(C2xQ8).8Q8 - GroupNames

G = (C₂×Q₈).₈Q₈ order 128 = 2⁷

8^th 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(C₈⋊₈D₄), 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(C₄²⋊₂C₂), 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

Derived series

C₁ — C₂²×C₄ — (C₂×Q₈).₈Q₈

Chief series

C₁ — C₂ — C₄ — C₂×C₄ — C₂²×C₄ — C₂²×Q₈ — C₂×Q₈⋊C₄ — (C₂×Q₈).₈Q₈

Lower central

C₁ — C₂ — C₂²×C₄ — (C₂×Q₈).₈Q₈

Upper central

C₁ — C₂³ — C₂×C₄² — (C₂×Q₈).₈Q₈

Jennings

C₁ — C₂ — C₂ — C₂²×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, C₄, C₄, C₂², C₈, C₂×C₄, C₂×C₄, C₂×C₄, Q₈, C₂³, C₄², C₄⋊C₄, C₄⋊C₄, C₂×C₈, C₂×C₈, C₂²×C₄, C₂²×C₄, C₂×Q₈, C₂×Q₈, C₂.C₄², Q₈⋊C₄, C₄⋊C₈, C₄.Q₈, C₂×C₄², C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₂²×C₈, C₂²×Q₈, C₂².₄Q₁₆, C₂³.₆₅C₂³, C₂³.₆₇C₂³, C₂×Q₈⋊C₄, C₂×C₄⋊C₈, C₂×C₄.Q₈, (C₂×Q₈).₈Q₈
Quotients: C₁, C₂, C₂², D₄, Q₈, C₂³, SD₁₆, C₂×D₄, C₂×Q₈, C₄○D₄, C₄⋊D₄, C₂²⋊Q₈, C₄²⋊₂C₂, C₂×SD₁₆, C₄○D₈, C₈.C₂², C₂³.Q₈, D₄.D₄, Q₈.D₄, C₈⋊₈D₄, 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)(9 115)(10 116)(11 117)(12 118)(13 119)(14 120)(15 113)(16 114)(17 28)(18 29)(19 30)(20 31)(21 32)(22 25)(23 26)(24 27)(33 125)(34 126)(35 127)(36 128)(37 121)(38 122)(39 123)(40 124)(41 105)(42 106)(43 107)(44 108)(45 109)(46 110)(47 111)(48 112)(49 64)(50 57)(51 58)(52 59)(53 60)(54 61)(55 62)(56 63)(65 102)(66 103)(67 104)(68 97)(69 98)(70 99)(71 100)(72 101)(81 96)(82 89)(83 90)(84 91)(85 92)(86 93)(87 94)(88 95)

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

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

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

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

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

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

32 conjugacy classes

class	1	2A	···	2G	4A	4B	4C	4D	4E	4F	4G	4H	4I	···	4P	8A	···	8H
order	1	2	···	2	4	4	4	4	4	4	4	4	4	···	4	8	···	8
size	1	1	···	1	2	2	2	2	4	4	4	4	8	···	8	4	···	4

32 irreducible representations

dim	1	1	1	1	1	1	1	2	2	2	2	2	2	2	4
type	+	+	+	+	+	+	+	+	+	+	-				-
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	D₄	D₄	D₄	Q₈	SD₁₆	C₄○D₄	C₄○D₈	C₈.C₂²
kernel	(C₂×Q₈).₈Q₈	C₂².₄Q₁₆	C₂³.₆₅C₂³	C₂³.₆₇C₂³	C₂×Q₈⋊C₄	C₂×C₄⋊C₈	C₂×C₄.Q₈	C₄⋊C₄	C₂×C₈	C₂²×C₄	C₂×Q₈	C₂×C₄	C₂×C₄	C₂²	C₂²
# reps	1	1	1	1	2	1	1	2	2	2	2	4	6	4	2

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

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	16
0	0	0	0	1	0

16	0	0	0	0	0
0	16	0	0	0	0
0	0	16	15	0	0
0	0	1	1	0	0
0	0	0	0	5	5
0	0	0	0	12	5

10	4	0	0	0	0
13	7	0	0	0	0
0	0	8	2	0	0
0	0	10	9	0	0
0	0	0	0	14	14
0	0	0	0	14	3

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

G = (C2×Q8).8Q8 order 128 = 27

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

G = (C₂×Q₈).₈Q₈ order 128 = 2⁷

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