Copied to
clipboard

G = C36.23D4order 288 = 25·32

23rd non-split extension by C36 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C36.23D4, (C2×Q8)⋊6D9, (Q8×C18)⋊4C2, D18⋊C416C2, (C4×Dic9)⋊7C2, (C2×D36).9C2, (C2×C4).57D18, (C2×C12).66D6, C18.58(C2×D4), C94(C4.4D4), (C6×Q8).17S3, C4.11(C9⋊D4), C18.37(C4○D4), C12.20(C3⋊D4), (C2×C36).64C22, (C2×C18).59C23, C3.(C12.23D4), C2.9(Q83D9), C6.45(Q83S3), C22.65(C22×D9), (C2×Dic9).43C22, (C22×D9).13C22, C2.22(C2×C9⋊D4), C6.106(C2×C3⋊D4), (C2×C6).216(C22×S3), SmallGroup(288,157)

Series: Derived Chief Lower central Upper central

C1C2×C18 — C36.23D4
C1C3C9C18C2×C18C22×D9C2×D36 — C36.23D4
C9C2×C18 — C36.23D4
C1C22C2×Q8

Generators and relations for C36.23D4
 G = < a,b,c | a36=b4=c2=1, bab-1=a17, cac=a-1, cbc=a18b-1 >

Subgroups: 564 in 114 conjugacy classes, 44 normal (18 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×2], C4 [×4], C22, C22 [×6], S3 [×2], C6, C6 [×2], C2×C4, C2×C4 [×2], C2×C4 [×2], D4 [×2], Q8 [×2], C23 [×2], C9, Dic3 [×2], C12 [×2], C12 [×2], D6 [×6], C2×C6, C42, C22⋊C4 [×4], C2×D4, C2×Q8, D9 [×2], C18, C18 [×2], D12 [×2], C2×Dic3 [×2], C2×C12, C2×C12 [×2], C3×Q8 [×2], C22×S3 [×2], C4.4D4, Dic9 [×2], C36 [×2], C36 [×2], D18 [×6], C2×C18, C4×Dic3, D6⋊C4 [×4], C2×D12, C6×Q8, D36 [×2], C2×Dic9 [×2], C2×C36, C2×C36 [×2], Q8×C9 [×2], C22×D9 [×2], C12.23D4, C4×Dic9, D18⋊C4 [×4], C2×D36, Q8×C18, C36.23D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], C2×D4, C4○D4 [×2], D9, C3⋊D4 [×2], C22×S3, C4.4D4, D18 [×3], Q83S3 [×2], C2×C3⋊D4, C9⋊D4 [×2], C22×D9, C12.23D4, Q83D9 [×2], C2×C9⋊D4, C36.23D4

Smallest permutation representation of C36.23D4
On 144 points
Generators in S144
(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)
(1 137 101 54)(2 118 102 71)(3 135 103 52)(4 116 104 69)(5 133 105 50)(6 114 106 67)(7 131 107 48)(8 112 108 65)(9 129 73 46)(10 110 74 63)(11 127 75 44)(12 144 76 61)(13 125 77 42)(14 142 78 59)(15 123 79 40)(16 140 80 57)(17 121 81 38)(18 138 82 55)(19 119 83 72)(20 136 84 53)(21 117 85 70)(22 134 86 51)(23 115 87 68)(24 132 88 49)(25 113 89 66)(26 130 90 47)(27 111 91 64)(28 128 92 45)(29 109 93 62)(30 126 94 43)(31 143 95 60)(32 124 96 41)(33 141 97 58)(34 122 98 39)(35 139 99 56)(36 120 100 37)
(2 36)(3 35)(4 34)(5 33)(6 32)(7 31)(8 30)(9 29)(10 28)(11 27)(12 26)(13 25)(14 24)(15 23)(16 22)(17 21)(18 20)(37 136)(38 135)(39 134)(40 133)(41 132)(42 131)(43 130)(44 129)(45 128)(46 127)(47 126)(48 125)(49 124)(50 123)(51 122)(52 121)(53 120)(54 119)(55 118)(56 117)(57 116)(58 115)(59 114)(60 113)(61 112)(62 111)(63 110)(64 109)(65 144)(66 143)(67 142)(68 141)(69 140)(70 139)(71 138)(72 137)(73 93)(74 92)(75 91)(76 90)(77 89)(78 88)(79 87)(80 86)(81 85)(82 84)(94 108)(95 107)(96 106)(97 105)(98 104)(99 103)(100 102)

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

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

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

54 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F4G4H6A6B6C9A9B9C12A···12F18A···18I36A···36R
order12222234444444466699912···1218···1836···36
size1111363622244181818182222224···42···24···4

54 irreducible representations

dim111112222222244
type++++++++++++
imageC1C2C2C2C2S3D4D6C4○D4D9C3⋊D4D18C9⋊D4Q83S3Q83D9
kernelC36.23D4C4×Dic9D18⋊C4C2×D36Q8×C18C6×Q8C36C2×C12C18C2×Q8C12C2×C4C4C6C2
# reps1141112343491226

Matrix representation of C36.23D4 in GL6(𝔽37)

3600000
0360000
0061100
00261700
000014
00001836
,
2160000
25160000
000100
001000
00003113
000036
,
100000
30360000
000100
001000
000010
00001836

G:=sub<GL(6,GF(37))| [36,0,0,0,0,0,0,36,0,0,0,0,0,0,6,26,0,0,0,0,11,17,0,0,0,0,0,0,1,18,0,0,0,0,4,36],[21,25,0,0,0,0,6,16,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,31,3,0,0,0,0,13,6],[1,30,0,0,0,0,0,36,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,18,0,0,0,0,0,36] >;

C36.23D4 in GAP, Magma, Sage, TeX

C_{36}._{23}D_4
% in TeX

G:=Group("C36.23D4");
// GroupNames label

G:=SmallGroup(288,157);
// by ID

G=gap.SmallGroup(288,157);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,120,254,219,100,6725,292,9414]);
// Polycyclic

G:=Group<a,b,c|a^36=b^4=c^2=1,b*a*b^-1=a^17,c*a*c=a^-1,c*b*c=a^18*b^-1>;
// generators/relations

׿
×
𝔽