p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.174D4, C23.465C24, C22.1892- 1+4, (C2×Q8).23Q8, C4.57(C22⋊Q8), C42⋊9C4.30C2, C42⋊8C4.33C2, C2.18(Q8⋊3Q8), (C2×C42).566C22, (C22×C4).841C23, C22.316(C22×D4), C22.106(C22×Q8), (C22×Q8).436C22, C23.81C23.16C2, C2.C42.201C22, C23.63C23.27C2, C23.67C23.40C2, C2.42(C22.26C24), C2.25(C23.38C23), C2.65(C22.46C24), (C4×C4⋊C4).66C2, (C2×C4×Q8).34C2, (C2×C4).54(C2×Q8), (C2×C4).359(C2×D4), C2.33(C2×C22⋊Q8), (C2×C4).150(C4○D4), (C2×C4⋊C4).313C22, C22.341(C2×C4○D4), (C2×C42.C2).21C2, SmallGroup(128,1297)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.174D4
G = < a,b,c,d | a4=b4=c4=1, d2=a2, ab=ba, cac-1=a-1, dad-1=a-1b2, bc=cb, dbd-1=b-1, dcd-1=c-1 >
Subgroups: 356 in 226 conjugacy classes, 112 normal (26 characteristic)
C1, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, Q8, C23, C42, C42, C4⋊C4, C22×C4, C22×C4, C2×Q8, C2×Q8, C2.C42, C2×C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C4×Q8, C42.C2, C22×Q8, C4×C4⋊C4, C42⋊8C4, C42⋊9C4, C23.63C23, C23.67C23, C23.81C23, C2×C4×Q8, C2×C42.C2, C42.174D4
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, C4○D4, C24, C22⋊Q8, C22×D4, C22×Q8, C2×C4○D4, 2- 1+4, C2×C22⋊Q8, C22.26C24, C23.38C23, C22.46C24, Q8⋊3Q8, C42.174D4
►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 13 9 41)(2 14 10 42)(3 15 11 43)(4 16 12 44)(5 48 38 20)(6 45 39 17)(7 46 40 18)(8 47 37 19)(21 53 49 25)(22 54 50 26)(23 55 51 27)(24 56 52 28)(29 61 57 69)(30 62 58 70)(31 63 59 71)(32 64 60 72)(33 123 127 91)(34 124 128 92)(35 121 125 89)(36 122 126 90)(65 111 100 87)(66 112 97 88)(67 109 98 85)(68 110 99 86)(73 101 105 77)(74 102 106 78)(75 103 107 79)(76 104 108 80)(81 93 113 117)(82 94 114 118)(83 95 115 119)(84 96 116 120)
(1 59 51 45)(2 58 52 48)(3 57 49 47)(4 60 50 46)(5 42 62 56)(6 41 63 55)(7 44 64 54)(8 43 61 53)(9 31 23 17)(10 30 24 20)(11 29 21 19)(12 32 22 18)(13 71 27 39)(14 70 28 38)(15 69 25 37)(16 72 26 40)(33 105 98 95)(34 108 99 94)(35 107 100 93)(36 106 97 96)(65 117 125 75)(66 120 126 74)(67 119 127 73)(68 118 128 76)(77 85 115 123)(78 88 116 122)(79 87 113 121)(80 86 114 124)(81 89 103 111)(82 92 104 110)(83 91 101 109)(84 90 102 112)
(1 89 3 91)(2 124 4 122)(5 76 7 74)(6 107 8 105)(9 121 11 123)(10 92 12 90)(13 125 15 127)(14 34 16 36)(17 79 19 77)(18 102 20 104)(21 85 23 87)(22 112 24 110)(25 67 27 65)(26 97 28 99)(29 115 31 113)(30 82 32 84)(33 41 35 43)(37 73 39 75)(38 108 40 106)(42 128 44 126)(45 103 47 101)(46 78 48 80)(49 109 51 111)(50 88 52 86)(53 98 55 100)(54 66 56 68)(57 83 59 81)(58 114 60 116)(61 95 63 93)(62 118 64 120)(69 119 71 117)(70 94 72 96)
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), (1,13,9,41)(2,14,10,42)(3,15,11,43)(4,16,12,44)(5,48,38,20)(6,45,39,17)(7,46,40,18)(8,47,37,19)(21,53,49,25)(22,54,50,26)(23,55,51,27)(24,56,52,28)(29,61,57,69)(30,62,58,70)(31,63,59,71)(32,64,60,72)(33,123,127,91)(34,124,128,92)(35,121,125,89)(36,122,126,90)(65,111,100,87)(66,112,97,88)(67,109,98,85)(68,110,99,86)(73,101,105,77)(74,102,106,78)(75,103,107,79)(76,104,108,80)(81,93,113,117)(82,94,114,118)(83,95,115,119)(84,96,116,120), (1,59,51,45)(2,58,52,48)(3,57,49,47)(4,60,50,46)(5,42,62,56)(6,41,63,55)(7,44,64,54)(8,43,61,53)(9,31,23,17)(10,30,24,20)(11,29,21,19)(12,32,22,18)(13,71,27,39)(14,70,28,38)(15,69,25,37)(16,72,26,40)(33,105,98,95)(34,108,99,94)(35,107,100,93)(36,106,97,96)(65,117,125,75)(66,120,126,74)(67,119,127,73)(68,118,128,76)(77,85,115,123)(78,88,116,122)(79,87,113,121)(80,86,114,124)(81,89,103,111)(82,92,104,110)(83,91,101,109)(84,90,102,112), (1,89,3,91)(2,124,4,122)(5,76,7,74)(6,107,8,105)(9,121,11,123)(10,92,12,90)(13,125,15,127)(14,34,16,36)(17,79,19,77)(18,102,20,104)(21,85,23,87)(22,112,24,110)(25,67,27,65)(26,97,28,99)(29,115,31,113)(30,82,32,84)(33,41,35,43)(37,73,39,75)(38,108,40,106)(42,128,44,126)(45,103,47,101)(46,78,48,80)(49,109,51,111)(50,88,52,86)(53,98,55,100)(54,66,56,68)(57,83,59,81)(58,114,60,116)(61,95,63,93)(62,118,64,120)(69,119,71,117)(70,94,72,96)>;
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), (1,13,9,41)(2,14,10,42)(3,15,11,43)(4,16,12,44)(5,48,38,20)(6,45,39,17)(7,46,40,18)(8,47,37,19)(21,53,49,25)(22,54,50,26)(23,55,51,27)(24,56,52,28)(29,61,57,69)(30,62,58,70)(31,63,59,71)(32,64,60,72)(33,123,127,91)(34,124,128,92)(35,121,125,89)(36,122,126,90)(65,111,100,87)(66,112,97,88)(67,109,98,85)(68,110,99,86)(73,101,105,77)(74,102,106,78)(75,103,107,79)(76,104,108,80)(81,93,113,117)(82,94,114,118)(83,95,115,119)(84,96,116,120), (1,59,51,45)(2,58,52,48)(3,57,49,47)(4,60,50,46)(5,42,62,56)(6,41,63,55)(7,44,64,54)(8,43,61,53)(9,31,23,17)(10,30,24,20)(11,29,21,19)(12,32,22,18)(13,71,27,39)(14,70,28,38)(15,69,25,37)(16,72,26,40)(33,105,98,95)(34,108,99,94)(35,107,100,93)(36,106,97,96)(65,117,125,75)(66,120,126,74)(67,119,127,73)(68,118,128,76)(77,85,115,123)(78,88,116,122)(79,87,113,121)(80,86,114,124)(81,89,103,111)(82,92,104,110)(83,91,101,109)(84,90,102,112), (1,89,3,91)(2,124,4,122)(5,76,7,74)(6,107,8,105)(9,121,11,123)(10,92,12,90)(13,125,15,127)(14,34,16,36)(17,79,19,77)(18,102,20,104)(21,85,23,87)(22,112,24,110)(25,67,27,65)(26,97,28,99)(29,115,31,113)(30,82,32,84)(33,41,35,43)(37,73,39,75)(38,108,40,106)(42,128,44,126)(45,103,47,101)(46,78,48,80)(49,109,51,111)(50,88,52,86)(53,98,55,100)(54,66,56,68)(57,83,59,81)(58,114,60,116)(61,95,63,93)(62,118,64,120)(69,119,71,117)(70,94,72,96) );
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)], [(1,13,9,41),(2,14,10,42),(3,15,11,43),(4,16,12,44),(5,48,38,20),(6,45,39,17),(7,46,40,18),(8,47,37,19),(21,53,49,25),(22,54,50,26),(23,55,51,27),(24,56,52,28),(29,61,57,69),(30,62,58,70),(31,63,59,71),(32,64,60,72),(33,123,127,91),(34,124,128,92),(35,121,125,89),(36,122,126,90),(65,111,100,87),(66,112,97,88),(67,109,98,85),(68,110,99,86),(73,101,105,77),(74,102,106,78),(75,103,107,79),(76,104,108,80),(81,93,113,117),(82,94,114,118),(83,95,115,119),(84,96,116,120)], [(1,59,51,45),(2,58,52,48),(3,57,49,47),(4,60,50,46),(5,42,62,56),(6,41,63,55),(7,44,64,54),(8,43,61,53),(9,31,23,17),(10,30,24,20),(11,29,21,19),(12,32,22,18),(13,71,27,39),(14,70,28,38),(15,69,25,37),(16,72,26,40),(33,105,98,95),(34,108,99,94),(35,107,100,93),(36,106,97,96),(65,117,125,75),(66,120,126,74),(67,119,127,73),(68,118,128,76),(77,85,115,123),(78,88,116,122),(79,87,113,121),(80,86,114,124),(81,89,103,111),(82,92,104,110),(83,91,101,109),(84,90,102,112)], [(1,89,3,91),(2,124,4,122),(5,76,7,74),(6,107,8,105),(9,121,11,123),(10,92,12,90),(13,125,15,127),(14,34,16,36),(17,79,19,77),(18,102,20,104),(21,85,23,87),(22,112,24,110),(25,67,27,65),(26,97,28,99),(29,115,31,113),(30,82,32,84),(33,41,35,43),(37,73,39,75),(38,108,40,106),(42,128,44,126),(45,103,47,101),(46,78,48,80),(49,109,51,111),(50,88,52,86),(53,98,55,100),(54,66,56,68),(57,83,59,81),(58,114,60,116),(61,95,63,93),(62,118,64,120),(69,119,71,117),(70,94,72,96)]])
38 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4H | 4I | ··· | 4Z | 4AA | 4AB | 4AC | 4AD |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | Q8 | C4○D4 | 2- 1+4 |
| kernel | C42.174D4 | C4×C4⋊C4 | C42⋊8C4 | C42⋊9C4 | C23.63C23 | C23.67C23 | C23.81C23 | C2×C4×Q8 | C2×C42.C2 | C42 | C2×Q8 | C2×C4 | C22 |
| # reps | 1 | 1 | 1 | 1 | 4 | 2 | 4 | 1 | 1 | 4 | 4 | 12 | 2 |
Matrix representation of C42.174D4 ►in GL6(𝔽5)
| 2 | 0 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 2 | 1 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 4 | 2 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 1 | 3 |
| 0 | 2 | 0 | 0 | 0 | 0 |
| 2 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 4 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(5))| [2,0,0,0,0,0,0,3,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,2,0,0,0,0,0,1],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,4,0,0,0,0,0,2],[0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,2,1,0,0,0,0,0,3],[0,2,0,0,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,0,4,1] >;
C42.174D4 in GAP, Magma, Sage, TeX
C_4^2._{174}D_4 % in TeX
G:=Group("C4^2.174D4"); // GroupNames label
G:=SmallGroup(128,1297);
// by ID
G=gap.SmallGroup(128,1297);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,456,758,723,184,675,80]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=a^2,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a^-1*b^2,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations