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G = C22.7M5(2)  order 128 = 27

1st central extension by C22 of M5(2)

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

Aliases: C22.7M5(2), (C2×C4)⋊2C16, (C2×C16)⋊9C4, (C2×C8).9C8, C2.2(C4×C16), C2.1(C4⋊C16), C8.40(C4⋊C4), C4.23(C4⋊C8), (C2×C8).64Q8, (C2×C8).398D4, (C22×C4).7C8, (C2×C4).83C42, C22.14(C4×C8), (C2×C42).27C4, C23.42(C2×C8), C22.6(C2×C16), (C22×C8).23C4, (C22×C16).1C2, C2.3(C165C4), C4.15(C8⋊C4), C22.18(C4⋊C8), C8.54(C22⋊C4), C2.1(C22⋊C16), C4.30(C22⋊C8), (C2×C4).86M4(2), C22.37(C22⋊C8), (C22×C8).588C22, C4.24(C2.C42), C2.2(C22.7C42), (C2×C4×C8).5C2, (C2×C4).94(C2×C8), (C2×C8).258(C2×C4), (C2×C4).158(C4⋊C4), (C22×C4).501(C2×C4), (C2×C4).384(C22⋊C4), SmallGroup(128,106)

Series: Derived Chief Lower central Upper central Jennings

C1C2 — C22.7M5(2)
C1C2C4C2×C4C2×C8C22×C8C2×C4×C8 — C22.7M5(2)
C1C2 — C22.7M5(2)
C1C22×C8 — C22.7M5(2)
C1C2C2C2C2C4C4C22×C8 — C22.7M5(2)

Generators and relations for C22.7M5(2)
 G = < a,b,c,d | a2=b2=c16=1, d2=b, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=ac9 >

Subgroups: 104 in 80 conjugacy classes, 56 normal (26 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×4], C22 [×3], C22 [×4], C8 [×4], C8 [×2], C2×C4 [×6], C2×C4 [×4], C2×C4 [×4], C23, C16 [×4], C42 [×2], C2×C8 [×2], C2×C8 [×6], C2×C8 [×2], C22×C4, C22×C4 [×2], C4×C8 [×2], C2×C16 [×4], C2×C16 [×4], C2×C42, C22×C8 [×2], C2×C4×C8, C22×C16 [×2], C22.7M5(2)
Quotients: C1, C2 [×3], C4 [×6], C22, C8 [×4], C2×C4 [×3], D4 [×3], Q8, C16 [×4], C42, C22⋊C4 [×3], C4⋊C4 [×3], C2×C8 [×2], M4(2) [×2], C2.C42, C4×C8, C8⋊C4, C22⋊C8 [×2], C4⋊C8 [×2], C2×C16 [×2], M5(2) [×2], C22.7C42, C4×C16, C165C4, C22⋊C16 [×2], C4⋊C16 [×2], C22.7M5(2)

Smallest permutation representation of C22.7M5(2)
Regular action on 128 points
Generators in S128
(1 108)(2 109)(3 110)(4 111)(5 112)(6 97)(7 98)(8 99)(9 100)(10 101)(11 102)(12 103)(13 104)(14 105)(15 106)(16 107)(17 95)(18 96)(19 81)(20 82)(21 83)(22 84)(23 85)(24 86)(25 87)(26 88)(27 89)(28 90)(29 91)(30 92)(31 93)(32 94)(33 128)(34 113)(35 114)(36 115)(37 116)(38 117)(39 118)(40 119)(41 120)(42 121)(43 122)(44 123)(45 124)(46 125)(47 126)(48 127)(49 75)(50 76)(51 77)(52 78)(53 79)(54 80)(55 65)(56 66)(57 67)(58 68)(59 69)(60 70)(61 71)(62 72)(63 73)(64 74)
(1 91)(2 92)(3 93)(4 94)(5 95)(6 96)(7 81)(8 82)(9 83)(10 84)(11 85)(12 86)(13 87)(14 88)(15 89)(16 90)(17 112)(18 97)(19 98)(20 99)(21 100)(22 101)(23 102)(24 103)(25 104)(26 105)(27 106)(28 107)(29 108)(30 109)(31 110)(32 111)(33 65)(34 66)(35 67)(36 68)(37 69)(38 70)(39 71)(40 72)(41 73)(42 74)(43 75)(44 76)(45 77)(46 78)(47 79)(48 80)(49 122)(50 123)(51 124)(52 125)(53 126)(54 127)(55 128)(56 113)(57 114)(58 115)(59 116)(60 117)(61 118)(62 119)(63 120)(64 121)
(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)
(1 125 91 52)(2 39 92 71)(3 127 93 54)(4 41 94 73)(5 113 95 56)(6 43 96 75)(7 115 81 58)(8 45 82 77)(9 117 83 60)(10 47 84 79)(11 119 85 62)(12 33 86 65)(13 121 87 64)(14 35 88 67)(15 123 89 50)(16 37 90 69)(17 66 112 34)(18 49 97 122)(19 68 98 36)(20 51 99 124)(21 70 100 38)(22 53 101 126)(23 72 102 40)(24 55 103 128)(25 74 104 42)(26 57 105 114)(27 76 106 44)(28 59 107 116)(29 78 108 46)(30 61 109 118)(31 80 110 48)(32 63 111 120)

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

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

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

80 conjugacy classes

class 1 2A···2G4A···4H4I···4P8A···8P8Q···8X16A···16AF
order12···24···44···48···88···816···16
size11···11···12···21···12···22···2

80 irreducible representations

dim1111111112222
type++++-
imageC1C2C2C4C4C4C8C8C16D4Q8M4(2)M5(2)
kernelC22.7M5(2)C2×C4×C8C22×C16C2×C16C2×C42C22×C8C2×C8C22×C4C2×C4C2×C8C2×C8C2×C4C22
# reps11282288323148

Matrix representation of C22.7M5(2) in GL4(𝔽17) generated by

16000
0100
00160
00016
,
16000
01600
00160
00016
,
3000
01300
00816
00149
,
4000
01300
00116
00216
G:=sub<GL(4,GF(17))| [16,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[3,0,0,0,0,13,0,0,0,0,8,14,0,0,16,9],[4,0,0,0,0,13,0,0,0,0,1,2,0,0,16,16] >;

C22.7M5(2) in GAP, Magma, Sage, TeX

C_2^2._7M_5(2)
% in TeX

G:=Group("C2^2.7M5(2)");
// GroupNames label

G:=SmallGroup(128,106);
// by ID

G=gap.SmallGroup(128,106);
# by ID

G:=PCGroup([7,-2,2,-2,2,2,-2,-2,56,85,120,136,124]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^16=1,d^2=b,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=a*c^9>;
// generators/relations

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