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G = C4×C4.Q8order 128 = 27

Direct product of C4 and C4.Q8

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

Aliases: C4×C4.Q8, C83C42, C42.52Q8, (C4×C8)⋊20C4, C4.3(C4×Q8), C2.4(C4×SD16), C4.23(C2×C42), C22.95(C4×D4), C42.316(C2×C4), (C2×C4).130SD16, C23.734(C2×D4), (C22×C4).814D4, C44(C22.4Q16), C22.38(C4○D8), C22.43(C2×SD16), C4.47(C42⋊C2), C22.4Q16.54C2, (C22×C8).561C22, (C22×C4).1315C23, (C2×C42).1051C22, C2.3(C23.25D4), (C2×C4×C8).53C2, (C4×C4⋊C4).9C2, C2.13(C4×C4⋊C4), C2.3(C2×C4.Q8), C4⋊C4.146(C2×C4), (C2×C8).205(C2×C4), C22.58(C2×C4⋊C4), (C2×C4).184(C2×Q8), (C2×C4.Q8).35C2, (C2×C4).164(C4⋊C4), (C2×C4).546(C4○D4), (C2×C4⋊C4).750C22, (C2×C4).356(C22×C4), (C2×C4)2(C22.4Q16), SmallGroup(128,506)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — C4×C4.Q8
C1C2C22C2×C4C22×C4C2×C42C4×C4⋊C4 — C4×C4.Q8
C1C2C4 — C4×C4.Q8
C1C22×C4C2×C42 — C4×C4.Q8
C1C2C2C22×C4 — C4×C4.Q8

Generators and relations for C4×C4.Q8
 G = < a,b,c,d | a4=b4=1, c4=b2, d2=b-1c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c3 >

Subgroups: 228 in 140 conjugacy classes, 92 normal (18 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×6], C4 [×10], C22, C22 [×6], C8 [×4], C8 [×2], C2×C4 [×2], C2×C4 [×12], C2×C4 [×18], C23, C42 [×4], C42 [×4], C4⋊C4 [×8], C4⋊C4 [×4], C2×C8 [×8], C2×C8 [×2], C22×C4 [×3], C22×C4 [×4], C2.C42 [×2], C4×C8 [×4], C4.Q8 [×8], C2×C42, C2×C42 [×2], C2×C4⋊C4 [×4], C22×C8 [×2], C22.4Q16 [×2], C4×C4⋊C4 [×2], C2×C4×C8, C2×C4.Q8 [×2], C4×C4.Q8
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], D4 [×2], Q8 [×2], C23, C42 [×4], C4⋊C4 [×4], SD16 [×4], C22×C4 [×3], C2×D4, C2×Q8, C4○D4 [×2], C4.Q8 [×4], C2×C42, C2×C4⋊C4, C42⋊C2, C4×D4 [×2], C4×Q8 [×2], C2×SD16 [×2], C4○D8 [×2], C4×C4⋊C4, C2×C4.Q8, C23.25D4, C4×SD16 [×4], C4×C4.Q8

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

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

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

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

56 conjugacy classes

class 1 2A···2G4A···4H4I···4P4Q···4AF8A···8P
order12···24···44···44···48···8
size11···11···12···24···42···2

56 irreducible representations

dim111111122222
type+++++-+
imageC1C2C2C2C2C4C4Q8D4SD16C4○D4C4○D8
kernelC4×C4.Q8C22.4Q16C4×C4⋊C4C2×C4×C8C2×C4.Q8C4×C8C4.Q8C42C22×C4C2×C4C2×C4C22
# reps1221281622848

Matrix representation of C4×C4.Q8 in GL4(𝔽17) generated by

4000
01600
00130
00013
,
16000
01600
00016
0010
,
16000
01600
001212
00512
,
13000
0400
00160
0001
G:=sub<GL(4,GF(17))| [4,0,0,0,0,16,0,0,0,0,13,0,0,0,0,13],[16,0,0,0,0,16,0,0,0,0,0,1,0,0,16,0],[16,0,0,0,0,16,0,0,0,0,12,5,0,0,12,12],[13,0,0,0,0,4,0,0,0,0,16,0,0,0,0,1] >;

C4×C4.Q8 in GAP, Magma, Sage, TeX

C_4\times C_4.Q_8
% in TeX

G:=Group("C4xC4.Q8");
// GroupNames label

G:=SmallGroup(128,506);
// by ID

G=gap.SmallGroup(128,506);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,142,1018,248,1411,172]);
// Polycyclic

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

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