Copied to
clipboard

G = C4⋊C4⋊Q8order 128 = 27

1st semidirect product of C4⋊C4 and Q8 acting via Q8/C2=C22

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

Aliases: C4⋊C41Q8, (C2×C8)⋊1Q8, C4⋊C4.99D4, (C2×C4).22D8, C2.6(C8⋊Q8), C4.21(C4⋊Q8), (C2×C4).17Q16, C2.4(C82Q8), C22.86(C2×D8), C4.6(C22⋊Q8), C2.7(C4.Q16), C2.7(D4⋊Q8), C23.919(C2×D4), (C22×C4).315D4, C22.48(C4⋊Q8), C2.23(C22⋊D8), C22.58(C2×Q16), C22.227C22≀C2, C22.4Q16.38C2, (C22×C8).114C22, (C2×C42).371C22, C2.23(C22⋊Q16), C22.142(C8⋊C22), (C22×C4).1453C23, C22.106(C22⋊Q8), C22.131(C8.C22), C22.7C42.24C2, C23.65C23.15C2, C2.4(C23.78C23), (C2×C4⋊Q8).21C2, (C2×C4).217(C2×Q8), (C2×C2.D8).12C2, (C2×C4).1048(C2×D4), (C2×C4).775(C4○D4), (C2×C4⋊C4).134C22, SmallGroup(128,789)

Series: Derived Chief Lower central Upper central Jennings

C1C22×C4 — C4⋊C4⋊Q8
C1C2C22C2×C4C22×C4C2×C4⋊C4C23.65C23 — C4⋊C4⋊Q8
C1C2C22×C4 — C4⋊C4⋊Q8
C1C23C2×C42 — C4⋊C4⋊Q8
C1C2C2C22×C4 — C4⋊C4⋊Q8

Generators and relations for C4⋊C4⋊Q8
 G = < a,b,c,d | a4=b4=c4=1, d2=c2, bab-1=dad-1=a-1, ac=ca, cbc-1=b-1, dbd-1=a-1b-1, dcd-1=c-1 >

Subgroups: 288 in 141 conjugacy classes, 58 normal (28 characteristic)
C1, C2 [×7], C4 [×4], C4 [×11], C22 [×7], C8 [×4], C2×C4 [×2], C2×C4 [×8], C2×C4 [×19], Q8 [×8], C23, C42 [×2], C4⋊C4 [×6], C4⋊C4 [×11], C2×C8 [×4], C2×C8 [×4], C22×C4 [×3], C22×C4 [×4], C2×Q8 [×8], C2.C42, C2.D8 [×4], C2×C42, C2×C4⋊C4 [×2], C2×C4⋊C4 [×2], C2×C4⋊C4 [×2], C4⋊Q8 [×4], C22×C8 [×2], C22×Q8, C22.7C42, C22.4Q16 [×2], C23.65C23, C2×C2.D8 [×2], C2×C4⋊Q8, C4⋊C4⋊Q8
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], Q8 [×6], C23, D8 [×2], Q16 [×2], C2×D4 [×3], C2×Q8 [×3], C4○D4, C22≀C2, C22⋊Q8 [×3], C4⋊Q8 [×3], C2×D8, C2×Q16, C8⋊C22, C8.C22, C23.78C23, C22⋊D8, C22⋊Q16, D4⋊Q8, C4.Q16, C82Q8, C8⋊Q8, C4⋊C4⋊Q8

Smallest permutation representation of C4⋊C4⋊Q8
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 58 23 71)(2 57 24 70)(3 60 21 69)(4 59 22 72)(5 61 11 49)(6 64 12 52)(7 63 9 51)(8 62 10 50)(13 75 124 87)(14 74 121 86)(15 73 122 85)(16 76 123 88)(17 43 29 54)(18 42 30 53)(19 41 31 56)(20 44 32 55)(25 34 37 47)(26 33 38 46)(27 36 39 45)(28 35 40 48)(65 113 77 126)(66 116 78 125)(67 115 79 128)(68 114 80 127)(81 117 94 105)(82 120 95 108)(83 119 96 107)(84 118 93 106)(89 110 104 98)(90 109 101 97)(91 112 102 100)(92 111 103 99)
(1 53 8 33)(2 54 5 34)(3 55 6 35)(4 56 7 36)(9 45 22 41)(10 46 23 42)(11 47 24 43)(12 48 21 44)(13 103 115 83)(14 104 116 84)(15 101 113 81)(16 102 114 82)(17 49 37 70)(18 50 38 71)(19 51 39 72)(20 52 40 69)(25 57 29 61)(26 58 30 62)(27 59 31 63)(28 60 32 64)(65 117 85 97)(66 118 86 98)(67 119 87 99)(68 120 88 100)(73 109 77 105)(74 110 78 106)(75 111 79 107)(76 112 80 108)(89 125 93 121)(90 126 94 122)(91 127 95 123)(92 128 96 124)
(1 66 8 86)(2 65 5 85)(3 68 6 88)(4 67 7 87)(9 75 22 79)(10 74 23 78)(11 73 24 77)(12 76 21 80)(13 69 115 52)(14 72 116 51)(15 71 113 50)(16 70 114 49)(17 102 37 82)(18 101 38 81)(19 104 39 84)(20 103 40 83)(25 95 29 91)(26 94 30 90)(27 93 31 89)(28 96 32 92)(33 118 53 98)(34 117 54 97)(35 120 55 100)(36 119 56 99)(41 111 45 107)(42 110 46 106)(43 109 47 105)(44 112 48 108)(57 127 61 123)(58 126 62 122)(59 125 63 121)(60 128 64 124)

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

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

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

32 conjugacy classes

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

32 irreducible representations

dim111111222222244
type+++++++--++-+-
imageC1C2C2C2C2C2D4Q8Q8D4D8Q16C4○D4C8⋊C22C8.C22
kernelC4⋊C4⋊Q8C22.7C42C22.4Q16C23.65C23C2×C2.D8C2×C4⋊Q8C4⋊C4C4⋊C4C2×C8C22×C4C2×C4C2×C4C2×C4C22C22
# reps112121424244211

Matrix representation of C4⋊C4⋊Q8 in GL6(𝔽17)

1600000
0160000
001000
000100
000001
0000160
,
1180000
260000
005500
0051200
0000613
00001311
,
10150000
870000
0051200
00121200
000010
000001
,
1170000
260000
0001600
001000
000064
0000411

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[11,2,0,0,0,0,8,6,0,0,0,0,0,0,5,5,0,0,0,0,5,12,0,0,0,0,0,0,6,13,0,0,0,0,13,11],[10,8,0,0,0,0,15,7,0,0,0,0,0,0,5,12,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[11,2,0,0,0,0,7,6,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,0,0,0,6,4,0,0,0,0,4,11] >;

C4⋊C4⋊Q8 in GAP, Magma, Sage, TeX

C_4\rtimes C_4\rtimes Q_8
% in TeX

G:=Group("C4:C4:Q8");
// GroupNames label

G:=SmallGroup(128,789);
// by ID

G=gap.SmallGroup(128,789);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,56,141,64,422,387,58,2804,718,172]);
// Polycyclic

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

׿
×
𝔽