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G = C42.33Q8  order 128 = 27

33rd non-split extension by C42 of Q8 acting via Q8/C2=C22

p-group, metabelian, nilpotent (class 2), monomial

Aliases: C42.33Q8, C23.212C24, C22.502+ 1+4, C22.342- 1+4, C4.19(C4×Q8), C42.C2⋊13C4, C42.181(C2×C4), C42⋊4C4.14C2, C42⋊9C4.22C2, C22.34(C22×Q8), (C22×C4).477C23, C22.103(C23×C4), (C2×C42).419C22, C2.C42.48C22, C23.63C23.3C2, C23.65C23.30C2, C2.5(C22.34C24), C2.5(C22.35C24), C2.4(C23.41C23), C2.17(C23.33C23), C2.12(C2×C4×Q8), (C4×C4⋊C4).35C2, C4⋊C4.105(C2×C4), (C2×C4).162(C2×Q8), (C2×C4).32(C22×C4), C22.97(C2×C4○D4), (C2×C4).651(C4○D4), (C2×C4⋊C4).182C22, (C2×C42.C2).11C2, SmallGroup(128,1062)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C22 — C42.33Q8
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C42⋊4C4 — C42.33Q8
Lower central
C1 — C22 — C42.33Q8
Upper central
C1 — C23 — C42.33Q8
Jennings
C1 — C23 — C42.33Q8

Generators and relations for C42.33Q8
 G = < a,b,c,d | a4=b4=c4=1, d2=a2c2, ab=ba, cac-1=dad-1=ab2, bc=cb, dbd-1=b-1, dcd-1=c-1 >

Subgroups: 332 in 222 conjugacy classes, 148 normal (18 characteristic)
C1, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, C23, C42, C42, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2.C42, C2×C42, C2×C42, C2×C4⋊C4, C42.C2, C42⋊4C4, C4×C4⋊C4, C42⋊9C4, C23.63C23, C23.65C23, C2×C42.C2, C42.33Q8
Quotients: C1, C2, C4, C22, C2×C4, Q8, C23, C22×C4, C2×Q8, C4○D4, C24, C4×Q8, C23×C4, C22×Q8, C2×C4○D4, 2+ 1+4, 2- 1+4, C2×C4×Q8, C23.33C23, C22.34C24, C22.35C24, C23.41C23, C42.33Q8

Smallest permutation representation of C42.33Q8
►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 27 23 55)(2 28 24 56)(3 25 21 53)(4 26 22 54)(5 41 37 9)(6 42 38 10)(7 43 39 11)(8 44 40 12)(13 49 45 17)(14 50 46 18)(15 51 47 19)(16 52 48 20)(29 61 57 69)(30 62 58 70)(31 63 59 71)(32 64 60 72)(33 121 125 89)(34 122 126 90)(35 123 127 91)(36 124 128 92)(65 102 78 74)(66 103 79 75)(67 104 80 76)(68 101 77 73)(81 93 113 117)(82 94 114 118)(83 95 115 119)(84 96 116 120)(85 98 105 109)(86 99 106 110)(87 100 107 111)(88 97 108 112)

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

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

44 conjugacy classes

class 1 2A···2G4A···4L4M···4AJ
order12···24···44···4
size11···12···24···4

44 irreducible representations

dim111111112244
type+++++++-+-
imageC1C2C2C2C2C2C2C4Q8C4○D42+ 1+42- 1+4
kernelC42.33Q8C42⋊4C4C4×C4⋊C4C42⋊9C4C23.63C23C23.65C23C2×C42.C2C42.C2C42C2×C4C22C22
# reps1121461164422

Matrix representation of C42.33Q8 ►in GL8(𝔽5)

10000000
01000000
00300000
00030000
00000100
00004000
00000004
00000010
,
40000000
04000000
00400000
00040000
00000100
00004000
00000001
00000040
,
20000000
13000000
00210000
00030000
00000010
00000001
00001000
00000100
,
11000000
34000000
00400000
00410000
00000300
00003000
00000003
00000030

G:=sub<GL(8,GF(5))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0],[2,1,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,1,3,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[1,3,0,0,0,0,0,0,1,4,0,0,0,0,0,0,0,0,4,4,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0] >;
 

C42.33Q8 in GAP, Magma, Sage, TeX 

C_4^2._{33}Q_8
 
% in TeX
 
G:=Group("C4^2.33Q8");
 
// GroupNames label
 
G:=SmallGroup(128,1062);
 
// by ID
 
G=gap.SmallGroup(128,1062);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,680,758,219,184,675,80]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=a^2*c^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a*b^2,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
 
// generators/relations
 

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