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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 [×2], C2 [×3], C3, C4 [×5], C22, C22 [×2], C22 [×5], S3, C6, C6 [×2], C6 [×2], C2×C4 [×2], C2×C4 [×5], D4 [×2], C23, C23, C9, Dic3 [×3], C12 [×2], D6 [×3], C2×C6, C2×C6 [×2], C2×C6 [×2], C22⋊C4, C22⋊C4 [×2], C4⋊C4 [×2], C22×C4, C2×D4, D9, C18, C18 [×2], C18 [×2], C2×Dic3 [×5], C3⋊D4 [×2], C2×C12 [×2], C22×S3, C22×C6, C22.D4, Dic9 [×3], C36 [×2], D18 [×3], C2×C18, C2×C18 [×2], C2×C18 [×2], C4⋊Dic3 [×2], D6⋊C4 [×2], C3×C22⋊C4, C22×Dic3, C2×C3⋊D4, C2×Dic9, C2×Dic9 [×2], C2×Dic9 [×2], C9⋊D4 [×2], C2×C36 [×2], C22×D9, C22×C18, C23.21D6, C4⋊Dic9 [×2], D18⋊C4 [×2], C9×C22⋊C4, C22×Dic9, C2×C9⋊D4, C22.4D36
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], C2×D4, C4○D4 [×2], D9, D12 [×2], C22×S3, C22.D4, D18 [×3], C2×D12, D42S3 [×2], D36 [×2], C22×D9, C23.21D6, C2×D36, D42D9 [×2], C22.4D36

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

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

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

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

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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