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G = C158(C23⋊C4)  order 480 = 25·3·5

2nd semidirect product of C15 and C23⋊C4 acting via C23⋊C4/C23=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C158(C23⋊C4), (C2×Dic3)⋊Dic5, (C2×C30).37D4, (C2×C10).2D12, (C22×S3)⋊Dic5, C23.D51S3, C23.9(S3×D5), (C10×Dic3)⋊1C4, C31(C23⋊Dic5), C10.47(D6⋊C4), (C22×C10).28D6, (C22×C6).13D10, C55(C23.6D6), C30.38D415C2, C22.5(S3×Dic5), C30.67(C22⋊C4), C2.11(D6⋊Dic5), C22.3(C15⋊D4), C22.8(C5⋊D12), C6.11(C23.D5), (C22×C30).27C22, (S3×C2×C10)⋊1C4, (C2×C3⋊D4).1D5, (C2×C30).91(C2×C4), (C2×C10).73(C4×S3), (C10×C3⋊D4).1C2, (C2×C6).5(C5⋊D4), (C3×C23.D5)⋊1C2, (C2×C6).3(C2×Dic5), (C2×C10).4(C3⋊D4), SmallGroup(480,72)

Series: Derived Chief Lower central Upper central

C1C2×C30 — C158(C23⋊C4)
C1C5C15C30C2×C30C22×C30C3×C23.D5 — C158(C23⋊C4)
C15C30C2×C30 — C158(C23⋊C4)
C1C2C23

Generators and relations for C158(C23⋊C4)
 G = < a,b,c,d,e | a15=b2=c2=d2=e4=1, bab=a11, ac=ca, ad=da, eae-1=a4, bc=cb, bd=db, ebe-1=bcd, ece-1=cd=dc, de=ed >

Subgroups: 492 in 104 conjugacy classes, 34 normal (all characteristic)
C1, C2, C2 [×4], C3, C4 [×3], C22 [×3], C22 [×3], C5, S3, C6, C6 [×3], C2×C4 [×3], D4 [×2], C23, C23, C10, C10 [×4], Dic3 [×2], C12, D6 [×2], C2×C6 [×3], C2×C6, C15, C22⋊C4 [×2], C2×D4, Dic5 [×2], C20, C2×C10 [×3], C2×C10 [×3], C2×Dic3, C2×Dic3, C3⋊D4 [×2], C2×C12, C22×S3, C22×C6, C5×S3, C30, C30 [×3], C23⋊C4, C2×Dic5 [×2], C2×C20, C5×D4 [×2], C22×C10, C22×C10, C6.D4, C3×C22⋊C4, C2×C3⋊D4, C5×Dic3, C3×Dic5, Dic15, S3×C10 [×2], C2×C30 [×3], C2×C30, C23.D5, C23.D5, D4×C10, C23.6D6, C6×Dic5, C10×Dic3, C5×C3⋊D4 [×2], C2×Dic15, S3×C2×C10, C22×C30, C23⋊Dic5, C3×C23.D5, C30.38D4, C10×C3⋊D4, C158(C23⋊C4)
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], D5, D6, C22⋊C4, Dic5 [×2], D10, C4×S3, D12, C3⋊D4, C23⋊C4, C2×Dic5, C5⋊D4 [×2], D6⋊C4, S3×D5, C23.D5, C23.6D6, S3×Dic5, C15⋊D4, C5⋊D12, C23⋊Dic5, D6⋊Dic5, C158(C23⋊C4)

Smallest permutation representation of C158(C23⋊C4)
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 64)(2 75)(3 71)(4 67)(5 63)(6 74)(7 70)(8 66)(9 62)(10 73)(11 69)(12 65)(13 61)(14 72)(15 68)(16 88)(17 84)(18 80)(19 76)(20 87)(21 83)(22 79)(23 90)(24 86)(25 82)(26 78)(27 89)(28 85)(29 81)(30 77)(31 94)(32 105)(33 101)(34 97)(35 93)(36 104)(37 100)(38 96)(39 92)(40 103)(41 99)(42 95)(43 91)(44 102)(45 98)(46 107)(47 118)(48 114)(49 110)(50 106)(51 117)(52 113)(53 109)(54 120)(55 116)(56 112)(57 108)(58 119)(59 115)(60 111)
(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 57)(17 58)(18 59)(19 60)(20 46)(21 47)(22 48)(23 49)(24 50)(25 51)(26 52)(27 53)(28 54)(29 55)(30 56)(61 104)(62 105)(63 91)(64 92)(65 93)(66 94)(67 95)(68 96)(69 97)(70 98)(71 99)(72 100)(73 101)(74 102)(75 103)(76 111)(77 112)(78 113)(79 114)(80 115)(81 116)(82 117)(83 118)(84 119)(85 120)(86 106)(87 107)(88 108)(89 109)(90 110)
(1 28)(2 29)(3 30)(4 16)(5 17)(6 18)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)(13 25)(14 26)(15 27)(31 46)(32 47)(33 48)(34 49)(35 50)(36 51)(37 52)(38 53)(39 54)(40 55)(41 56)(42 57)(43 58)(44 59)(45 60)(61 82)(62 83)(63 84)(64 85)(65 86)(66 87)(67 88)(68 89)(69 90)(70 76)(71 77)(72 78)(73 79)(74 80)(75 81)(91 119)(92 120)(93 106)(94 107)(95 108)(96 109)(97 110)(98 111)(99 112)(100 113)(101 114)(102 115)(103 116)(104 117)(105 118)
(1 64 39 120)(2 68 40 109)(3 72 41 113)(4 61 42 117)(5 65 43 106)(6 69 44 110)(7 73 45 114)(8 62 31 118)(9 66 32 107)(10 70 33 111)(11 74 34 115)(12 63 35 119)(13 67 36 108)(14 71 37 112)(15 75 38 116)(16 82 57 104)(17 86 58 93)(18 90 59 97)(19 79 60 101)(20 83 46 105)(21 87 47 94)(22 76 48 98)(23 80 49 102)(24 84 50 91)(25 88 51 95)(26 77 52 99)(27 81 53 103)(28 85 54 92)(29 89 55 96)(30 78 56 100)

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

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

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

57 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E5A5B6A6B6C6D6E10A···10F10G10H10I10J10K10L10M10N12A12B12C12D15A15B20A20B20C20D30A···30N
order122222344444556666610···1010101010101010101212121215152020202030···30
size11222122122020606022222442···24444121212122020202044121212124···4

57 irreducible representations

dim1111112222222222244444444
type++++++++--++++--+
imageC1C2C2C2C4C4S3D4D5D6Dic5Dic5D10C4×S3D12C3⋊D4C5⋊D4C23⋊C4S3×D5C23.6D6S3×Dic5C15⋊D4C5⋊D12C23⋊Dic5C158(C23⋊C4)
kernelC158(C23⋊C4)C3×C23.D5C30.38D4C10×C3⋊D4C10×Dic3S3×C2×C10C23.D5C2×C30C2×C3⋊D4C22×C10C2×Dic3C22×S3C22×C6C2×C10C2×C10C2×C10C2×C6C15C23C5C22C22C22C3C1
# reps1111221221222222812222248

Matrix representation of C158(C23⋊C4) in GL6(𝔽61)

0170000
43430000
000100
00606000
00006060
000010
,
30160000
1310000
000010
000001
001000
000100
,
6000000
0600000
0091800
00435200
0000918
00004352
,
100000
010000
0060000
0006000
0000600
0000060
,
27590000
59340000
00005243
0000529
001000
00606000

G:=sub<GL(6,GF(61))| [0,43,0,0,0,0,17,43,0,0,0,0,0,0,0,60,0,0,0,0,1,60,0,0,0,0,0,0,60,1,0,0,0,0,60,0],[30,1,0,0,0,0,16,31,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,9,43,0,0,0,0,18,52,0,0,0,0,0,0,9,43,0,0,0,0,18,52],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[27,59,0,0,0,0,59,34,0,0,0,0,0,0,0,0,1,60,0,0,0,0,0,60,0,0,52,52,0,0,0,0,43,9,0,0] >;

C158(C23⋊C4) in GAP, Magma, Sage, TeX

C_{15}\rtimes_8(C_2^3\rtimes C_4)
% in TeX

G:=Group("C15:8(C2^3:C4)");
// GroupNames label

G:=SmallGroup(480,72);
// by ID

G=gap.SmallGroup(480,72);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,219,675,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d,e|a^15=b^2=c^2=d^2=e^4=1,b*a*b=a^11,a*c=c*a,a*d=d*a,e*a*e^-1=a^4,b*c=c*b,b*d=d*b,e*b*e^-1=b*c*d,e*c*e^-1=c*d=d*c,d*e=e*d>;
// generators/relations

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