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G = C22×C4.Q8order 128 = 27

Direct product of C22 and C4.Q8

direct product, p-group, metabelian, nilpotent (class 3), monomial

Aliases: C22×C4.Q8, C24.191D4, C23.48SD16, C89(C22×C4), (C22×C8)⋊18C4, C4.1(C22×Q8), (C23×C8).21C2, C4.45(C23×C4), C4⋊C4.346C23, C23.85(C4⋊C4), (C2×C4).183C24, (C2×C8).587C23, C23.838(C2×D4), (C22×C4).603D4, (C22×C4).101Q8, C2.3(C22×SD16), C22.80(C2×SD16), (C22×C8).564C22, (C23×C4).694C22, C22.130(C22×D4), (C22×C4).1504C23, (C2×C8)⋊38(C2×C4), C4.63(C2×C4⋊C4), (C2×C4).841(C2×D4), C2.22(C22×C4⋊C4), C22.74(C2×C4⋊C4), (C2×C4).237(C2×Q8), (C2×C4).149(C4⋊C4), (C22×C4⋊C4).42C2, (C2×C4⋊C4).901C22, (C22×C4).494(C2×C4), (C2×C4).571(C22×C4), SmallGroup(128,1639)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — C22×C4.Q8
C1C2C22C2×C4C22×C4C23×C4C23×C8 — C22×C4.Q8
C1C2C4 — C22×C4.Q8
C1C24C23×C4 — C22×C4.Q8
C1C2C2C2×C4 — C22×C4.Q8

Generators and relations for C22×C4.Q8
 G = < a,b,c,d,e | a2=b2=c4=1, d4=c2, e2=c-1d2, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=c-1, ede-1=d3 >

Subgroups: 460 in 300 conjugacy classes, 220 normal (10 characteristic)
C1, C2, C2 [×14], C4 [×2], C4 [×6], C4 [×8], C22, C22 [×34], C8 [×8], C2×C4, C2×C4 [×27], C2×C4 [×32], C23 [×15], C4⋊C4 [×8], C4⋊C4 [×12], C2×C8 [×28], C22×C4 [×14], C22×C4 [×20], C24, C4.Q8 [×16], C2×C4⋊C4 [×12], C2×C4⋊C4 [×6], C22×C8 [×14], C23×C4, C23×C4 [×2], C2×C4.Q8 [×12], C22×C4⋊C4 [×2], C23×C8, C22×C4.Q8
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×4], Q8 [×4], C23 [×15], C4⋊C4 [×16], SD16 [×8], C22×C4 [×14], C2×D4 [×6], C2×Q8 [×6], C24, C4.Q8 [×16], C2×C4⋊C4 [×12], C2×SD16 [×12], C23×C4, C22×D4, C22×Q8, C2×C4.Q8 [×12], C22×C4⋊C4, C22×SD16 [×2], C22×C4.Q8

Smallest permutation representation of C22×C4.Q8
Regular action on 128 points
Generators in S128
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 9)(8 10)(17 46)(18 47)(19 48)(20 41)(21 42)(22 43)(23 44)(24 45)(25 85)(26 86)(27 87)(28 88)(29 81)(30 82)(31 83)(32 84)(33 64)(34 57)(35 58)(36 59)(37 60)(38 61)(39 62)(40 63)(49 71)(50 72)(51 65)(52 66)(53 67)(54 68)(55 69)(56 70)(73 89)(74 90)(75 91)(76 92)(77 93)(78 94)(79 95)(80 96)(97 118)(98 119)(99 120)(100 113)(101 114)(102 115)(103 116)(104 117)(105 124)(106 125)(107 126)(108 127)(109 128)(110 121)(111 122)(112 123)
(1 22)(2 23)(3 24)(4 17)(5 18)(6 19)(7 20)(8 21)(9 41)(10 42)(11 43)(12 44)(13 45)(14 46)(15 47)(16 48)(25 34)(26 35)(27 36)(28 37)(29 38)(30 39)(31 40)(32 33)(49 123)(50 124)(51 125)(52 126)(53 127)(54 128)(55 121)(56 122)(57 85)(58 86)(59 87)(60 88)(61 81)(62 82)(63 83)(64 84)(65 106)(66 107)(67 108)(68 109)(69 110)(70 111)(71 112)(72 105)(73 117)(74 118)(75 119)(76 120)(77 113)(78 114)(79 115)(80 116)(89 104)(90 97)(91 98)(92 99)(93 100)(94 101)(95 102)(96 103)
(1 57 5 61)(2 58 6 62)(3 59 7 63)(4 60 8 64)(9 40 13 36)(10 33 14 37)(11 34 15 38)(12 35 16 39)(17 88 21 84)(18 81 22 85)(19 82 23 86)(20 83 24 87)(25 47 29 43)(26 48 30 44)(27 41 31 45)(28 42 32 46)(49 100 53 104)(50 101 54 97)(51 102 55 98)(52 103 56 99)(65 115 69 119)(66 116 70 120)(67 117 71 113)(68 118 72 114)(73 112 77 108)(74 105 78 109)(75 106 79 110)(76 107 80 111)(89 123 93 127)(90 124 94 128)(91 125 95 121)(92 126 96 122)
(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)
(1 117 63 69)(2 120 64 72)(3 115 57 67)(4 118 58 70)(5 113 59 65)(6 116 60 68)(7 119 61 71)(8 114 62 66)(9 98 38 49)(10 101 39 52)(11 104 40 55)(12 99 33 50)(13 102 34 53)(14 97 35 56)(15 100 36 51)(16 103 37 54)(17 74 86 111)(18 77 87 106)(19 80 88 109)(20 75 81 112)(21 78 82 107)(22 73 83 110)(23 76 84 105)(24 79 85 108)(25 127 45 95)(26 122 46 90)(27 125 47 93)(28 128 48 96)(29 123 41 91)(30 126 42 94)(31 121 43 89)(32 124 44 92)

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

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

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

56 conjugacy classes

class 1 2A···2O4A···4H4I···4X8A···8P
order12···24···44···48···8
size11···12···24···42···2

56 irreducible representations

dim111112222
type+++++-+
imageC1C2C2C2C4D4Q8D4SD16
kernelC22×C4.Q8C2×C4.Q8C22×C4⋊C4C23×C8C22×C8C22×C4C22×C4C24C23
# reps112211634116

Matrix representation of C22×C4.Q8 in GL6(𝔽17)

1600000
010000
0016000
0001600
000010
000001
,
100000
0160000
0016000
0001600
000010
000001
,
100000
0160000
0016000
0001600
0000016
000010
,
1600000
0160000
000100
0016000
0000125
00001212
,
100000
0130000
006400
0041100
00001212
0000125

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,1,0,0,0,0,16,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,12,12,0,0,0,0,5,12],[1,0,0,0,0,0,0,13,0,0,0,0,0,0,6,4,0,0,0,0,4,11,0,0,0,0,0,0,12,12,0,0,0,0,12,5] >;

C22×C4.Q8 in GAP, Magma, Sage, TeX

C_2^2\times C_4.Q_8
% in TeX

G:=Group("C2^2xC4.Q8");
// GroupNames label

G:=SmallGroup(128,1639);
// by ID

G=gap.SmallGroup(128,1639);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,120,2804,172]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^4=1,d^4=c^2,e^2=c^-1*d^2,a*b=b*a,a*c=c*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,e*c*e^-1=c^-1,e*d*e^-1=d^3>;
// generators/relations

׿
×
𝔽