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G = (C2×Q8).8Q8order 128 = 27

8th non-split extension by C2×Q8 of Q8 acting via Q8/C2=C22

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

Aliases: (C2×Q8).8Q8, C4⋊C4.107D4, (C2×C8).163D4, (C2×C4).38SD16, C2.21(C88D4), C2.8(Q8⋊Q8), C23.923(C2×D4), (C22×C4).155D4, C2.10(Q8.Q8), C4.36(C22⋊Q8), C4.154(C4⋊D4), C2.15(C8.D4), (C22×C8).81C22, C4.34(C422C2), C2.15(D4.D4), C22.117(C4○D8), C22.4Q16.23C2, (C2×C42).377C22, C2.20(Q8.D4), C22.102(C2×SD16), C2.7(C23.Q8), (C22×Q8).75C22, C22.244(C4⋊D4), (C22×C4).1457C23, C22.110(C22⋊Q8), C22.134(C8.C22), C23.65C23.18C2, C23.67C23.17C2, (C2×C4⋊C8).48C2, (C2×C4).284(C2×Q8), (C2×C4.Q8).24C2, (C2×C4).1056(C2×D4), (C2×C4).621(C4○D4), (C2×C4⋊C4).142C22, (C2×Q8⋊C4).14C2, SmallGroup(128,798)

Series: Derived Chief Lower central Upper central Jennings

C1C22×C4 — (C2×Q8).8Q8
C1C2C4C2×C4C22×C4C22×Q8C2×Q8⋊C4 — (C2×Q8).8Q8
C1C2C22×C4 — (C2×Q8).8Q8
C1C23C2×C42 — (C2×Q8).8Q8
C1C2C2C22×C4 — (C2×Q8).8Q8

Generators and relations for (C2×Q8).8Q8
 G = < a,b,c,d,e | a2=b4=1, c2=d4=b2, e2=b2cd2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, dbd-1=abc, ebe-1=b-1c, cd=dc, ece-1=b2c, ede-1=d3 >

Subgroups: 248 in 124 conjugacy classes, 52 normal (44 characteristic)
C1, C2 [×7], C4 [×4], C4 [×9], C22 [×7], C8 [×3], C2×C4 [×6], C2×C4 [×2], C2×C4 [×19], Q8 [×6], C23, C42 [×2], C4⋊C4 [×2], C4⋊C4 [×7], C2×C8 [×2], C2×C8 [×5], C22×C4 [×3], C22×C4 [×4], C2×Q8 [×2], C2×Q8 [×5], C2.C42 [×3], Q8⋊C4 [×4], C4⋊C8 [×2], C4.Q8 [×2], C2×C42, C2×C4⋊C4 [×3], C2×C4⋊C4, C22×C8 [×2], C22×Q8, C22.4Q16, C23.65C23, C23.67C23, C2×Q8⋊C4 [×2], C2×C4⋊C8, C2×C4.Q8, (C2×Q8).8Q8
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], Q8 [×2], C23, SD16 [×2], C2×D4 [×3], C2×Q8, C4○D4 [×3], C4⋊D4 [×3], C22⋊Q8 [×3], C422C2, C2×SD16, C4○D8, C8.C22 [×2], C23.Q8, D4.D4, Q8.D4, C88D4, C8.D4, Q8⋊Q8, Q8.Q8, (C2×Q8).8Q8

Smallest permutation representation of (C2×Q8).8Q8
Regular action on 128 points
Generators in S128
(1 75)(2 76)(3 77)(4 78)(5 79)(6 80)(7 73)(8 74)(9 86)(10 87)(11 88)(12 81)(13 82)(14 83)(15 84)(16 85)(17 61)(18 62)(19 63)(20 64)(21 57)(22 58)(23 59)(24 60)(25 55)(26 56)(27 49)(28 50)(29 51)(30 52)(31 53)(32 54)(33 125)(34 126)(35 127)(36 128)(37 121)(38 122)(39 123)(40 124)(41 120)(42 113)(43 114)(44 115)(45 116)(46 117)(47 118)(48 119)(65 102)(66 103)(67 104)(68 97)(69 98)(70 99)(71 100)(72 101)(89 105)(90 106)(91 107)(92 108)(93 109)(94 110)(95 111)(96 112)
(1 71 5 67)(2 30 6 26)(3 69 7 65)(4 28 8 32)(9 35 13 39)(10 112 14 108)(11 33 15 37)(12 110 16 106)(17 29 21 25)(18 68 22 72)(19 27 23 31)(20 66 24 70)(34 46 38 42)(36 44 40 48)(41 105 45 109)(43 111 47 107)(49 59 53 63)(50 74 54 78)(51 57 55 61)(52 80 56 76)(58 101 62 97)(60 99 64 103)(73 102 77 98)(75 100 79 104)(81 94 85 90)(82 123 86 127)(83 92 87 96)(84 121 88 125)(89 116 93 120)(91 114 95 118)(113 126 117 122)(115 124 119 128)
(1 57 5 61)(2 58 6 62)(3 59 7 63)(4 60 8 64)(9 118 13 114)(10 119 14 115)(11 120 15 116)(12 113 16 117)(17 75 21 79)(18 76 22 80)(19 77 23 73)(20 78 24 74)(25 100 29 104)(26 101 30 97)(27 102 31 98)(28 103 32 99)(33 93 37 89)(34 94 38 90)(35 95 39 91)(36 96 40 92)(41 84 45 88)(42 85 46 81)(43 86 47 82)(44 87 48 83)(49 65 53 69)(50 66 54 70)(51 67 55 71)(52 68 56 72)(105 125 109 121)(106 126 110 122)(107 127 111 123)(108 128 112 124)
(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)
(1 109 63 127)(2 112 64 122)(3 107 57 125)(4 110 58 128)(5 105 59 123)(6 108 60 126)(7 111 61 121)(8 106 62 124)(9 29 116 102)(10 32 117 97)(11 27 118 100)(12 30 119 103)(13 25 120 98)(14 28 113 101)(15 31 114 104)(16 26 115 99)(17 37 73 95)(18 40 74 90)(19 35 75 93)(20 38 76 96)(21 33 77 91)(22 36 78 94)(23 39 79 89)(24 34 80 92)(41 69 82 55)(42 72 83 50)(43 67 84 53)(44 70 85 56)(45 65 86 51)(46 68 87 54)(47 71 88 49)(48 66 81 52)

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

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

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

32 conjugacy classes

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

32 irreducible representations

dim111111122222224
type++++++++++--
imageC1C2C2C2C2C2C2D4D4D4Q8SD16C4○D4C4○D8C8.C22
kernel(C2×Q8).8Q8C22.4Q16C23.65C23C23.67C23C2×Q8⋊C4C2×C4⋊C8C2×C4.Q8C4⋊C4C2×C8C22×C4C2×Q8C2×C4C2×C4C22C22
# reps111121122224642

Matrix representation of (C2×Q8).8Q8 in GL6(𝔽17)

1600000
0160000
001000
000100
000010
000001
,
0160000
1600000
0016000
0001600
000004
000040
,
1600000
0160000
001000
000100
0000016
000010
,
1600000
0160000
00161500
001100
000055
0000125
,
1040000
1370000
008200
0010900
00001414
0000143

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,16,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,4,0,0,0,0,4,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,16,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,1,0,0,0,0,15,1,0,0,0,0,0,0,5,12,0,0,0,0,5,5],[10,13,0,0,0,0,4,7,0,0,0,0,0,0,8,10,0,0,0,0,2,9,0,0,0,0,0,0,14,14,0,0,0,0,14,3] >;

(C2×Q8).8Q8 in GAP, Magma, Sage, TeX

(C_2\times Q_8)._8Q_8
% in TeX

G:=Group("(C2xQ8).8Q8");
// GroupNames label

G:=SmallGroup(128,798);
// by ID

G=gap.SmallGroup(128,798);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,168,141,64,422,387,352,2019,521,248,2804,718,172,4037,1027,124]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^4=1,c^2=d^4=b^2,e^2=b^2*c*d^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,d*b*d^-1=a*b*c,e*b*e^-1=b^-1*c,c*d=d*c,e*c*e^-1=b^2*c,e*d*e^-1=d^3>;
// generators/relations

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