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G = C4219Q8order 128 = 27

6th semidirect product of C42 and Q8 acting via Q8/C4=C2

p-group, metabelian, nilpotent (class 2), monomial, rational

Aliases: C4219Q8, C43.19C2, C42.348D4, C23.768C24, C41(C4⋊Q8), C4.10(C41D4), C429C4.42C2, (C22×C4).270C23, C22.474(C22×D4), C22.186(C22×Q8), (C2×C42).1099C22, (C22×Q8).254C22, C2.24(C2×C4⋊Q8), (C2×C4⋊Q8).41C2, (C2×C4).838(C2×D4), C2.19(C2×C41D4), (C2×C4).234(C2×Q8), (C2×C4⋊C4).570C22, SmallGroup(128,1600)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C4219Q8
C1C2C22C23C22×C4C2×C42C43 — C4219Q8
C1C23 — C4219Q8
C1C23 — C4219Q8
C1C23 — C4219Q8

Generators and relations for C4219Q8
 G = < a,b,c,d | a4=b4=c4=1, d2=c2, ab=ba, ac=ca, dad-1=a-1, bc=cb, dbd-1=b-1, dcd-1=c-1 >

Subgroups: 516 in 324 conjugacy classes, 180 normal (6 characteristic)
C1, C2, C2 [×6], C4 [×28], C4 [×8], C22 [×7], C2×C4 [×42], C2×C4 [×24], Q8 [×16], C23, C42 [×28], C4⋊C4 [×48], C22×C4, C22×C4 [×14], C2×Q8 [×24], C2×C42 [×7], C2×C4⋊C4 [×24], C4⋊Q8 [×24], C22×Q8 [×4], C43, C429C4 [×8], C2×C4⋊Q8 [×6], C4219Q8
Quotients: C1, C2 [×15], C22 [×35], D4 [×12], Q8 [×16], C23 [×15], C2×D4 [×18], C2×Q8 [×24], C24, C41D4 [×4], C4⋊Q8 [×24], C22×D4 [×3], C22×Q8 [×4], C2×C41D4, C2×C4⋊Q8 [×6], C4219Q8

Smallest permutation representation of C4219Q8
Regular action on 128 points
Generators in S128
(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)
(1 79 22 29)(2 80 23 30)(3 77 24 31)(4 78 21 32)(5 69 104 88)(6 70 101 85)(7 71 102 86)(8 72 103 87)(9 61 60 28)(10 62 57 25)(11 63 58 26)(12 64 59 27)(13 100 19 37)(14 97 20 38)(15 98 17 39)(16 99 18 40)(33 51 75 54)(34 52 76 55)(35 49 73 56)(36 50 74 53)(41 114 107 83)(42 115 108 84)(43 116 105 81)(44 113 106 82)(45 66 110 118)(46 67 111 119)(47 68 112 120)(48 65 109 117)(89 121 126 96)(90 122 127 93)(91 123 128 94)(92 124 125 95)
(1 14 11 49)(2 15 12 50)(3 16 9 51)(4 13 10 52)(5 44 48 126)(6 41 45 127)(7 42 46 128)(8 43 47 125)(17 59 53 23)(18 60 54 24)(19 57 55 21)(20 58 56 22)(25 34 32 37)(26 35 29 38)(27 36 30 39)(28 33 31 40)(61 75 77 99)(62 76 78 100)(63 73 79 97)(64 74 80 98)(65 96 69 113)(66 93 70 114)(67 94 71 115)(68 95 72 116)(81 120 124 87)(82 117 121 88)(83 118 122 85)(84 119 123 86)(89 104 106 109)(90 101 107 110)(91 102 108 111)(92 103 105 112)
(1 95 11 116)(2 94 12 115)(3 93 9 114)(4 96 10 113)(5 100 48 76)(6 99 45 75)(7 98 46 74)(8 97 47 73)(13 65 52 69)(14 68 49 72)(15 67 50 71)(16 66 51 70)(17 119 53 86)(18 118 54 85)(19 117 55 88)(20 120 56 87)(21 121 57 82)(22 124 58 81)(23 123 59 84)(24 122 60 83)(25 106 32 89)(26 105 29 92)(27 108 30 91)(28 107 31 90)(33 101 40 110)(34 104 37 109)(35 103 38 112)(36 102 39 111)(41 77 127 61)(42 80 128 64)(43 79 125 63)(44 78 126 62)

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

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

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

44 conjugacy classes

class 1 2A···2G4A···4AB4AC···4AJ
order12···24···44···4
size11···12···28···8

44 irreducible representations

dim111122
type+++++-
imageC1C2C2C2D4Q8
kernelC4219Q8C43C429C4C2×C4⋊Q8C42C42
# reps11861216

Matrix representation of C4219Q8 in GL6(𝔽5)

040000
100000
004300
001100
000040
000004
,
100000
010000
004000
000400
000001
000040
,
100000
010000
004300
001100
000040
000004
,
010000
100000
003100
000200
000040
000001

G:=sub<GL(6,GF(5))| [0,1,0,0,0,0,4,0,0,0,0,0,0,0,4,1,0,0,0,0,3,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,1,0,0,0,0,3,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,3,0,0,0,0,0,1,2,0,0,0,0,0,0,4,0,0,0,0,0,0,1] >;

C4219Q8 in GAP, Magma, Sage, TeX

C_4^2\rtimes_{19}Q_8
% in TeX

G:=Group("C4^2:19Q8");
// GroupNames label

G:=SmallGroup(128,1600);
// by ID

G=gap.SmallGroup(128,1600);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,120,758,184,2019,248]);
// Polycyclic

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

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