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G = C4⋊C4.95D4order 128 = 27

50th non-split extension by C4⋊C4 of D4 acting via D4/C2=C22

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

Aliases: C4⋊C4.95D4, (C2×Q8).97D4, (C2×C4).16Q16, (C2×C4).35SD16, C4.24(C4⋊D4), (C22×C4).314D4, C23.915(C2×D4), C22.56(C2×Q16), C2.22(Q8⋊D4), C2.13(C42Q16), (C22×C8).75C22, C2.6(C4.SD16), C22.98(C2×SD16), C22.225C22≀C2, C2.13(D4.D4), (C2×C42).366C22, C2.22(C22⋊Q16), (C22×Q8).67C22, C22.232(C4⋊D4), (C22×C4).1449C23, C22.92(C4.4D4), C4.73(C22.D4), C2.16(C23.10D4), C22.128(C8.C22), C22.7C42.12C2, C23.65C23.14C2, C2.6(C42.30C22), (C2×C4⋊Q8).20C2, (C2×C4).1041(C2×D4), (C2×C4).880(C4○D4), (C2×C4⋊C4).128C22, (C2×Q8⋊C4).12C2, SmallGroup(128,775)

Series: Derived Chief Lower central Upper central Jennings

C1C22×C4 — C4⋊C4.95D4
C1C2C4C2×C4C22×C4C2×C4⋊C4C2×Q8⋊C4 — C4⋊C4.95D4
C1C2C22×C4 — C4⋊C4.95D4
C1C23C2×C42 — C4⋊C4.95D4
C1C2C2C22×C4 — C4⋊C4.95D4

Generators and relations for C4⋊C4.95D4
 G = < a,b,c,d | a4=b4=c4=1, d2=a2, bab-1=dad-1=a-1, ac=ca, cbc-1=b-1, dbd-1=ab, dcd-1=c-1 >

Subgroups: 296 in 149 conjugacy classes, 54 normal (28 characteristic)
C1, C2 [×7], C4 [×4], C4 [×11], C22 [×7], C8 [×2], C2×C4 [×2], C2×C4 [×8], C2×C4 [×19], Q8 [×12], C23, C42 [×2], C4⋊C4 [×2], C4⋊C4 [×13], C2×C8 [×6], C22×C4 [×3], C22×C4 [×4], C2×Q8 [×4], C2×Q8 [×10], C2.C42, Q8⋊C4 [×8], C2×C42, C2×C4⋊C4 [×2], C2×C4⋊C4 [×3], C4⋊Q8 [×4], C22×C8 [×2], C22×Q8 [×2], C22.7C42, C23.65C23, C2×Q8⋊C4 [×4], C2×C4⋊Q8, C4⋊C4.95D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×8], C23, SD16 [×2], Q16 [×2], C2×D4 [×4], C4○D4 [×3], C22≀C2, C4⋊D4 [×3], C22.D4 [×2], C4.4D4, C2×SD16, C2×Q16, C8.C22 [×2], C23.10D4, Q8⋊D4, C22⋊Q16, D4.D4, C42Q16, C4.SD16, C42.30C22, C4⋊C4.95D4

Smallest permutation representation of C4⋊C4.95D4
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)
(1 115 11 15)(2 114 12 14)(3 113 9 13)(4 116 10 16)(5 124 21 125)(6 123 22 128)(7 122 23 127)(8 121 24 126)(17 99 39 118)(18 98 40 117)(19 97 37 120)(20 100 38 119)(25 106 30 111)(26 105 31 110)(27 108 32 109)(28 107 29 112)(33 102 54 84)(34 101 55 83)(35 104 56 82)(36 103 53 81)(41 96 46 91)(42 95 47 90)(43 94 48 89)(44 93 45 92)(49 79 72 74)(50 78 69 73)(51 77 70 76)(52 80 71 75)(57 86 62 65)(58 85 63 68)(59 88 64 67)(60 87 61 66)
(1 97 8 106)(2 98 5 107)(3 99 6 108)(4 100 7 105)(9 118 22 109)(10 119 23 110)(11 120 24 111)(12 117 21 112)(13 39 128 32)(14 40 125 29)(15 37 126 30)(16 38 127 31)(17 123 27 113)(18 124 28 114)(19 121 25 115)(20 122 26 116)(33 73 43 67)(34 74 44 68)(35 75 41 65)(36 76 42 66)(45 85 55 79)(46 86 56 80)(47 87 53 77)(48 88 54 78)(49 92 58 83)(50 89 59 84)(51 90 60 81)(52 91 57 82)(61 103 70 95)(62 104 71 96)(63 101 72 93)(64 102 69 94)
(1 74 3 76)(2 73 4 75)(5 67 7 65)(6 66 8 68)(9 77 11 79)(10 80 12 78)(13 69 15 71)(14 72 16 70)(17 84 19 82)(18 83 20 81)(21 88 23 86)(22 87 24 85)(25 91 27 89)(26 90 28 92)(29 93 31 95)(30 96 32 94)(33 100 35 98)(34 99 36 97)(37 104 39 102)(38 103 40 101)(41 107 43 105)(42 106 44 108)(45 109 47 111)(46 112 48 110)(49 116 51 114)(50 115 52 113)(53 120 55 118)(54 119 56 117)(57 123 59 121)(58 122 60 124)(61 125 63 127)(62 128 64 126)

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

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

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

32 conjugacy classes

class 1 2A···2G4A4B4C4D4E4F4G4H4I···4P8A···8H
order12···2444444444···48···8
size11···1222244448···84···4

32 irreducible representations

dim111112222224
type++++++++--
imageC1C2C2C2C2D4D4D4SD16Q16C4○D4C8.C22
kernelC4⋊C4.95D4C22.7C42C23.65C23C2×Q8⋊C4C2×C4⋊Q8C4⋊C4C22×C4C2×Q8C2×C4C2×C4C2×C4C22
# reps111412244462

Matrix representation of C4⋊C4.95D4 in GL6(𝔽17)

1600000
0160000
001000
000100
0000115
0000116
,
10130000
470000
004000
0001300
0000130
0000134
,
470000
10130000
000400
004000
0000162
0000161
,
13100000
740000
001000
0001600
0000710
00001210

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,15,16],[10,4,0,0,0,0,13,7,0,0,0,0,0,0,4,0,0,0,0,0,0,13,0,0,0,0,0,0,13,13,0,0,0,0,0,4],[4,10,0,0,0,0,7,13,0,0,0,0,0,0,0,4,0,0,0,0,4,0,0,0,0,0,0,0,16,16,0,0,0,0,2,1],[13,7,0,0,0,0,10,4,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,7,12,0,0,0,0,10,10] >;

C4⋊C4.95D4 in GAP, Magma, Sage, TeX

C_4\rtimes C_4._{95}D_4
% in TeX

G:=Group("C4:C4.95D4");
// GroupNames label

G:=SmallGroup(128,775);
// by ID

G=gap.SmallGroup(128,775);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,448,141,456,422,387,58,2804,1411,718,172,4037,2028,1027,124]);
// Polycyclic

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

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