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G = C8×C4⋊C4order 128 = 27

Direct product of C8 and C4⋊C4

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

Aliases: C8×C4⋊C4, C42(C4×C8), C85(C4⋊C8), C4⋊C815C4, (C4×C8)⋊13C4, C2.2(C8×D4), C2.1(C8×Q8), C4.44(C4×Q8), (C2×C8).65Q8, C4.168(C4×D4), (C2×C8).400D4, (C2×C4).48C42, C22.90(C4×D4), C22.22(C4×Q8), C42.313(C2×C4), C83(C2.C42), C22.26(C8○D4), C22.22(C22×C8), C22.30(C2×C42), C4.73(C42⋊C2), C2.5(C82M4(2)), C2.C42.28C4, (C2×C42).994C22, (C22×C8).590C22, C23.259(C22×C4), C82(C22.7C42), (C22×C4).1612C23, C22.7C42.49C2, C8(C2×C4⋊C8), C2.9(C2×C4×C8), C2.3(C4×C4⋊C4), C4⋊C8(C22×C8), (C2×C4×C8).16C2, (C2×C8)2(C4⋊C8), C4.73(C2×C4⋊C4), (C2×C4⋊C8).60C2, (C4×C4⋊C4).80C2, (C2×C4⋊C4).82C4, (C2×C4).39(C2×C8), (C2×C8).204(C2×C4), (C2×C4).332(C2×Q8), (C2×C4).1504(C2×D4), (C2×C4).922(C4○D4), (C2×C4).602(C22×C4), (C22×C4).377(C2×C4), (C2×C8)2(C2.C42), (C2×C8)2(C22.7C42), (C2×C8)(C4×C4⋊C4), (C2×C8)(C2×C4⋊C8), (C22×C8)(C4×C4⋊C4), SmallGroup(128,501)

Series: Derived Chief Lower central Upper central Jennings

C1C2 — C8×C4⋊C4
C1C2C22C2×C4C22×C4C22×C8C2×C4×C8 — C8×C4⋊C4
C1C2 — C8×C4⋊C4
C1C22×C8 — C8×C4⋊C4
C1C2C2C22×C4 — C8×C4⋊C4

Generators and relations for C8×C4⋊C4
 G = < a,b,c | a8=b4=c4=1, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 188 in 148 conjugacy classes, 108 normal (28 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×4], C4 [×10], C22 [×3], C22 [×4], C8 [×4], C8 [×6], C2×C4 [×6], C2×C4 [×16], C2×C4 [×10], C23, C42 [×4], C42 [×4], C4⋊C4 [×8], C2×C8 [×12], C2×C8 [×6], C22×C4 [×3], C22×C4 [×4], C2.C42 [×2], C4×C8 [×4], C4×C8 [×4], C4⋊C8 [×4], C2×C42, C2×C42 [×2], C2×C4⋊C4 [×2], C22×C8 [×2], C22×C8 [×2], C22.7C42 [×2], C4×C4⋊C4, C2×C4×C8, C2×C4×C8 [×2], C2×C4⋊C8, C8×C4⋊C4
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C8 [×8], C2×C4 [×18], D4 [×2], Q8 [×2], C23, C42 [×4], C4⋊C4 [×4], C2×C8 [×12], C22×C4 [×3], C2×D4, C2×Q8, C4○D4 [×2], C4×C8 [×4], C2×C42, C2×C4⋊C4, C42⋊C2, C4×D4 [×2], C4×Q8 [×2], C22×C8 [×2], C8○D4 [×2], C4×C4⋊C4, C2×C4×C8, C82M4(2), C8×D4 [×2], C8×Q8 [×2], C8×C4⋊C4

Smallest permutation representation of C8×C4⋊C4
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)
(1 108 33 120)(2 109 34 113)(3 110 35 114)(4 111 36 115)(5 112 37 116)(6 105 38 117)(7 106 39 118)(8 107 40 119)(9 69 25 63)(10 70 26 64)(11 71 27 57)(12 72 28 58)(13 65 29 59)(14 66 30 60)(15 67 31 61)(16 68 32 62)(17 83 125 79)(18 84 126 80)(19 85 127 73)(20 86 128 74)(21 87 121 75)(22 88 122 76)(23 81 123 77)(24 82 124 78)(41 101 53 95)(42 102 54 96)(43 103 55 89)(44 104 56 90)(45 97 49 91)(46 98 50 92)(47 99 51 93)(48 100 52 94)
(1 81 43 58)(2 82 44 59)(3 83 45 60)(4 84 46 61)(5 85 47 62)(6 86 48 63)(7 87 41 64)(8 88 42 57)(9 117 128 94)(10 118 121 95)(11 119 122 96)(12 120 123 89)(13 113 124 90)(14 114 125 91)(15 115 126 92)(16 116 127 93)(17 97 30 110)(18 98 31 111)(19 99 32 112)(20 100 25 105)(21 101 26 106)(22 102 27 107)(23 103 28 108)(24 104 29 109)(33 77 55 72)(34 78 56 65)(35 79 49 66)(36 80 50 67)(37 73 51 68)(38 74 52 69)(39 75 53 70)(40 76 54 71)

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

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

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

80 conjugacy classes

class 1 2A···2G4A···4H4I···4AF8A···8P8Q···8AN
order12···24···44···48···88···8
size11···11···12···21···12···2

80 irreducible representations

dim11111111112222
type++++++-
imageC1C2C2C2C2C4C4C4C4C8D4Q8C4○D4C8○D4
kernelC8×C4⋊C4C22.7C42C4×C4⋊C4C2×C4×C8C2×C4⋊C8C2.C42C4×C8C4⋊C8C2×C4⋊C4C4⋊C4C2×C8C2×C8C2×C4C22
# reps121314884322248

Matrix representation of C8×C4⋊C4 in GL4(𝔽17) generated by

4000
0200
00130
00013
,
16000
01600
0001
00160
,
4000
0400
0001
0010
G:=sub<GL(4,GF(17))| [4,0,0,0,0,2,0,0,0,0,13,0,0,0,0,13],[16,0,0,0,0,16,0,0,0,0,0,16,0,0,1,0],[4,0,0,0,0,4,0,0,0,0,0,1,0,0,1,0] >;

C8×C4⋊C4 in GAP, Magma, Sage, TeX

C_8\times C_4\rtimes C_4
% in TeX

G:=Group("C8xC4:C4");
// GroupNames label

G:=SmallGroup(128,501);
// by ID

G=gap.SmallGroup(128,501);
# by ID

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

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

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