M4(2)xC17 - GroupNames

(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)(120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136)

(1 71 130 65 31 99 119 48)(2 72 131 66 32 100 103 49)(3 73 132 67 33 101 104 50)(4 74 133 68 34 102 105 51)(5 75 134 52 18 86 106 35)(6 76 135 53 19 87 107 36)(7 77 136 54 20 88 108 37)(8 78 120 55 21 89 109 38)(9 79 121 56 22 90 110 39)(10 80 122 57 23 91 111 40)(11 81 123 58 24 92 112 41)(12 82 124 59 25 93 113 42)(13 83 125 60 26 94 114 43)(14 84 126 61 27 95 115 44)(15 85 127 62 28 96 116 45)(16 69 128 63 29 97 117 46)(17 70 129 64 30 98 118 47)

(35 52)(36 53)(37 54)(38 55)(39 56)(40 57)(41 58)(42 59)(43 60)(44 61)(45 62)(46 63)(47 64)(48 65)(49 66)(50 67)(51 68)(69 97)(70 98)(71 99)(72 100)(73 101)(74 102)(75 86)(76 87)(77 88)(78 89)(79 90)(80 91)(81 92)(82 93)(83 94)(84 95)(85 96)

G:=sub<Sym(136)| (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)(120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136), (1,71,130,65,31,99,119,48)(2,72,131,66,32,100,103,49)(3,73,132,67,33,101,104,50)(4,74,133,68,34,102,105,51)(5,75,134,52,18,86,106,35)(6,76,135,53,19,87,107,36)(7,77,136,54,20,88,108,37)(8,78,120,55,21,89,109,38)(9,79,121,56,22,90,110,39)(10,80,122,57,23,91,111,40)(11,81,123,58,24,92,112,41)(12,82,124,59,25,93,113,42)(13,83,125,60,26,94,114,43)(14,84,126,61,27,95,115,44)(15,85,127,62,28,96,116,45)(16,69,128,63,29,97,117,46)(17,70,129,64,30,98,118,47), (35,52)(36,53)(37,54)(38,55)(39,56)(40,57)(41,58)(42,59)(43,60)(44,61)(45,62)(46,63)(47,64)(48,65)(49,66)(50,67)(51,68)(69,97)(70,98)(71,99)(72,100)(73,101)(74,102)(75,86)(76,87)(77,88)(78,89)(79,90)(80,91)(81,92)(82,93)(83,94)(84,95)(85,96)>;

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)(120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136), (1,71,130,65,31,99,119,48)(2,72,131,66,32,100,103,49)(3,73,132,67,33,101,104,50)(4,74,133,68,34,102,105,51)(5,75,134,52,18,86,106,35)(6,76,135,53,19,87,107,36)(7,77,136,54,20,88,108,37)(8,78,120,55,21,89,109,38)(9,79,121,56,22,90,110,39)(10,80,122,57,23,91,111,40)(11,81,123,58,24,92,112,41)(12,82,124,59,25,93,113,42)(13,83,125,60,26,94,114,43)(14,84,126,61,27,95,115,44)(15,85,127,62,28,96,116,45)(16,69,128,63,29,97,117,46)(17,70,129,64,30,98,118,47), (35,52)(36,53)(37,54)(38,55)(39,56)(40,57)(41,58)(42,59)(43,60)(44,61)(45,62)(46,63)(47,64)(48,65)(49,66)(50,67)(51,68)(69,97)(70,98)(71,99)(72,100)(73,101)(74,102)(75,86)(76,87)(77,88)(78,89)(79,90)(80,91)(81,92)(82,93)(83,94)(84,95)(85,96) );

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),(120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136)], [(1,71,130,65,31,99,119,48),(2,72,131,66,32,100,103,49),(3,73,132,67,33,101,104,50),(4,74,133,68,34,102,105,51),(5,75,134,52,18,86,106,35),(6,76,135,53,19,87,107,36),(7,77,136,54,20,88,108,37),(8,78,120,55,21,89,109,38),(9,79,121,56,22,90,110,39),(10,80,122,57,23,91,111,40),(11,81,123,58,24,92,112,41),(12,82,124,59,25,93,113,42),(13,83,125,60,26,94,114,43),(14,84,126,61,27,95,115,44),(15,85,127,62,28,96,116,45),(16,69,128,63,29,97,117,46),(17,70,129,64,30,98,118,47)], [(35,52),(36,53),(37,54),(38,55),(39,56),(40,57),(41,58),(42,59),(43,60),(44,61),(45,62),(46,63),(47,64),(48,65),(49,66),(50,67),(51,68),(69,97),(70,98),(71,99),(72,100),(73,101),(74,102),(75,86),(76,87),(77,88),(78,89),(79,90),(80,91),(81,92),(82,93),(83,94),(84,95),(85,96)]])

170 conjugacy classes

class	1	2A	2B	4A	4B	4C	8A	8B	8C	8D	17A	···	17P	34A	···	34P	34Q	···	34AF	68A	···	68AF	68AG	···	68AV	136A	···	136BL
order	1	2	2	4	4	4	8	8	8	8	17	···	17	34	···	34	34	···	34	68	···	68	68	···	68	136	···	136
size	1	1	2	1	1	2	2	2	2	2	1	···	1	1	···	1	2	···	2	1	···	1	2	···	2	2	···	2

170 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	2	2
type	+	+	+
image	C₁	C₂	C₂	C₄	C₄	C₁₇	C₃₄	C₃₄	C₆₈	C₆₈	M₄(2)	M₄(2)×C₁₇
kernel	M₄(2)×C₁₇	C₁₃₆	C₂×C₆₈	C₆₈	C₂×C₃₄	M₄(2)	C₈	C₂×C₄	C₄	C₂²	C₁₇	C₁
# reps	1	2	1	2	2	16	32	16	32	32	2	32

Matrix representation of M₄(2)×C₁₇ ►in GL₂(𝔽₁₃₇) generated by

34	0
0	34

6	87
63	131

1	52
0	136

G:=sub<GL(2,GF(137))| [34,0,0,34],[6,63,87,131],[1,0,52,136] >;

M₄(2)×C₁₇ in GAP, Magma, Sage, TeX

M_4(2)\times C_{17}

% in TeX

G:=Group("M4(2)xC17");

// GroupNames label

G:=SmallGroup(272,24);

// by ID

G=gap.SmallGroup(272,24);

# by ID

G:=PCGroup([5,-2,-2,-17,-2,-2,340,1381,58]);

// Polycyclic

G:=Group<a,b,c|a^17=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^5>;

// generators/relations

Export

Subgroup lattice of M₄(2)×C₁₇ in TeX

G = M4(2)×C17 order 272 = 24·17

Direct product of C17 and M4(2)

G = M₄(2)×C₁₇ order 272 = 2⁴·17

Direct product of C₁₇ and M₄(2)