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G = C22.4D36order 288 = 25·32

3rd non-split extension by C22 of D36 acting via D36/D18=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C22.4D36, C23.21D18, D18⋊C46C2, C4⋊Dic95C2, C22⋊C46D9, C18.6(C2×D4), (C2×C12).5D6, C2.8(C2×D36), (C2×C18).4D4, (C2×C4).7D18, (C2×C6).5D12, C6.35(C2×D12), (C2×C36).6C22, (C22×C6).46D6, C18.23(C4○D4), (C2×C18).27C23, (C22×Dic9)⋊2C2, C92(C22.D4), C6.78(D42S3), C2.10(D42D9), C3.(C23.21D6), (C22×D9).6C22, C22.45(C22×D9), (C22×C18).16C22, (C2×Dic9).27C22, (C9×C22⋊C4)⋊4C2, (C2×C9⋊D4).5C2, (C3×C22⋊C4).4S3, (C2×C6).184(C22×S3), SmallGroup(288,96)

Series: Derived Chief Lower central Upper central

C1C2×C18 — C22.4D36
C1C3C9C18C2×C18C22×D9C2×C9⋊D4 — C22.4D36
C9C2×C18 — C22.4D36
C1C22C22⋊C4

Generators and relations for C22.4D36
 G = < a,b,c,d | a2=b2=c36=1, d2=b, cac-1=ab=ba, ad=da, bc=cb, bd=db, dcd-1=bc-1 >

Subgroups: 524 in 117 conjugacy classes, 44 normal (20 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, D4, C23, C23, C9, Dic3, C12, D6, C2×C6, C2×C6, C2×C6, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, D9, C18, C18, C18, C2×Dic3, C3⋊D4, C2×C12, C22×S3, C22×C6, C22.D4, Dic9, C36, D18, C2×C18, C2×C18, C2×C18, C4⋊Dic3, D6⋊C4, C3×C22⋊C4, C22×Dic3, C2×C3⋊D4, C2×Dic9, C2×Dic9, C2×Dic9, C9⋊D4, C2×C36, C22×D9, C22×C18, C23.21D6, C4⋊Dic9, D18⋊C4, C9×C22⋊C4, C22×Dic9, C2×C9⋊D4, C22.4D36
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, D9, D12, C22×S3, C22.D4, D18, C2×D12, D42S3, D36, C22×D9, C23.21D6, C2×D36, D42D9, C22.4D36

Smallest permutation representation of C22.4D36
On 144 points
Generators in S144
(1 89)(2 136)(3 91)(4 138)(5 93)(6 140)(7 95)(8 142)(9 97)(10 144)(11 99)(12 110)(13 101)(14 112)(15 103)(16 114)(17 105)(18 116)(19 107)(20 118)(21 73)(22 120)(23 75)(24 122)(25 77)(26 124)(27 79)(28 126)(29 81)(30 128)(31 83)(32 130)(33 85)(34 132)(35 87)(36 134)(37 88)(38 135)(39 90)(40 137)(41 92)(42 139)(43 94)(44 141)(45 96)(46 143)(47 98)(48 109)(49 100)(50 111)(51 102)(52 113)(53 104)(54 115)(55 106)(56 117)(57 108)(58 119)(59 74)(60 121)(61 76)(62 123)(63 78)(64 125)(65 80)(66 127)(67 82)(68 129)(69 84)(70 131)(71 86)(72 133)
(1 38)(2 39)(3 40)(4 41)(5 42)(6 43)(7 44)(8 45)(9 46)(10 47)(11 48)(12 49)(13 50)(14 51)(15 52)(16 53)(17 54)(18 55)(19 56)(20 57)(21 58)(22 59)(23 60)(24 61)(25 62)(26 63)(27 64)(28 65)(29 66)(30 67)(31 68)(32 69)(33 70)(34 71)(35 72)(36 37)(73 119)(74 120)(75 121)(76 122)(77 123)(78 124)(79 125)(80 126)(81 127)(82 128)(83 129)(84 130)(85 131)(86 132)(87 133)(88 134)(89 135)(90 136)(91 137)(92 138)(93 139)(94 140)(95 141)(96 142)(97 143)(98 144)(99 109)(100 110)(101 111)(102 112)(103 113)(104 114)(105 115)(106 116)(107 117)(108 118)
(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)(109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144)
(1 37 38 36)(2 35 39 72)(3 71 40 34)(4 33 41 70)(5 69 42 32)(6 31 43 68)(7 67 44 30)(8 29 45 66)(9 65 46 28)(10 27 47 64)(11 63 48 26)(12 25 49 62)(13 61 50 24)(14 23 51 60)(15 59 52 22)(16 21 53 58)(17 57 54 20)(18 19 55 56)(73 104 119 114)(74 113 120 103)(75 102 121 112)(76 111 122 101)(77 100 123 110)(78 109 124 99)(79 98 125 144)(80 143 126 97)(81 96 127 142)(82 141 128 95)(83 94 129 140)(84 139 130 93)(85 92 131 138)(86 137 132 91)(87 90 133 136)(88 135 134 89)(105 108 115 118)(106 117 116 107)

G:=sub<Sym(144)| (1,89)(2,136)(3,91)(4,138)(5,93)(6,140)(7,95)(8,142)(9,97)(10,144)(11,99)(12,110)(13,101)(14,112)(15,103)(16,114)(17,105)(18,116)(19,107)(20,118)(21,73)(22,120)(23,75)(24,122)(25,77)(26,124)(27,79)(28,126)(29,81)(30,128)(31,83)(32,130)(33,85)(34,132)(35,87)(36,134)(37,88)(38,135)(39,90)(40,137)(41,92)(42,139)(43,94)(44,141)(45,96)(46,143)(47,98)(48,109)(49,100)(50,111)(51,102)(52,113)(53,104)(54,115)(55,106)(56,117)(57,108)(58,119)(59,74)(60,121)(61,76)(62,123)(63,78)(64,125)(65,80)(66,127)(67,82)(68,129)(69,84)(70,131)(71,86)(72,133), (1,38)(2,39)(3,40)(4,41)(5,42)(6,43)(7,44)(8,45)(9,46)(10,47)(11,48)(12,49)(13,50)(14,51)(15,52)(16,53)(17,54)(18,55)(19,56)(20,57)(21,58)(22,59)(23,60)(24,61)(25,62)(26,63)(27,64)(28,65)(29,66)(30,67)(31,68)(32,69)(33,70)(34,71)(35,72)(36,37)(73,119)(74,120)(75,121)(76,122)(77,123)(78,124)(79,125)(80,126)(81,127)(82,128)(83,129)(84,130)(85,131)(86,132)(87,133)(88,134)(89,135)(90,136)(91,137)(92,138)(93,139)(94,140)(95,141)(96,142)(97,143)(98,144)(99,109)(100,110)(101,111)(102,112)(103,113)(104,114)(105,115)(106,116)(107,117)(108,118), (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)(109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144), (1,37,38,36)(2,35,39,72)(3,71,40,34)(4,33,41,70)(5,69,42,32)(6,31,43,68)(7,67,44,30)(8,29,45,66)(9,65,46,28)(10,27,47,64)(11,63,48,26)(12,25,49,62)(13,61,50,24)(14,23,51,60)(15,59,52,22)(16,21,53,58)(17,57,54,20)(18,19,55,56)(73,104,119,114)(74,113,120,103)(75,102,121,112)(76,111,122,101)(77,100,123,110)(78,109,124,99)(79,98,125,144)(80,143,126,97)(81,96,127,142)(82,141,128,95)(83,94,129,140)(84,139,130,93)(85,92,131,138)(86,137,132,91)(87,90,133,136)(88,135,134,89)(105,108,115,118)(106,117,116,107)>;

G:=Group( (1,89)(2,136)(3,91)(4,138)(5,93)(6,140)(7,95)(8,142)(9,97)(10,144)(11,99)(12,110)(13,101)(14,112)(15,103)(16,114)(17,105)(18,116)(19,107)(20,118)(21,73)(22,120)(23,75)(24,122)(25,77)(26,124)(27,79)(28,126)(29,81)(30,128)(31,83)(32,130)(33,85)(34,132)(35,87)(36,134)(37,88)(38,135)(39,90)(40,137)(41,92)(42,139)(43,94)(44,141)(45,96)(46,143)(47,98)(48,109)(49,100)(50,111)(51,102)(52,113)(53,104)(54,115)(55,106)(56,117)(57,108)(58,119)(59,74)(60,121)(61,76)(62,123)(63,78)(64,125)(65,80)(66,127)(67,82)(68,129)(69,84)(70,131)(71,86)(72,133), (1,38)(2,39)(3,40)(4,41)(5,42)(6,43)(7,44)(8,45)(9,46)(10,47)(11,48)(12,49)(13,50)(14,51)(15,52)(16,53)(17,54)(18,55)(19,56)(20,57)(21,58)(22,59)(23,60)(24,61)(25,62)(26,63)(27,64)(28,65)(29,66)(30,67)(31,68)(32,69)(33,70)(34,71)(35,72)(36,37)(73,119)(74,120)(75,121)(76,122)(77,123)(78,124)(79,125)(80,126)(81,127)(82,128)(83,129)(84,130)(85,131)(86,132)(87,133)(88,134)(89,135)(90,136)(91,137)(92,138)(93,139)(94,140)(95,141)(96,142)(97,143)(98,144)(99,109)(100,110)(101,111)(102,112)(103,113)(104,114)(105,115)(106,116)(107,117)(108,118), (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)(109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144), (1,37,38,36)(2,35,39,72)(3,71,40,34)(4,33,41,70)(5,69,42,32)(6,31,43,68)(7,67,44,30)(8,29,45,66)(9,65,46,28)(10,27,47,64)(11,63,48,26)(12,25,49,62)(13,61,50,24)(14,23,51,60)(15,59,52,22)(16,21,53,58)(17,57,54,20)(18,19,55,56)(73,104,119,114)(74,113,120,103)(75,102,121,112)(76,111,122,101)(77,100,123,110)(78,109,124,99)(79,98,125,144)(80,143,126,97)(81,96,127,142)(82,141,128,95)(83,94,129,140)(84,139,130,93)(85,92,131,138)(86,137,132,91)(87,90,133,136)(88,135,134,89)(105,108,115,118)(106,117,116,107) );

G=PermutationGroup([[(1,89),(2,136),(3,91),(4,138),(5,93),(6,140),(7,95),(8,142),(9,97),(10,144),(11,99),(12,110),(13,101),(14,112),(15,103),(16,114),(17,105),(18,116),(19,107),(20,118),(21,73),(22,120),(23,75),(24,122),(25,77),(26,124),(27,79),(28,126),(29,81),(30,128),(31,83),(32,130),(33,85),(34,132),(35,87),(36,134),(37,88),(38,135),(39,90),(40,137),(41,92),(42,139),(43,94),(44,141),(45,96),(46,143),(47,98),(48,109),(49,100),(50,111),(51,102),(52,113),(53,104),(54,115),(55,106),(56,117),(57,108),(58,119),(59,74),(60,121),(61,76),(62,123),(63,78),(64,125),(65,80),(66,127),(67,82),(68,129),(69,84),(70,131),(71,86),(72,133)], [(1,38),(2,39),(3,40),(4,41),(5,42),(6,43),(7,44),(8,45),(9,46),(10,47),(11,48),(12,49),(13,50),(14,51),(15,52),(16,53),(17,54),(18,55),(19,56),(20,57),(21,58),(22,59),(23,60),(24,61),(25,62),(26,63),(27,64),(28,65),(29,66),(30,67),(31,68),(32,69),(33,70),(34,71),(35,72),(36,37),(73,119),(74,120),(75,121),(76,122),(77,123),(78,124),(79,125),(80,126),(81,127),(82,128),(83,129),(84,130),(85,131),(86,132),(87,133),(88,134),(89,135),(90,136),(91,137),(92,138),(93,139),(94,140),(95,141),(96,142),(97,143),(98,144),(99,109),(100,110),(101,111),(102,112),(103,113),(104,114),(105,115),(106,116),(107,117),(108,118)], [(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),(109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144)], [(1,37,38,36),(2,35,39,72),(3,71,40,34),(4,33,41,70),(5,69,42,32),(6,31,43,68),(7,67,44,30),(8,29,45,66),(9,65,46,28),(10,27,47,64),(11,63,48,26),(12,25,49,62),(13,61,50,24),(14,23,51,60),(15,59,52,22),(16,21,53,58),(17,57,54,20),(18,19,55,56),(73,104,119,114),(74,113,120,103),(75,102,121,112),(76,111,122,101),(77,100,123,110),(78,109,124,99),(79,98,125,144),(80,143,126,97),(81,96,127,142),(82,141,128,95),(83,94,129,140),(84,139,130,93),(85,92,131,138),(86,137,132,91),(87,90,133,136),(88,135,134,89),(105,108,115,118),(106,117,116,107)]])

54 conjugacy classes

class 1 2A2B2C2D2E2F 3 4A4B4C4D4E4F4G6A6B6C6D6E9A9B9C12A12B12C12D18A···18I18J···18O36A···36L
order122222234444444666669991212121218···1818···1836···36
size1111223624418181818362224422244442···24···44···4

54 irreducible representations

dim111111222222222244
type+++++++++++++++--
imageC1C2C2C2C2C2S3D4D6D6C4○D4D9D12D18D18D36D42S3D42D9
kernelC22.4D36C4⋊Dic9D18⋊C4C9×C22⋊C4C22×Dic9C2×C9⋊D4C3×C22⋊C4C2×C18C2×C12C22×C6C18C22⋊C4C2×C6C2×C4C23C22C6C2
# reps1221111221434631226

Matrix representation of C22.4D36 in GL4(𝔽37) generated by

36000
03600
003125
0066
,
1000
0100
00360
00036
,
252900
83300
003635
0001
,
273200
51000
003635
0011
G:=sub<GL(4,GF(37))| [36,0,0,0,0,36,0,0,0,0,31,6,0,0,25,6],[1,0,0,0,0,1,0,0,0,0,36,0,0,0,0,36],[25,8,0,0,29,33,0,0,0,0,36,0,0,0,35,1],[27,5,0,0,32,10,0,0,0,0,36,1,0,0,35,1] >;

C22.4D36 in GAP, Magma, Sage, TeX

C_2^2._4D_{36}
% in TeX

G:=Group("C2^2.4D36");
// GroupNames label

G:=SmallGroup(288,96);
// by ID

G=gap.SmallGroup(288,96);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,254,219,142,6725,292,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^36=1,d^2=b,c*a*c^-1=a*b=b*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=b*c^-1>;
// generators/relations

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