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G = C8.M4(2)  order 128 = 27

4th non-split extension by C8 of M4(2) acting via M4(2)/C4=C22

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

Aliases: C8.4M4(2), C42.650C23, C8⋊C8.3C2, C82C8.7C2, (C2×C8).184D4, Q8⋊C8.10C2, C2.D8.20C4, C4.17(C8○D4), (C2×Q16).12C4, (C4×Q16).16C2, C84Q8.14C2, C2.10(C86D4), C4⋊C8.228C22, (C4×C8).319C22, Q8⋊C4.12C4, C22.141(C4×D4), C4.11(C2×M4(2)), (C4×Q8).16C22, C2.11(C8.26D4), C2.5(Q16⋊C4), C4.141(C8.C22), C4⋊C4.63(C2×C4), (C2×C8).58(C2×C4), (C2×Q8).56(C2×C4), (C2×C4).1486(C2×D4), (C2×C4).511(C4○D4), (C2×C4).342(C22×C4), SmallGroup(128,325)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C8.M4(2)
C1C2C22C2×C4C42C4×C8C84Q8 — C8.M4(2)
C1C2C2×C4 — C8.M4(2)
C1C2×C4C4×C8 — C8.M4(2)
C1C22C22C42 — C8.M4(2)

Generators and relations for C8.M4(2)
 G = < a,b,c | a8=b8=1, c2=a4, bab-1=a3, cac-1=a-1, cbc-1=a6b5 >

Subgroups: 120 in 74 conjugacy classes, 42 normal (22 characteristic)
C1, C2 [×3], C4 [×2], C4 [×2], C4 [×5], C22, C8 [×2], C8 [×6], C2×C4 [×3], C2×C4 [×4], Q8 [×4], C42, C42 [×2], C4⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], C2×C8 [×2], C2×C8 [×3], Q16 [×2], C2×Q8 [×2], C4×C8 [×3], C8⋊C4 [×2], Q8⋊C4 [×2], C4⋊C8 [×2], C4⋊C8 [×2], C2.D8, C4×Q8 [×2], C2×Q16, C8⋊C8, Q8⋊C8 [×2], C82C8, C4×Q16, C84Q8 [×2], C8.M4(2)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, M4(2) [×2], C22×C4, C2×D4, C4○D4, C4×D4, C2×M4(2), C8○D4, C8.C22 [×2], C86D4, Q16⋊C4, C8.26D4, C8.M4(2)

Smallest permutation representation of C8.M4(2)
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 86 15 24 54 47 124 92)(2 81 16 19 55 42 125 95)(3 84 9 22 56 45 126 90)(4 87 10 17 49 48 127 93)(5 82 11 20 50 43 128 96)(6 85 12 23 51 46 121 91)(7 88 13 18 52 41 122 94)(8 83 14 21 53 44 123 89)(25 119 79 67 97 109 33 60)(26 114 80 70 98 112 34 63)(27 117 73 65 99 107 35 58)(28 120 74 68 100 110 36 61)(29 115 75 71 101 105 37 64)(30 118 76 66 102 108 38 59)(31 113 77 69 103 111 39 62)(32 116 78 72 104 106 40 57)
(1 34 5 38)(2 33 6 37)(3 40 7 36)(4 39 8 35)(9 32 13 28)(10 31 14 27)(11 30 15 26)(12 29 16 25)(17 105 21 109)(18 112 22 108)(19 111 23 107)(20 110 24 106)(41 63 45 59)(42 62 46 58)(43 61 47 57)(44 60 48 64)(49 77 53 73)(50 76 54 80)(51 75 55 79)(52 74 56 78)(65 81 69 85)(66 88 70 84)(67 87 71 83)(68 86 72 82)(89 119 93 115)(90 118 94 114)(91 117 95 113)(92 116 96 120)(97 121 101 125)(98 128 102 124)(99 127 103 123)(100 126 104 122)

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

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

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

32 conjugacy classes

class 1 2A2B2C4A4B4C4D4E4F4G4H4I4J4K4L8A···8L8M8N8O8P
order12224444444444448···88888
size11111111222288884···48888

32 irreducible representations

dim111111111222244
type+++++++-
imageC1C2C2C2C2C2C4C4C4D4M4(2)C4○D4C8○D4C8.C22C8.26D4
kernelC8.M4(2)C8⋊C8Q8⋊C8C82C8C4×Q16C84Q8Q8⋊C4C2.D8C2×Q16C2×C8C8C2×C4C4C4C2
# reps112112422242422

Matrix representation of C8.M4(2) in GL6(𝔽17)

1600000
0160000
0012800
008500
000032
0000214
,
1130000
1260000
0000153
000032
008500
005900
,
1470000
1130000
000010
000001
0016000
0001600

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,12,8,0,0,0,0,8,5,0,0,0,0,0,0,3,2,0,0,0,0,2,14],[11,12,0,0,0,0,3,6,0,0,0,0,0,0,0,0,8,5,0,0,0,0,5,9,0,0,15,3,0,0,0,0,3,2,0,0],[14,11,0,0,0,0,7,3,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,1,0,0,0,0,0,0,1,0,0] >;

C8.M4(2) in GAP, Magma, Sage, TeX

C_8.M_4(2)
% in TeX

G:=Group("C8.M4(2)");
// GroupNames label

G:=SmallGroup(128,325);
// by ID

G=gap.SmallGroup(128,325);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,232,2102,723,100,1123,570,136,172]);
// Polycyclic

G:=Group<a,b,c|a^8=b^8=1,c^2=a^4,b*a*b^-1=a^3,c*a*c^-1=a^-1,c*b*c^-1=a^6*b^5>;
// generators/relations

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