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G = C4⋊C814C4order 128 = 27

10th semidirect product of C4⋊C8 and C4 acting via C4/C2=C2

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

Aliases: C4⋊C814C4, C89(C4⋊C4), C8⋊C415C4, C4.46(C4×Q8), (C2×C8).59Q8, C4.170(C4×D4), (C2×C8).385D4, C2.3(C89D4), (C2×C4).26C42, C2.2(C84Q8), C22.92(C4×D4), C22.24(C4×Q8), C42.132(C2×C4), C2.13(C4×M4(2)), (C2×C4).35M4(2), C22.28(C8○D4), C22.56(C2×C42), C4.75(C42⋊C2), C2.C42.16C4, (C2×C42).239C22, C23.261(C22×C4), (C22×C8).477C22, C22.43(C2×M4(2)), C2.17(C82M4(2)), (C22×C4).1614C23, C22.7C42.43C2, (C2×C4×C8).63C2, (C4×C4⋊C4).8C2, C4.75(C2×C4⋊C4), C2.10(C4×C4⋊C4), (C2×C4⋊C8).54C2, (C2×C4⋊C4).49C4, (C2×C8).138(C2×C4), (C2×C4).334(C2×Q8), (C2×C8⋊C4).27C2, (C2×C4).1506(C2×D4), (C2×C4).924(C4○D4), (C22×C4).111(C2×C4), (C2×C4).604(C22×C4), SmallGroup(128,503)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C4⋊C814C4
C1C2C22C2×C4C22×C4C22×C8C2×C4×C8 — C4⋊C814C4
C1C22 — C4⋊C814C4
C1C22×C4 — C4⋊C814C4
C1C2C2C22×C4 — C4⋊C814C4

Generators and relations for C4⋊C814C4
 G = < a,b,c | a4=b8=c4=1, bab-1=a-1, cac-1=a-1b4, bc=cb >

Subgroups: 188 in 136 conjugacy classes, 88 normal (36 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×10], C22 [×3], C22 [×4], C8 [×4], C8 [×6], C2×C4 [×6], C2×C4 [×8], C2×C4 [×14], C23, C42 [×4], C42 [×4], C4⋊C4 [×4], C2×C8 [×12], C2×C8 [×6], C22×C4 [×3], C22×C4 [×4], C2.C42 [×2], C4×C8 [×2], C8⋊C4 [×4], C8⋊C4 [×2], C4⋊C8 [×4], C2×C42 [×3], C2×C4⋊C4 [×2], C22×C8 [×4], C22.7C42 [×2], C4×C4⋊C4, C2×C4×C8, C2×C8⋊C4 [×2], C2×C4⋊C8, C4⋊C814C4
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], D4 [×2], Q8 [×2], C23, C42 [×4], C4⋊C4 [×4], M4(2) [×4], C22×C4 [×3], C2×D4, C2×Q8, C4○D4 [×2], C2×C42, C2×C4⋊C4, C42⋊C2, C4×D4 [×2], C4×Q8 [×2], C2×M4(2) [×2], C8○D4 [×2], C4×C4⋊C4, C4×M4(2), C82M4(2), C89D4 [×2], C84Q8 [×2], C4⋊C814C4

Smallest permutation representation of C4⋊C814C4
Regular action on 128 points
Generators in S128
(1 80 34 65)(2 66 35 73)(3 74 36 67)(4 68 37 75)(5 76 38 69)(6 70 39 77)(7 78 40 71)(8 72 33 79)(9 81 125 94)(10 95 126 82)(11 83 127 96)(12 89 128 84)(13 85 121 90)(14 91 122 86)(15 87 123 92)(16 93 124 88)(17 57 32 51)(18 52 25 58)(19 59 26 53)(20 54 27 60)(21 61 28 55)(22 56 29 62)(23 63 30 49)(24 50 31 64)(41 116 103 110)(42 111 104 117)(43 118 97 112)(44 105 98 119)(45 120 99 106)(46 107 100 113)(47 114 101 108)(48 109 102 115)
(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 18 96 104)(2 19 89 97)(3 20 90 98)(4 21 91 99)(5 22 92 100)(6 23 93 101)(7 24 94 102)(8 17 95 103)(9 105 78 60)(10 106 79 61)(11 107 80 62)(12 108 73 63)(13 109 74 64)(14 110 75 57)(15 111 76 58)(16 112 77 59)(25 83 42 34)(26 84 43 35)(27 85 44 36)(28 86 45 37)(29 87 46 38)(30 88 47 39)(31 81 48 40)(32 82 41 33)(49 128 114 66)(50 121 115 67)(51 122 116 68)(52 123 117 69)(53 124 118 70)(54 125 119 71)(55 126 120 72)(56 127 113 65)

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

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

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

56 conjugacy classes

class 1 2A···2G4A···4H4I···4P4Q···4X8A···8P8Q···8X
order12···24···44···44···48···88···8
size11···11···12···24···42···24···4

56 irreducible representations

dim111111111122222
type+++++++-
imageC1C2C2C2C2C2C4C4C4C4D4Q8M4(2)C4○D4C8○D4
kernelC4⋊C814C4C22.7C42C4×C4⋊C4C2×C4×C8C2×C8⋊C4C2×C4⋊C8C2.C42C8⋊C4C4⋊C8C2×C4⋊C4C2×C8C2×C8C2×C4C2×C4C22
# reps121121488422848

Matrix representation of C4⋊C814C4 in GL6(𝔽17)

13120000
340000
003700
00111400
000079
0000210
,
800000
080000
009000
000900
000004
000010
,
0150000
900000
000200
009000
0000130
0000013

G:=sub<GL(6,GF(17))| [13,3,0,0,0,0,12,4,0,0,0,0,0,0,3,11,0,0,0,0,7,14,0,0,0,0,0,0,7,2,0,0,0,0,9,10],[8,0,0,0,0,0,0,8,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,0,1,0,0,0,0,4,0],[0,9,0,0,0,0,15,0,0,0,0,0,0,0,0,9,0,0,0,0,2,0,0,0,0,0,0,0,13,0,0,0,0,0,0,13] >;

C4⋊C814C4 in GAP, Magma, Sage, TeX

C_4\rtimes C_8\rtimes_{14}C_4
% in TeX

G:=Group("C4:C8:14C4");
// GroupNames label

G:=SmallGroup(128,503);
// by ID

G=gap.SmallGroup(128,503);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,1430,142,172]);
// Polycyclic

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

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