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G = C22×Q8⋊C4order 128 = 27

Direct product of C22 and Q8⋊C4

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

Aliases: C22×Q8⋊C4, C24.190D4, C23.23Q16, C23.47SD16, C4.2(C23×C4), Q84(C22×C4), (C23×C8).10C2, (C22×Q8)⋊20C4, C2.1(C22×Q16), C4⋊C4.339C23, (C2×C4).172C24, (C2×C8).466C23, (C22×C4).601D4, C23.837(C2×D4), C4.137(C22×D4), (Q8×C23).10C2, C22.44(C2×Q16), C2.2(C22×SD16), (C2×Q8).330C23, C22.79(C2×SD16), (C22×C8).503C22, (C23×C4).689C22, C22.122(C22×D4), C23.233(C22⋊C4), (C22×C4).1496C23, (C22×Q8).454C22, (C2×Q8)⋊37(C2×C4), C4.72(C2×C22⋊C4), (C2×C4).1403(C2×D4), (C22×C4⋊C4).41C2, (C2×C4⋊C4).898C22, (C22×C4).414(C2×C4), (C2×C4).457(C22×C4), C2.34(C22×C22⋊C4), (C2×C4).283(C22⋊C4), C22.137(C2×C22⋊C4), SmallGroup(128,1623)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — C22×Q8⋊C4
C1C2C22C2×C4C22×C4C23×C4Q8×C23 — C22×Q8⋊C4
C1C2C4 — C22×Q8⋊C4
C1C24C23×C4 — C22×Q8⋊C4
C1C2C2C2×C4 — C22×Q8⋊C4

Generators and relations for C22×Q8⋊C4
 G = < a,b,c,d,e | a2=b2=c4=e4=1, d2=c2, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=ece-1=c-1, ede-1=c-1d >

Subgroups: 636 in 408 conjugacy classes, 220 normal (14 characteristic)
C1, C2 [×3], C2 [×12], C4 [×2], C4 [×6], C4 [×12], C22, C22 [×34], C8 [×4], C2×C4, C2×C4 [×27], C2×C4 [×44], Q8 [×8], Q8 [×28], C23 [×15], C4⋊C4 [×4], C4⋊C4 [×6], C2×C8 [×4], C2×C8 [×12], C22×C4 [×14], C22×C4 [×24], C2×Q8 [×28], C2×Q8 [×42], C24, Q8⋊C4 [×16], C2×C4⋊C4 [×6], C2×C4⋊C4 [×3], C22×C8 [×6], C22×C8 [×4], C23×C4, C23×C4 [×2], C22×Q8 [×14], C22×Q8 [×7], C2×Q8⋊C4 [×12], C22×C4⋊C4, C23×C8, Q8×C23, C22×Q8⋊C4
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×8], C23 [×15], C22⋊C4 [×16], SD16 [×4], Q16 [×4], C22×C4 [×14], C2×D4 [×12], C24, Q8⋊C4 [×16], C2×C22⋊C4 [×12], C2×SD16 [×6], C2×Q16 [×6], C23×C4, C22×D4 [×2], C2×Q8⋊C4 [×12], C22×C22⋊C4, C22×SD16, C22×Q16, C22×Q8⋊C4

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

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

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

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

56 conjugacy classes

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

56 irreducible representations

dim1111112222
type+++++++-
imageC1C2C2C2C2C4D4D4SD16Q16
kernelC22×Q8⋊C4C2×Q8⋊C4C22×C4⋊C4C23×C8Q8×C23C22×Q8C22×C4C24C23C23
# reps112111167188

Matrix representation of C22×Q8⋊C4 in GL7(𝔽17)

16000000
0100000
0010000
00016000
00001600
00000160
00000016
,
16000000
01600000
00160000
0001000
0000100
0000010
0000001
,
1000000
01600000
00160000
00016000
00001600
0000001
00000160
,
1000000
00160000
01600000
0001200
00001600
0000071
00000110
,
16000000
0470000
010130000
00012500
0005500
0000064
00000411

G:=sub<GL(7,GF(17))| [16,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,2,16,0,0,0,0,0,0,0,7,1,0,0,0,0,0,1,10],[16,0,0,0,0,0,0,0,4,10,0,0,0,0,0,7,13,0,0,0,0,0,0,0,12,5,0,0,0,0,0,5,5,0,0,0,0,0,0,0,6,4,0,0,0,0,0,4,11] >;

C22×Q8⋊C4 in GAP, Magma, Sage, TeX

C_2^2\times Q_8\rtimes C_4
% in TeX

G:=Group("C2^2xQ8:C4");
// GroupNames label

G:=SmallGroup(128,1623);
// by ID

G=gap.SmallGroup(128,1623);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,456,2804,1411,172]);
// Polycyclic

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

׿
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