C157D4

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

Aliases: C2×C157D4, C307D4, C232D15, C223D30, D307C22, C30.37C23, Dic154C22, (C2×C6)⋊8D10, C1516(C2×D4), (C2×C10)⋊11D6, C63(C5⋊D4), (C22×C6)⋊2D5, C103(C3⋊D4), (C2×C30)⋊9C22, (C22×C10)⋊4S3, (C22×C30)⋊2C2, (C2×Dic15)⋊4C2, (C22×D15)⋊3C2, C6.37(C22×D5), C10.37(C22×S3), C2.10(C22×D15), C54(C2×C3⋊D4), C34(C2×C5⋊D4), SmallGroup(240,184)

Series: Derived Chief Lower central Upper central

C1C30 — C2×C157D4
C1C5C15C30D30C22×D15 — C2×C157D4
C15C30 — C2×C157D4
C1C22C23

Generators and relations for C2×C157D4
 G = < a,b,c,d | a2=b15=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 520 in 108 conjugacy classes, 43 normal (19 characteristic)
C1, C2, C2 [×2], C2 [×4], C3, C4 [×2], C22, C22 [×2], C22 [×6], C5, S3 [×2], C6, C6 [×2], C6 [×2], C2×C4, D4 [×4], C23, C23, D5 [×2], C10, C10 [×2], C10 [×2], Dic3 [×2], D6 [×4], C2×C6, C2×C6 [×2], C2×C6 [×2], C15, C2×D4, Dic5 [×2], D10 [×4], C2×C10, C2×C10 [×2], C2×C10 [×2], C2×Dic3, C3⋊D4 [×4], C22×S3, C22×C6, D15 [×2], C30, C30 [×2], C30 [×2], C2×Dic5, C5⋊D4 [×4], C22×D5, C22×C10, C2×C3⋊D4, Dic15 [×2], D30 [×2], D30 [×2], C2×C30, C2×C30 [×2], C2×C30 [×2], C2×C5⋊D4, C2×Dic15, C157D4 [×4], C22×D15, C22×C30, C2×C157D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], C3⋊D4 [×2], C22×S3, D15, C5⋊D4 [×2], C22×D5, C2×C3⋊D4, D30 [×3], C2×C5⋊D4, C157D4 [×2], C22×D15, C2×C157D4

Smallest permutation representation of C2×C157D4
On 120 points
Generators in S120
(1 72)(2 73)(3 74)(4 75)(5 61)(6 62)(7 63)(8 64)(9 65)(10 66)(11 67)(12 68)(13 69)(14 70)(15 71)(16 84)(17 85)(18 86)(19 87)(20 88)(21 89)(22 90)(23 76)(24 77)(25 78)(26 79)(27 80)(28 81)(29 82)(30 83)(31 101)(32 102)(33 103)(34 104)(35 105)(36 91)(37 92)(38 93)(39 94)(40 95)(41 96)(42 97)(43 98)(44 99)(45 100)(46 110)(47 111)(48 112)(49 113)(50 114)(51 115)(52 116)(53 117)(54 118)(55 119)(56 120)(57 106)(58 107)(59 108)(60 109)
(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 41 30 49)(2 40 16 48)(3 39 17 47)(4 38 18 46)(5 37 19 60)(6 36 20 59)(7 35 21 58)(8 34 22 57)(9 33 23 56)(10 32 24 55)(11 31 25 54)(12 45 26 53)(13 44 27 52)(14 43 28 51)(15 42 29 50)(61 92 87 109)(62 91 88 108)(63 105 89 107)(64 104 90 106)(65 103 76 120)(66 102 77 119)(67 101 78 118)(68 100 79 117)(69 99 80 116)(70 98 81 115)(71 97 82 114)(72 96 83 113)(73 95 84 112)(74 94 85 111)(75 93 86 110)
(1 72)(2 71)(3 70)(4 69)(5 68)(6 67)(7 66)(8 65)(9 64)(10 63)(11 62)(12 61)(13 75)(14 74)(15 73)(16 82)(17 81)(18 80)(19 79)(20 78)(21 77)(22 76)(23 90)(24 89)(25 88)(26 87)(27 86)(28 85)(29 84)(30 83)(31 108)(32 107)(33 106)(34 120)(35 119)(36 118)(37 117)(38 116)(39 115)(40 114)(41 113)(42 112)(43 111)(44 110)(45 109)(46 99)(47 98)(48 97)(49 96)(50 95)(51 94)(52 93)(53 92)(54 91)(55 105)(56 104)(57 103)(58 102)(59 101)(60 100)

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

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

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

C2×C157D4 is a maximal subgroup of
C159(C23⋊C4)  C23.6D30  Dic15.19D4  D306D4  C10.(C2×D12)  C6.D4⋊D5  Dic153D4  C1526(C4×D4)  C1528(C4×D4)  D307D4  Dic154D4  D30.16D4  (C2×C6)⋊8D20  (C2×C10)⋊4D12  Dic155D4  (C2×C6)⋊D20  (C2×C10)⋊11D12  D308D4  Dic1519D4  D3016D4  D30.28D4  D309D4  C23.11D30  C22.D60  C23.28D30  C6029D4  D3017D4  C602D4  Dic1512D4  C603D4  C245D15  C2×D5×C3⋊D4  C2×S3×C5⋊D4  C15⋊2+ 1+4  C2×D4×D15  D46D30
C2×C157D4 is a maximal quotient of
C60.205D4  C23.28D30  C6029D4  D4.D30  C23.22D30  C60.17D4  D3017D4  C602D4  Dic1512D4  C603D4  Q8.11D30  Dic154Q8  D307Q8  C60.23D4  D4⋊D30  D4.8D30  D4.9D30  C245D15

66 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A4B5A5B6A···6G10A···10N15A15B15C15D30A···30AB
order12222222344556···610···101515151530···30
size111122303023030222···22···222222···2

66 irreducible representations

dim111112222222222
type++++++++++++
imageC1C2C2C2C2S3D4D5D6D10C3⋊D4D15C5⋊D4D30C157D4
kernelC2×C157D4C2×Dic15C157D4C22×D15C22×C30C22×C10C30C22×C6C2×C10C2×C6C10C23C6C22C2
# reps11411122364481216

Matrix representation of C2×C157D4 in GL4(𝔽61) generated by

60000
06000
0010
0001
,
181800
436000
004731
003025
,
1000
436000
00378
002724
,
60000
18100
002533
003136
G:=sub<GL(4,GF(61))| [60,0,0,0,0,60,0,0,0,0,1,0,0,0,0,1],[18,43,0,0,18,60,0,0,0,0,47,30,0,0,31,25],[1,43,0,0,0,60,0,0,0,0,37,27,0,0,8,24],[60,18,0,0,0,1,0,0,0,0,25,31,0,0,33,36] >;

C2×C157D4 in GAP, Magma, Sage, TeX

C_2\times C_{15}\rtimes_7D_4
% in TeX

G:=Group("C2xC15:7D4");
// GroupNames label

G:=SmallGroup(240,184);
// by ID

G=gap.SmallGroup(240,184);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-5,218,964,6917]);
// Polycyclic

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

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