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G = C42.122D4order 128 = 27

104th non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.122D4, C4⋊C411Q8, C4⋊Q824C4, C4.30(C4×Q8), (C2×C4).136D8, C4.31(C4⋊Q8), C22.49(C2×D8), C42.164(C2×C4), C2.5(D4⋊Q8), C2.5(Q8⋊Q8), (C2×C4).121SD16, (C22×C4).765D4, C23.811(C2×D4), C4.10(D4⋊C4), C4.50(C4.4D4), (C22×C8).62C22, C22.76(C2×SD16), C22.4Q16.17C2, (C2×C42).334C22, C22.83(C22⋊Q8), (C22×C4).1422C23, C22.89(C8.C22), C2.24(C23.38D4), C2.10(C23.67C23), (C4×C4⋊C4).28C2, (C2×C4⋊C8).34C2, C4⋊C4.95(C2×C4), (C2×C4⋊Q8).14C2, (C2×C4).277(C2×Q8), C2.24(C2×D4⋊C4), (C2×C4).1362(C2×D4), (C2×C4).767(C4○D4), (C2×C4⋊C4).781C22, (C2×C4).436(C22×C4), (C2×C4).263(C22⋊C4), C22.297(C2×C22⋊C4), SmallGroup(128,720)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.122D4
C1C2C4C2×C4C22×C4C2×C42C4×C4⋊C4 — C42.122D4
C1C2C2×C4 — C42.122D4
C1C23C2×C42 — C42.122D4
C1C2C2C22×C4 — C42.122D4

Generators and relations for C42.122D4
 G = < a,b,c,d | a4=b4=c4=1, d2=a2, ab=ba, ac=ca, dad-1=a-1, cbc-1=dbd-1=b-1, dcd-1=a2b-1c-1 >

Subgroups: 284 in 146 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 [×8], C23, C42 [×4], C42 [×2], C4⋊C4 [×8], C4⋊C4 [×8], C2×C8 [×6], C22×C4 [×3], C22×C4 [×4], C2×Q8 [×8], C2.C42, C4⋊C8 [×2], C2×C42, C2×C42, C2×C4⋊C4 [×4], C2×C4⋊C4, C4⋊Q8 [×4], C4⋊Q8 [×2], C22×C8 [×2], C22×Q8, C22.4Q16 [×4], C4×C4⋊C4, C2×C4⋊C8, C2×C4⋊Q8, C42.122D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], Q8 [×4], C23, C22⋊C4 [×4], D8 [×2], SD16 [×2], C22×C4, C2×D4 [×2], C2×Q8 [×2], C4○D4 [×2], D4⋊C4 [×4], C2×C22⋊C4, C4×Q8 [×2], C22⋊Q8 [×2], C4.4D4, C4⋊Q8, C2×D8, C2×SD16, C8.C22 [×2], C23.67C23, C2×D4⋊C4, C23.38D4, D4⋊Q8 [×2], Q8⋊Q8 [×2], C42.122D4

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

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

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

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++++++-++-
imageC1C2C2C2C2C4D4Q8D4D8SD16C4○D4C8.C22
kernelC42.122D4C22.4Q16C4×C4⋊C4C2×C4⋊C8C2×C4⋊Q8C4⋊Q8C42C4⋊C4C22×C4C2×C4C2×C4C2×C4C22
# reps1411182424442

Matrix representation of C42.122D4 in GL5(𝔽17)

160000
00100
016000
00010
00001
,
160000
016000
001600
00001
000160
,
40000
016000
001600
000314
0001414
,
10000
001300
013000
00001
00010

G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1],[16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,1,0],[4,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,3,14,0,0,0,14,14],[1,0,0,0,0,0,0,13,0,0,0,13,0,0,0,0,0,0,0,1,0,0,0,1,0] >;

C42.122D4 in GAP, Magma, Sage, TeX

C_4^2._{122}D_4
% in TeX

G:=Group("C4^2.122D4");
// GroupNames label

G:=SmallGroup(128,720);
// by ID

G=gap.SmallGroup(128,720);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,848,422,100,1018,248,2028,124]);
// Polycyclic

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

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