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G = C42.117D4order 128 = 27

99th non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.117D4, C4⋊Q821C4, C4.95(C4×D4), C4⋊C4.225D4, (C2×C4).62Q16, C41(Q8⋊C4), C4.23(C41D4), C42.159(C2×C4), C2.5(C42Q16), (C2×C4).105SD16, (C22×C4).762D4, C23.807(C2×D4), C22.41(C2×Q16), C2.5(D4.D4), C4.66(C4.4D4), (C22×C8).56C22, C22.73(C2×SD16), (C2×C42).327C22, (C22×Q8).44C22, C22.152(C4⋊D4), (C22×C4).1417C23, C22.87(C8.C22), C2.23(C23.38D4), C2.33(C24.3C22), (C2×C4⋊C8).30C2, (C4×C4⋊C4).25C2, (C2×C4⋊Q8).12C2, (C2×C4).1359(C2×D4), (C2×Q8⋊C4).9C2, (C2×Q8).104(C2×C4), C2.23(C2×Q8⋊C4), (C2×C4).869(C4○D4), (C2×C4⋊C4).777C22, (C2×C4).431(C22×C4), (C2×C4).259(C22⋊C4), C22.291(C2×C22⋊C4), SmallGroup(128,713)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.117D4
C1C2C4C2×C4C22×C4C2×C42C4×C4⋊C4 — C42.117D4
C1C2C2×C4 — C42.117D4
C1C23C2×C42 — C42.117D4
C1C2C2C22×C4 — C42.117D4

Generators and relations for C42.117D4
 G = < a,b,c,d | a4=b4=c4=1, d2=b2, ab=ba, ac=ca, dad-1=a-1, cbc-1=dbd-1=b-1, dcd-1=bc-1 >

Subgroups: 300 in 162 conjugacy classes, 72 normal (20 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×6], C4 [×10], C22 [×3], C22 [×4], C8 [×2], C2×C4 [×2], C2×C4 [×12], C2×C4 [×18], Q8 [×12], C23, C42 [×4], C42 [×2], C4⋊C4 [×4], C4⋊C4 [×10], C2×C8 [×6], C22×C4 [×3], C22×C4 [×4], C2×Q8 [×4], C2×Q8 [×10], C2.C42, Q8⋊C4 [×8], C4⋊C8 [×2], C2×C42, C2×C42, C2×C4⋊C4 [×2], C2×C4⋊C4 [×2], C4⋊Q8 [×4], C4⋊Q8 [×2], C22×C8 [×2], C22×Q8 [×2], C4×C4⋊C4, C2×Q8⋊C4 [×4], C2×C4⋊C8, C2×C4⋊Q8, C42.117D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×8], C23, C22⋊C4 [×4], SD16 [×2], Q16 [×2], C22×C4, C2×D4 [×4], C4○D4 [×2], Q8⋊C4 [×4], C2×C22⋊C4, C4×D4 [×2], C4⋊D4 [×2], C4.4D4, C41D4, C2×SD16, C2×Q16, C8.C22 [×2], C24.3C22, C2×Q8⋊C4, C23.38D4, D4.D4 [×2], C42Q16 [×2], C42.117D4

Smallest permutation representation of C42.117D4
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 51 11 25)(2 52 12 26)(3 49 9 27)(4 50 10 28)(5 126 118 122)(6 127 119 123)(7 128 120 124)(8 125 117 121)(13 33 39 31)(14 34 40 32)(15 35 37 29)(16 36 38 30)(17 112 108 23)(18 109 105 24)(19 110 106 21)(20 111 107 22)(41 67 69 45)(42 68 70 46)(43 65 71 47)(44 66 72 48)(53 83 61 57)(54 84 62 58)(55 81 63 59)(56 82 64 60)(73 77 101 99)(74 78 102 100)(75 79 103 97)(76 80 104 98)(85 93 115 89)(86 94 116 90)(87 95 113 91)(88 96 114 92)
(1 45 13 61)(2 46 14 62)(3 47 15 63)(4 48 16 64)(5 78 19 94)(6 79 20 95)(7 80 17 96)(8 77 18 93)(9 65 37 55)(10 66 38 56)(11 67 39 53)(12 68 40 54)(21 116 122 102)(22 113 123 103)(23 114 124 104)(24 115 121 101)(25 41 31 57)(26 42 32 58)(27 43 29 59)(28 44 30 60)(33 83 51 69)(34 84 52 70)(35 81 49 71)(36 82 50 72)(73 109 85 125)(74 110 86 126)(75 111 87 127)(76 112 88 128)(89 117 99 105)(90 118 100 106)(91 119 97 107)(92 120 98 108)
(1 107 11 20)(2 106 12 19)(3 105 9 18)(4 108 10 17)(5 14 118 40)(6 13 119 39)(7 16 120 38)(8 15 117 37)(21 52 110 26)(22 51 111 25)(23 50 112 28)(24 49 109 27)(29 121 35 125)(30 124 36 128)(31 123 33 127)(32 122 34 126)(41 79 69 97)(42 78 70 100)(43 77 71 99)(44 80 72 98)(45 103 67 75)(46 102 68 74)(47 101 65 73)(48 104 66 76)(53 87 61 113)(54 86 62 116)(55 85 63 115)(56 88 64 114)(57 95 83 91)(58 94 84 90)(59 93 81 89)(60 96 82 92)

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

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

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

38 conjugacy classes

class 1 2A···2G4A···4H4I···4R4S4T4U4V8A···8H
order12···24···44···444448···8
size11···12···24···488884···4

38 irreducible representations

dim1111112222224
type++++++++--
imageC1C2C2C2C2C4D4D4D4SD16Q16C4○D4C8.C22
kernelC42.117D4C4×C4⋊C4C2×Q8⋊C4C2×C4⋊C8C2×C4⋊Q8C4⋊Q8C42C4⋊C4C22×C4C2×C4C2×C4C2×C4C22
# reps1141182424442

Matrix representation of C42.117D4 in GL5(𝔽17)

160000
011500
011600
000160
000016
,
160000
016000
001600
0001615
00011
,
130000
04900
041300
000814
000169
,
160000
01000
011600
000614
000111

G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,1,1,0,0,0,15,16,0,0,0,0,0,16,0,0,0,0,0,16],[16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16,1,0,0,0,15,1],[13,0,0,0,0,0,4,4,0,0,0,9,13,0,0,0,0,0,8,16,0,0,0,14,9],[16,0,0,0,0,0,1,1,0,0,0,0,16,0,0,0,0,0,6,1,0,0,0,14,11] >;

C42.117D4 in GAP, Magma, Sage, TeX

C_4^2._{117}D_4
% in TeX

G:=Group("C4^2.117D4");
// GroupNames label

G:=SmallGroup(128,713);
// by ID

G=gap.SmallGroup(128,713);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,100,1018,248,2028,124]);
// Polycyclic

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

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