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G = C425C8order 128 = 27

2nd semidirect product of C42 and C8 acting via C8/C4=C2

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

Aliases: C425C8, C43.4C2, (C2×C42).38C4, (C2×C4).62M4(2), C2.2(C425C4), (C22×C8).21C22, C22.37(C22×C8), C4.44(C422C2), C2.7(C42.6C4), C23.266(C22×C4), (C2×C42).997C22, C22.48(C2×M4(2)), C2.8(C42.12C4), (C22×C4).1623C23, C22.7C42.5C2, C22.57(C42⋊C2), (C2×C4).62(C2×C8), (C2×C4).931(C4○D4), (C22×C4).444(C2×C4), SmallGroup(128,571)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C425C8
C1C2C4C2×C4C22×C4C2×C42C43 — C425C8
C1C22 — C425C8
C1C22×C4 — C425C8
C1C2C2C22×C4 — C425C8

Generators and relations for C425C8
 G = < a,b,c | a4=b4=c8=1, ab=ba, cac-1=ab2, cbc-1=a2b-1 >

Subgroups: 180 in 120 conjugacy classes, 68 normal (8 characteristic)
C1, C2, C2 [×6], C4 [×4], C4 [×12], C22, C22 [×6], C8 [×4], C2×C4 [×18], C2×C4 [×12], C23, C42 [×4], C42 [×12], C2×C8 [×12], C22×C4, C22×C4 [×6], C2×C42, C2×C42 [×6], C22×C8 [×4], C22.7C42 [×6], C43, C425C8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], C23, C2×C8 [×6], M4(2) [×6], C22×C4, C4○D4 [×6], C42⋊C2 [×3], C422C2 [×4], C22×C8, C2×M4(2) [×3], C425C4, C42.12C4 [×3], C42.6C4 [×3], C425C8

Smallest permutation representation of C425C8
Regular action on 128 points
Generators in S128
(1 59 47 93)(2 116 48 38)(3 61 41 95)(4 118 42 40)(5 63 43 89)(6 120 44 34)(7 57 45 91)(8 114 46 36)(9 30 68 108)(10 87 69 53)(11 32 70 110)(12 81 71 55)(13 26 72 112)(14 83 65 49)(15 28 66 106)(16 85 67 51)(17 115 103 37)(18 60 104 94)(19 117 97 39)(20 62 98 96)(21 119 99 33)(22 64 100 90)(23 113 101 35)(24 58 102 92)(25 123 111 73)(27 125 105 75)(29 127 107 77)(31 121 109 79)(50 76 84 126)(52 78 86 128)(54 80 88 122)(56 74 82 124)
(1 29 17 85)(2 52 18 108)(3 31 19 87)(4 54 20 110)(5 25 21 81)(6 56 22 112)(7 27 23 83)(8 50 24 106)(9 116 78 60)(10 95 79 39)(11 118 80 62)(12 89 73 33)(13 120 74 64)(14 91 75 35)(15 114 76 58)(16 93 77 37)(26 44 82 100)(28 46 84 102)(30 48 86 104)(32 42 88 98)(34 124 90 72)(36 126 92 66)(38 128 94 68)(40 122 96 70)(41 109 97 53)(43 111 99 55)(45 105 101 49)(47 107 103 51)(57 125 113 65)(59 127 115 67)(61 121 117 69)(63 123 119 71)
(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)

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

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

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

56 conjugacy classes

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

56 irreducible representations

dim1111122
type+++
imageC1C2C2C4C8M4(2)C4○D4
kernelC425C8C22.7C42C43C2×C42C42C2×C4C2×C4
# reps1618161212

Matrix representation of C425C8 in GL5(𝔽17)

10000
001600
01000
000130
000013
,
10000
04000
00400
000132
00014
,
150000
010700
07700
00007
00050

G:=sub<GL(5,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,16,0,0,0,0,0,0,13,0,0,0,0,0,13],[1,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,13,1,0,0,0,2,4],[15,0,0,0,0,0,10,7,0,0,0,7,7,0,0,0,0,0,0,5,0,0,0,7,0] >;

C425C8 in GAP, Magma, Sage, TeX

C_4^2\rtimes_5C_8
% in TeX

G:=Group("C4^2:5C8");
// GroupNames label

G:=SmallGroup(128,571);
// by ID

G=gap.SmallGroup(128,571);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,176,422,58,124]);
// Polycyclic

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

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