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G = Q8xC22xC4  order 128 = 27

Direct product of C22xC4 and Q8

direct product, p-group, metabelian, nilpotent (class 2), monomial

Aliases: Q8xC22xC4, C22.9C25, C23.267C24, C24.655C23, C42.736C23, C2.5(C24xC4), C2.2(Q8xC23), C4.35(C23xC4), C4:C4.511C23, (C2xC4).156C24, (Q8xC23).16C2, C23.148(C2xQ8), (C22xC42).35C2, C22.48(C23xC4), (C2xQ8).477C23, C23.376(C4oD4), C22.49(C22xQ8), C23.296(C22xC4), (C23xC4).720C22, (C22xC4).1582C23, (C2xC42).1138C22, (C22xQ8).511C22, C4o(C2xC4xQ8), (C2xC4)o2(C4xQ8), C4o3(C22xC4:C4), C4:C4o4(C22xC4), (C22xC4)o(C4xQ8), C2.3(C22xC4oD4), (C22xC4:C4).51C2, (C2xC4:C4).980C22, (C22xC4).422(C2xC4), (C2xC4).478(C22xC4), C22.144(C2xC4oD4), (C2xC4)o(C2xC4xQ8), (C2xC4)o6(C2xC4:C4), (C22xC4)o(C2xC4xQ8), (C22xC4)o4(C2xC4:C4), (C2xC4)o3(C22xC4:C4), (C22xC4)o3(C22xC4:C4), SmallGroup(128,2155)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C2 — Q8xC22xC4
Chief series
C1 — C2 — C22 — C23 — C24 — C23xC4 — C22xC42 — Q8xC22xC4
Lower central
C1 — C2 — Q8xC22xC4
Upper central
C1 — C23xC4 — Q8xC22xC4
Jennings
C1 — C22 — Q8xC22xC4

Generators and relations for Q8xC22xC4
 G = < a,b,c,d,e | a2=b2=c4=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 892 in 832 conjugacy classes, 772 normal (9 characteristic)
C1, C2, C2, C4, C4, C22, C22, C2xC4, C2xC4, Q8, C23, C42, C4:C4, C22xC4, C22xC4, C2xQ8, C24, C2xC42, C2xC4:C4, C4xQ8, C23xC4, C23xC4, C22xQ8, C22xC42, C22xC4:C4, C2xC4xQ8, Q8xC23, Q8xC22xC4
Quotients: C1, C2, C4, C22, C2xC4, Q8, C23, C22xC4, C2xQ8, C4oD4, C24, C4xQ8, C23xC4, C22xQ8, C2xC4oD4, C25, C2xC4xQ8, C24xC4, Q8xC23, C22xC4oD4, Q8xC22xC4

Smallest permutation representation of Q8xC22xC4
►Regular action on 128 points
Generators in S128
(1 21)(2 22)(3 23)(4 24)(5 43)(6 44)(7 41)(8 42)(9 15)(10 16)(11 13)(12 14)(17 32)(18 29)(19 30)(20 31)(25 103)(26 104)(27 101)(28 102)(33 47)(34 48)(35 45)(36 46)(37 75)(38 76)(39 73)(40 74)(49 55)(50 56)(51 53)(52 54)(57 63)(58 64)(59 61)(60 62)(65 79)(66 80)(67 77)(68 78)(69 107)(70 108)(71 105)(72 106)(81 87)(82 88)(83 85)(84 86)(89 95)(90 96)(91 93)(92 94)(97 111)(98 112)(99 109)(100 110)(113 119)(114 120)(115 117)(116 118)(121 127)(122 128)(123 125)(124 126)

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

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

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

80 conjugacy classes

class 1 2A···2O4A···4P4Q···4BL
order12···24···44···4
size11···11···12···2

80 irreducible representations

dim11111122
type+++++-
imageC1C2C2C2C2C4Q8C4oD4
kernelQ8xC22xC4C22xC42C22xC4:C4C2xC4xQ8Q8xC23C22xQ8C22xC4C23
# reps1332413288

Matrix representation of Q8xC22xC4 ►in GL5(F5)

10000
04000
00400
00010
00001
,
10000
01000
00400
00040
00004
,
20000
04000
00100
00020
00002
,
40000
01000
00400
00030
00012
,
40000
01000
00100
00014
00024

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

Q8xC22xC4 in GAP, Magma, Sage, TeX 

Q_8\times C_2^2\times C_4
 
% in TeX
 
G:=Group("Q8xC2^2xC4");
 
// GroupNames label
 
G:=SmallGroup(128,2155);
 
// by ID
 
G=gap.SmallGroup(128,2155);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,2,-2,2,448,477,232,520]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^2=c^4=d^4=1,e^2=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,c*e=e*c,e*d*e^-1=d^-1>;
 
// generators/relations
 

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