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G = C23.7D30order 480 = 25·3·5

2nd non-split extension by C23 of D30 acting via D30/C15=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C30 — C23.7D30
 Chief series C1 — C5 — C15 — C30 — C2×C30 — C22×C30 — C30.38D4 — C23.7D30
 Lower central C15 — C30 — C2×C30 — C23.7D30
 Upper central C1 — C2 — C23 — C2×D4

Generators and relations for C23.7D30
G = < a,b,c,d,e | a2=b2=c2=d30=1, e2=ba=ab, dad-1=eae-1=ac=ca, ebe-1=bc=cb, bd=db, cd=dc, ce=ec, ede-1=bcd-1 >

Subgroups: 500 in 104 conjugacy classes, 39 normal (29 characteristic)
C1, C2, C2, C3, C4, C22, C22, C22, C5, C6, C6, C2×C4, C2×C4, D4, C23, C10, C10, Dic3, C12, C2×C6, C2×C6, C2×C6, C15, C22⋊C4, C2×D4, Dic5, C20, C2×C10, C2×C10, C2×C10, C2×Dic3, C2×C12, C3×D4, C22×C6, C30, C30, C23⋊C4, C2×Dic5, C2×C20, C5×D4, C22×C10, C6.D4, C6×D4, Dic15, C60, C2×C30, C2×C30, C2×C30, C23.D5, D4×C10, C23.7D6, C2×Dic15, C2×C60, D4×C15, C22×C30, C23⋊Dic5, C30.38D4, D4×C30, C23.7D30
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D5, Dic3, D6, C22⋊C4, Dic5, D10, C2×Dic3, C3⋊D4, D15, C23⋊C4, C2×Dic5, C5⋊D4, C6.D4, Dic15, D30, C23.D5, C23.7D6, C2×Dic15, C157D4, C23⋊Dic5, C30.38D4, C23.7D30

Smallest permutation representation of C23.7D30
On 120 points
Generators in S120
```(1 106)(2 34)(3 108)(4 36)(5 110)(6 38)(7 112)(8 40)(9 114)(10 42)(11 116)(12 44)(13 118)(14 46)(15 120)(16 48)(17 92)(18 50)(19 94)(20 52)(21 96)(22 54)(23 98)(24 56)(25 100)(26 58)(27 102)(28 60)(29 104)(30 32)(31 73)(33 75)(35 77)(37 79)(39 81)(41 83)(43 85)(45 87)(47 89)(49 61)(51 63)(53 65)(55 67)(57 69)(59 71)(62 93)(64 95)(66 97)(68 99)(70 101)(72 103)(74 105)(76 107)(78 109)(80 111)(82 113)(84 115)(86 117)(88 119)(90 91)
(1 106)(2 107)(3 108)(4 109)(5 110)(6 111)(7 112)(8 113)(9 114)(10 115)(11 116)(12 117)(13 118)(14 119)(15 120)(16 91)(17 92)(18 93)(19 94)(20 95)(21 96)(22 97)(23 98)(24 99)(25 100)(26 101)(27 102)(28 103)(29 104)(30 105)(31 73)(32 74)(33 75)(34 76)(35 77)(36 78)(37 79)(38 80)(39 81)(40 82)(41 83)(42 84)(43 85)(44 86)(45 87)(46 88)(47 89)(48 90)(49 61)(50 62)(51 63)(52 64)(53 65)(54 66)(55 67)(56 68)(57 69)(58 70)(59 71)(60 72)
(1 75)(2 76)(3 77)(4 78)(5 79)(6 80)(7 81)(8 82)(9 83)(10 84)(11 85)(12 86)(13 87)(14 88)(15 89)(16 90)(17 61)(18 62)(19 63)(20 64)(21 65)(22 66)(23 67)(24 68)(25 69)(26 70)(27 71)(28 72)(29 73)(30 74)(31 104)(32 105)(33 106)(34 107)(35 108)(36 109)(37 110)(38 111)(39 112)(40 113)(41 114)(42 115)(43 116)(44 117)(45 118)(46 119)(47 120)(48 91)(49 92)(50 93)(51 94)(52 95)(53 96)(54 97)(55 98)(56 99)(57 100)(58 101)(59 102)(60 103)
(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 15)(2 119 76 46)(3 13)(4 117 78 44)(5 11)(6 115 80 42)(7 9)(8 113 82 40)(10 111 84 38)(12 109 86 36)(14 107 88 34)(16 105 90 32)(17 29)(18 103 62 60)(19 27)(20 101 64 58)(21 25)(22 99 66 56)(24 97 68 54)(26 95 70 52)(28 93 72 50)(30 91 74 48)(31 92)(33 120)(35 118)(37 116)(39 114)(41 112)(43 110)(45 108)(47 106)(49 104)(51 102)(53 100)(55 98)(57 96)(59 94)(61 73)(63 71)(65 69)(75 89)(77 87)(79 85)(81 83)```

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

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

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

81 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 4E 5A 5B 6A 6B 6C 6D 6E 6F 6G 10A ··· 10F 10G ··· 10N 12A 12B 15A 15B 15C 15D 20A 20B 20C 20D 30A ··· 30L 30M ··· 30AB 60A ··· 60H order 1 2 2 2 2 2 3 4 4 4 4 4 5 5 6 6 6 6 6 6 6 10 ··· 10 10 ··· 10 12 12 15 15 15 15 20 20 20 20 30 ··· 30 30 ··· 30 60 ··· 60 size 1 1 2 2 2 4 2 4 60 60 60 60 2 2 2 2 2 4 4 4 4 2 ··· 2 4 ··· 4 4 4 2 2 2 2 4 4 4 4 2 ··· 2 4 ··· 4 4 ··· 4

81 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + - - + - - + + - - + + image C1 C2 C2 C4 C4 S3 D4 D5 Dic3 Dic3 D6 Dic5 Dic5 D10 C3⋊D4 D15 C5⋊D4 Dic15 Dic15 D30 C15⋊7D4 C23⋊C4 C23.7D6 C23⋊Dic5 C23.7D30 kernel C23.7D30 C30.38D4 D4×C30 C2×C60 C22×C30 D4×C10 C2×C30 C6×D4 C2×C20 C22×C10 C22×C10 C2×C12 C22×C6 C22×C6 C2×C10 C2×D4 C2×C6 C2×C4 C23 C23 C22 C15 C5 C3 C1 # reps 1 2 1 2 2 1 2 2 1 1 1 2 2 2 4 4 8 4 4 4 16 1 2 4 8

Matrix representation of C23.7D30 in GL4(𝔽61) generated by

 14 44 0 0 33 47 0 0 59 59 31 17 2 0 44 30
,
 14 44 0 0 33 47 0 0 28 0 30 44 33 33 17 31
,
 60 0 0 0 0 60 0 0 0 0 60 0 0 0 0 60
,
 4 39 25 11 34 11 48 14 58 35 11 22 58 23 12 35
,
 33 56 0 0 59 28 0 0 47 34 56 29 3 51 58 5
`G:=sub<GL(4,GF(61))| [14,33,59,2,44,47,59,0,0,0,31,44,0,0,17,30],[14,33,28,33,44,47,0,33,0,0,30,17,0,0,44,31],[60,0,0,0,0,60,0,0,0,0,60,0,0,0,0,60],[4,34,58,58,39,11,35,23,25,48,11,12,11,14,22,35],[33,59,47,3,56,28,34,51,0,0,56,58,0,0,29,5] >;`

C23.7D30 in GAP, Magma, Sage, TeX

`C_2^3._7D_{30}`
`% in TeX`

`G:=Group("C2^3.7D30");`
`// GroupNames label`

`G:=SmallGroup(480,194);`
`// by ID`

`G=gap.SmallGroup(480,194);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,219,675,2693,18822]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^30=1,e^2=b*a=a*b,d*a*d^-1=e*a*e^-1=a*c=c*a,e*b*e^-1=b*c=c*b,b*d=d*b,c*d=d*c,c*e=e*c,e*d*e^-1=b*c*d^-1>;`
`// generators/relations`

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