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## G = C22⋊C4×D15order 480 = 25·3·5

### Direct product of C22⋊C4 and D15

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — C22⋊C4×D15
 Chief series C1 — C5 — C15 — C30 — C2×C30 — C22×D15 — C23×D15 — C22⋊C4×D15
 Lower central C15 — C30 — C22⋊C4×D15
 Upper central C1 — C22 — C22⋊C4

Generators and relations for C22⋊C4×D15
G = < a,b,c,d,e | a2=b2=c4=d15=e2=1, cac-1=ab=ba, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 1796 in 264 conjugacy classes, 71 normal (25 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C5, S3, C6, C6, C6, C2×C4, C2×C4, C23, C23, D5, C10, C10, C10, Dic3, C12, D6, C2×C6, C2×C6, C2×C6, C15, C22⋊C4, C22⋊C4, C22×C4, C24, Dic5, C20, D10, C2×C10, C2×C10, C2×C10, C4×S3, C2×Dic3, C2×C12, C22×S3, C22×C6, D15, D15, C30, C30, C30, C2×C22⋊C4, C4×D5, C2×Dic5, C2×C20, C22×D5, C22×C10, D6⋊C4, C6.D4, C3×C22⋊C4, S3×C2×C4, S3×C23, Dic15, C60, D30, D30, C2×C30, C2×C30, C2×C30, D10⋊C4, C23.D5, C5×C22⋊C4, C2×C4×D5, C23×D5, S3×C22⋊C4, C4×D15, C2×Dic15, C2×C60, C22×D15, C22×D15, C22×D15, C22×C30, D5×C22⋊C4, D303C4, C30.38D4, C15×C22⋊C4, C2×C4×D15, C23×D15, C22⋊C4×D15
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D5, D6, C22⋊C4, C22×C4, C2×D4, D10, C4×S3, C22×S3, D15, C2×C22⋊C4, C4×D5, C22×D5, S3×C2×C4, S3×D4, D30, C2×C4×D5, D4×D5, S3×C22⋊C4, C4×D15, C22×D15, D5×C22⋊C4, C2×C4×D15, D4×D15, C22⋊C4×D15

Smallest permutation representation of C22⋊C4×D15
On 120 points
Generators in S120
(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)(61 109)(62 110)(63 111)(64 112)(65 113)(66 114)(67 115)(68 116)(69 117)(70 118)(71 119)(72 120)(73 106)(74 107)(75 108)(76 98)(77 99)(78 100)(79 101)(80 102)(81 103)(82 104)(83 105)(84 91)(85 92)(86 93)(87 94)(88 95)(89 96)(90 97)
(1 24)(2 25)(3 26)(4 27)(5 28)(6 29)(7 30)(8 16)(9 17)(10 18)(11 19)(12 20)(13 21)(14 22)(15 23)(31 57)(32 58)(33 59)(34 60)(35 46)(36 47)(37 48)(38 49)(39 50)(40 51)(41 52)(42 53)(43 54)(44 55)(45 56)(61 76)(62 77)(63 78)(64 79)(65 80)(66 81)(67 82)(68 83)(69 84)(70 85)(71 86)(72 87)(73 88)(74 89)(75 90)(91 117)(92 118)(93 119)(94 120)(95 106)(96 107)(97 108)(98 109)(99 110)(100 111)(101 112)(102 113)(103 114)(104 115)(105 116)
(1 99 39 62)(2 100 40 63)(3 101 41 64)(4 102 42 65)(5 103 43 66)(6 104 44 67)(7 105 45 68)(8 91 31 69)(9 92 32 70)(10 93 33 71)(11 94 34 72)(12 95 35 73)(13 96 36 74)(14 97 37 75)(15 98 38 61)(16 117 57 84)(17 118 58 85)(18 119 59 86)(19 120 60 87)(20 106 46 88)(21 107 47 89)(22 108 48 90)(23 109 49 76)(24 110 50 77)(25 111 51 78)(26 112 52 79)(27 113 53 80)(28 114 54 81)(29 115 55 82)(30 116 56 83)
(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 23)(2 22)(3 21)(4 20)(5 19)(6 18)(7 17)(8 16)(9 30)(10 29)(11 28)(12 27)(13 26)(14 25)(15 24)(31 57)(32 56)(33 55)(34 54)(35 53)(36 52)(37 51)(38 50)(39 49)(40 48)(41 47)(42 46)(43 60)(44 59)(45 58)(61 77)(62 76)(63 90)(64 89)(65 88)(66 87)(67 86)(68 85)(69 84)(70 83)(71 82)(72 81)(73 80)(74 79)(75 78)(91 117)(92 116)(93 115)(94 114)(95 113)(96 112)(97 111)(98 110)(99 109)(100 108)(101 107)(102 106)(103 120)(104 119)(105 118)

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

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

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

90 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 3 4A 4B 4C 4D 4E 4F 4G 4H 5A 5B 6A 6B 6C 6D 6E 10A ··· 10F 10G 10H 10I 10J 12A 12B 12C 12D 15A 15B 15C 15D 20A ··· 20H 30A ··· 30L 30M ··· 30T 60A ··· 60P order 1 2 2 2 2 2 2 2 2 2 2 2 3 4 4 4 4 4 4 4 4 5 5 6 6 6 6 6 10 ··· 10 10 10 10 10 12 12 12 12 15 15 15 15 20 ··· 20 30 ··· 30 30 ··· 30 60 ··· 60 size 1 1 1 1 2 2 15 15 15 15 30 30 2 2 2 2 2 30 30 30 30 2 2 2 2 2 4 4 2 ··· 2 4 4 4 4 4 4 4 4 2 2 2 2 4 ··· 4 2 ··· 2 4 ··· 4 4 ··· 4

90 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 S3 D4 D5 D6 D6 D10 D10 C4×S3 D15 C4×D5 D30 D30 C4×D15 S3×D4 D4×D5 D4×D15 kernel C22⋊C4×D15 D30⋊3C4 C30.38D4 C15×C22⋊C4 C2×C4×D15 C23×D15 C22×D15 C5×C22⋊C4 D30 C3×C22⋊C4 C2×C20 C22×C10 C2×C12 C22×C6 C2×C10 C22⋊C4 C2×C6 C2×C4 C23 C22 C10 C6 C2 # reps 1 2 1 1 2 1 8 1 4 2 2 1 4 2 4 4 8 8 4 16 2 4 8

Matrix representation of C22⋊C4×D15 in GL6(𝔽61)

 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 1 0 0 0 0 0 0 60
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 60 0 0 0 0 0 0 60
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 11 0 0 0 0 0 0 11 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 30 23 0 0 0 0 2 28 0 0 0 0 0 0 0 1 0 0 0 0 60 60 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 37 8 0 0 0 0 12 24 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 60 0 0 0 0 0 0 60

G:=sub<GL(6,GF(61))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,60],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[30,2,0,0,0,0,23,28,0,0,0,0,0,0,0,60,0,0,0,0,1,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[37,12,0,0,0,0,8,24,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,60,0,0,0,0,0,0,60] >;

C22⋊C4×D15 in GAP, Magma, Sage, TeX

C_2^2\rtimes C_4\times D_{15}
% in TeX

G:=Group("C2^2:C4xD15");
// GroupNames label

G:=SmallGroup(480,845);
// by ID

G=gap.SmallGroup(480,845);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,219,58,2693,18822]);
// Polycyclic

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

׿
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