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G = C4⋊C4.85D4order 128 = 27

40th non-split extension by C4⋊C4 of D4 acting via D4/C2=C22

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

Aliases: C4⋊C4.85D4, (C2×Q8).6Q8, C2.8(Q8.Q8), (C22×C4).144D4, C23.908(C2×D4), C4.32(C22⋊Q8), C4.33(C4.4D4), (C22×C8).68C22, C2.7(C23⋊Q8), C2.30(D4.7D4), C22.210C22≀C2, C22.106(C4○D8), C22.4Q16.22C2, (C2×C42).356C22, (C22×Q8).61C22, (C22×C4).1442C23, C22.89(C4.4D4), C22.105(C22⋊Q8), C22.123(C8.C22), C22.7C42.11C2, C23.67C23.14C2, C2.5(C42.30C22), C2.7(C42.78C22), (C2×C4).282(C2×Q8), (C2×C4).1032(C2×D4), (C2×C42.C2).9C2, (C2×C4).619(C4○D4), (C2×C4⋊C4).115C22, (C2×Q8⋊C4).11C2, SmallGroup(128,758)

Series: Derived Chief Lower central Upper central Jennings

C1C22×C4 — C4⋊C4.85D4
C1C2C22C2×C4C22×C4C22×Q8C23.67C23 — C4⋊C4.85D4
C1C2C22×C4 — C4⋊C4.85D4
C1C23C2×C42 — C4⋊C4.85D4
C1C2C2C22×C4 — C4⋊C4.85D4

Generators and relations for C4⋊C4.85D4
 G = < a,b,c,d | a4=b4=c4=1, d2=a2b2, bab-1=cac-1=a-1, ad=da, cbc-1=ab, dbd-1=b-1, dcd-1=b2c-1 >

Subgroups: 256 in 129 conjugacy classes, 50 normal (18 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×9], C22 [×3], C22 [×4], C8 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×21], Q8 [×6], C23, C42 [×2], C4⋊C4 [×4], C4⋊C4 [×12], C2×C8 [×6], C22×C4, C22×C4 [×2], C22×C4 [×4], C2×Q8 [×2], C2×Q8 [×5], C2.C42 [×2], Q8⋊C4 [×4], C2×C42, C2×C4⋊C4, C2×C4⋊C4 [×2], C2×C4⋊C4 [×2], C42.C2 [×4], C22×C8 [×2], C22×Q8, C22.7C42, C22.4Q16 [×2], C23.67C23, C2×Q8⋊C4 [×2], C2×C42.C2, C4⋊C4.85D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], Q8 [×2], C23, C2×D4 [×3], C2×Q8, C4○D4 [×3], C22≀C2, C22⋊Q8 [×3], C4.4D4 [×3], C4○D8 [×2], C8.C22 [×2], C23⋊Q8, D4.7D4 [×2], Q8.Q8 [×2], C42.78C22, C42.30C22, C4⋊C4.85D4

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

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

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

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

32 conjugacy classes

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

32 irreducible representations

dim111111222224
type++++++++--
imageC1C2C2C2C2C2D4D4Q8C4○D4C4○D8C8.C22
kernelC4⋊C4.85D4C22.7C42C22.4Q16C23.67C23C2×Q8⋊C4C2×C42.C2C4⋊C4C22×C4C2×Q8C2×C4C22C22
# reps112121422682

Matrix representation of C4⋊C4.85D4 in GL6(𝔽17)

1600000
0160000
0016000
0001600
0000016
000010
,
12150000
1250000
000400
004000
000055
0000512
,
12150000
1350000
00141200
005300
000010
0000016
,
520000
5120000
005300
00141200
000004
0000130

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,1,0,0,0,0,16,0],[12,12,0,0,0,0,15,5,0,0,0,0,0,0,0,4,0,0,0,0,4,0,0,0,0,0,0,0,5,5,0,0,0,0,5,12],[12,13,0,0,0,0,15,5,0,0,0,0,0,0,14,5,0,0,0,0,12,3,0,0,0,0,0,0,1,0,0,0,0,0,0,16],[5,5,0,0,0,0,2,12,0,0,0,0,0,0,5,14,0,0,0,0,3,12,0,0,0,0,0,0,0,13,0,0,0,0,4,0] >;

C4⋊C4.85D4 in GAP, Magma, Sage, TeX

C_4\rtimes C_4._{85}D_4
% in TeX

G:=Group("C4:C4.85D4");
// GroupNames label

G:=SmallGroup(128,758);
// by ID

G=gap.SmallGroup(128,758);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,56,141,512,422,387,352,2804,718,172]);
// Polycyclic

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

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