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## G = C7×M4(2).8C22order 448 = 26·7

### Direct product of C7 and M4(2).8C22

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C7×M4(2).8C22
 Chief series C1 — C2 — C4 — C2×C4 — C2×C28 — C7×M4(2) — C7×C4.D4 — C7×M4(2).8C22
 Lower central C1 — C2 — C22 — C7×M4(2).8C22
 Upper central C1 — C28 — C22×C28 — C7×M4(2).8C22

Generators and relations for C7×M4(2).8C22
G = < a,b,c,d,e | a7=b8=c2=d2=1, e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b5, dbd=bc, cd=dc, ece-1=b4c, ede-1=b4cd >

Subgroups: 242 in 150 conjugacy classes, 78 normal (26 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C14, C14, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C28, C28, C28, C2×C14, C2×C14, C2×C14, C4.D4, C4.10D4, C2×M4(2), C2×C4○D4, C56, C2×C28, C2×C28, C2×C28, C7×D4, C7×Q8, C22×C14, C22×C14, M4(2).8C22, C2×C56, C7×M4(2), C7×M4(2), C22×C28, C22×C28, D4×C14, D4×C14, Q8×C14, C7×C4○D4, C7×C4.D4, C7×C4.10D4, C14×M4(2), C14×C4○D4, C7×M4(2).8C22
Quotients: C1, C2, C4, C22, C7, C2×C4, D4, C23, C14, C22⋊C4, C22×C4, C2×D4, C28, C2×C14, C2×C22⋊C4, C2×C28, C7×D4, C22×C14, M4(2).8C22, C7×C22⋊C4, C22×C28, D4×C14, C14×C22⋊C4, C7×M4(2).8C22

Smallest permutation representation of C7×M4(2).8C22
On 112 points
Generators in S112
(1 75 46 24 89 69 38)(2 76 47 17 90 70 39)(3 77 48 18 91 71 40)(4 78 41 19 92 72 33)(5 79 42 20 93 65 34)(6 80 43 21 94 66 35)(7 73 44 22 95 67 36)(8 74 45 23 96 68 37)(9 112 86 53 27 104 58)(10 105 87 54 28 97 59)(11 106 88 55 29 98 60)(12 107 81 56 30 99 61)(13 108 82 49 31 100 62)(14 109 83 50 32 101 63)(15 110 84 51 25 102 64)(16 111 85 52 26 103 57)
(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)
(2 6)(4 8)(9 13)(11 15)(17 21)(19 23)(25 29)(27 31)(33 37)(35 39)(41 45)(43 47)(49 53)(51 55)(58 62)(60 64)(66 70)(68 72)(74 78)(76 80)(82 86)(84 88)(90 94)(92 96)(98 102)(100 104)(106 110)(108 112)
(1 97)(2 102)(3 103)(4 100)(5 101)(6 98)(7 99)(8 104)(9 45)(10 46)(11 43)(12 44)(13 41)(14 42)(15 47)(16 48)(17 110)(18 111)(19 108)(20 109)(21 106)(22 107)(23 112)(24 105)(25 39)(26 40)(27 37)(28 38)(29 35)(30 36)(31 33)(32 34)(49 72)(50 65)(51 70)(52 71)(53 68)(54 69)(55 66)(56 67)(57 77)(58 74)(59 75)(60 80)(61 73)(62 78)(63 79)(64 76)(81 95)(82 92)(83 93)(84 90)(85 91)(86 96)(87 89)(88 94)
(1 8 3 2 5 4 7 6)(9 16 11 10 13 12 15 14)(17 20 19 22 21 24 23 18)(25 32 27 26 29 28 31 30)(33 36 35 38 37 40 39 34)(41 44 43 46 45 48 47 42)(49 56 51 50 53 52 55 54)(57 60 59 62 61 64 63 58)(65 72 67 66 69 68 71 70)(73 80 75 74 77 76 79 78)(81 84 83 86 85 88 87 82)(89 96 91 90 93 92 95 94)(97 100 99 102 101 104 103 98)(105 108 107 110 109 112 111 106)

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

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

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

154 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 4E 4F 4G 7A ··· 7F 8A ··· 8H 14A ··· 14F 14G ··· 14X 14Y ··· 14AJ 28A ··· 28L 28M ··· 28AD 28AE ··· 28AP 56A ··· 56AV order 1 2 2 2 2 2 2 4 4 4 4 4 4 4 7 ··· 7 8 ··· 8 14 ··· 14 14 ··· 14 14 ··· 14 28 ··· 28 28 ··· 28 28 ··· 28 56 ··· 56 size 1 1 2 2 2 4 4 1 1 2 2 2 4 4 1 ··· 1 4 ··· 4 1 ··· 1 2 ··· 2 4 ··· 4 1 ··· 1 2 ··· 2 4 ··· 4 4 ··· 4

154 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 4 4 type + + + + + + image C1 C2 C2 C2 C2 C4 C4 C7 C14 C14 C14 C14 C28 C28 D4 C7×D4 M4(2).8C22 C7×M4(2).8C22 kernel C7×M4(2).8C22 C7×C4.D4 C7×C4.10D4 C14×M4(2) C14×C4○D4 C22×C28 D4×C14 M4(2).8C22 C4.D4 C4.10D4 C2×M4(2) C2×C4○D4 C22×C4 C2×D4 C2×C28 C2×C4 C7 C1 # reps 1 2 2 2 1 4 4 6 12 12 12 6 24 24 4 24 2 12

Matrix representation of C7×M4(2).8C22 in GL6(𝔽113)

 30 0 0 0 0 0 0 30 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 1 0 0 0 0 112 0 0 0 0 0 0 0 0 0 30 0 0 0 27 1 0 30 0 0 57 0 0 0 0 0 33 7 86 112
,
 112 0 0 0 0 0 0 112 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 112 0 0 0 66 15 0 112
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 80 49 0 1 0 0 64 33 1 0
,
 0 1 0 0 0 0 112 0 0 0 0 0 0 0 0 0 111 0 0 0 66 15 0 111 0 0 49 0 0 0 0 0 43 7 47 98

G:=sub<GL(6,GF(113))| [30,0,0,0,0,0,0,30,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,112,0,0,0,0,1,0,0,0,0,0,0,0,0,27,57,33,0,0,0,1,0,7,0,0,30,0,0,86,0,0,0,30,0,112],[112,0,0,0,0,0,0,112,0,0,0,0,0,0,1,0,0,66,0,0,0,1,0,15,0,0,0,0,112,0,0,0,0,0,0,112],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,80,64,0,0,1,0,49,33,0,0,0,0,0,1,0,0,0,0,1,0],[0,112,0,0,0,0,1,0,0,0,0,0,0,0,0,66,49,43,0,0,0,15,0,7,0,0,111,0,0,47,0,0,0,111,0,98] >;

C7×M4(2).8C22 in GAP, Magma, Sage, TeX

C_7\times M_4(2)._8C_2^2
% in TeX

G:=Group("C7xM4(2).8C2^2");
// GroupNames label

G:=SmallGroup(448,821);
// by ID

G=gap.SmallGroup(448,821);
# by ID

G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,784,813,1192,9804,7068,124]);
// Polycyclic

G:=Group<a,b,c,d,e|a^7=b^8=c^2=d^2=1,e^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=e*b*e^-1=b^5,d*b*d=b*c,c*d=d*c,e*c*e^-1=b^4*c,e*d*e^-1=b^4*c*d>;
// generators/relations

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