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## G = C7×C23.C8order 448 = 26·7

### Direct product of C7 and C23.C8

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C7×C23.C8
 Chief series C1 — C2 — C4 — C8 — C2×C8 — C2×C56 — C7×M5(2) — C7×C23.C8
 Lower central C1 — C2 — C22 — C7×C23.C8
 Upper central C1 — C28 — C2×C56 — C7×C23.C8

Generators and relations for C7×C23.C8
G = < a,b,c,d,e | a7=b2=c2=d2=1, e8=d, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe-1=bcd, ece-1=cd=dc, de=ed >

Subgroups: 90 in 58 conjugacy classes, 34 normal (26 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, C23, C14, C14, C16, C2×C8, M4(2), C22×C4, C28, C28, C2×C14, C2×C14, M5(2), C2×M4(2), C56, C56, C2×C28, C2×C28, C22×C14, C23.C8, C112, C2×C56, C7×M4(2), C22×C28, C7×M5(2), C14×M4(2), C7×C23.C8
Quotients: C1, C2, C4, C22, C7, C8, C2×C4, D4, C14, C22⋊C4, C2×C8, M4(2), C28, C2×C14, C22⋊C8, C56, C2×C28, C7×D4, C23.C8, C7×C22⋊C4, C2×C56, C7×M4(2), C7×C22⋊C8, C7×C23.C8

Smallest permutation representation of C7×C23.C8
On 112 points
Generators in S112
(1 66 82 102 18 35 61)(2 67 83 103 19 36 62)(3 68 84 104 20 37 63)(4 69 85 105 21 38 64)(5 70 86 106 22 39 49)(6 71 87 107 23 40 50)(7 72 88 108 24 41 51)(8 73 89 109 25 42 52)(9 74 90 110 26 43 53)(10 75 91 111 27 44 54)(11 76 92 112 28 45 55)(12 77 93 97 29 46 56)(13 78 94 98 30 47 57)(14 79 95 99 31 48 58)(15 80 96 100 32 33 59)(16 65 81 101 17 34 60)
(2 10)(3 11)(6 14)(7 15)(19 27)(20 28)(23 31)(24 32)(33 41)(36 44)(37 45)(40 48)(50 58)(51 59)(54 62)(55 63)(67 75)(68 76)(71 79)(72 80)(83 91)(84 92)(87 95)(88 96)(99 107)(100 108)(103 111)(104 112)
(2 10)(4 12)(6 14)(8 16)(17 25)(19 27)(21 29)(23 31)(34 42)(36 44)(38 46)(40 48)(50 58)(52 60)(54 62)(56 64)(65 73)(67 75)(69 77)(71 79)(81 89)(83 91)(85 93)(87 95)(97 105)(99 107)(101 109)(103 111)
(1 9)(2 10)(3 11)(4 12)(5 13)(6 14)(7 15)(8 16)(17 25)(18 26)(19 27)(20 28)(21 29)(22 30)(23 31)(24 32)(33 41)(34 42)(35 43)(36 44)(37 45)(38 46)(39 47)(40 48)(49 57)(50 58)(51 59)(52 60)(53 61)(54 62)(55 63)(56 64)(65 73)(66 74)(67 75)(68 76)(69 77)(70 78)(71 79)(72 80)(81 89)(82 90)(83 91)(84 92)(85 93)(86 94)(87 95)(88 96)(97 105)(98 106)(99 107)(100 108)(101 109)(102 110)(103 111)(104 112)
(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)

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

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

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

154 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 7A ··· 7F 8A 8B 8C 8D 8E 8F 14A ··· 14F 14G ··· 14L 14M ··· 14R 16A ··· 16H 28A ··· 28L 28M ··· 28R 28S ··· 28X 56A ··· 56X 56Y ··· 56AJ 112A ··· 112AV order 1 2 2 2 4 4 4 4 7 ··· 7 8 8 8 8 8 8 14 ··· 14 14 ··· 14 14 ··· 14 16 ··· 16 28 ··· 28 28 ··· 28 28 ··· 28 56 ··· 56 56 ··· 56 112 ··· 112 size 1 1 2 4 1 1 2 4 1 ··· 1 2 2 2 2 4 4 1 ··· 1 2 ··· 2 4 ··· 4 4 ··· 4 1 ··· 1 2 ··· 2 4 ··· 4 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 2 2 4 4 type + + + + image C1 C2 C2 C4 C4 C7 C8 C8 C14 C14 C28 C28 C56 C56 D4 M4(2) C7×D4 C7×M4(2) C23.C8 C7×C23.C8 kernel C7×C23.C8 C7×M5(2) C14×M4(2) C2×C56 C22×C28 C23.C8 C2×C28 C22×C14 M5(2) C2×M4(2) C2×C8 C22×C4 C2×C4 C23 C56 C28 C8 C4 C7 C1 # reps 1 2 1 2 2 6 4 4 12 6 12 12 24 24 2 2 12 12 2 12

Matrix representation of C7×C23.C8 in GL4(𝔽113) generated by

 106 0 0 0 0 106 0 0 0 0 106 0 0 0 0 106
,
 1 0 0 0 0 112 0 0 0 0 1 0 30 0 98 112
,
 1 0 0 0 0 1 0 0 0 0 112 0 30 100 0 112
,
 112 0 0 0 0 112 0 0 0 0 112 0 0 0 0 112
,
 0 0 1 0 30 100 98 111 0 1 0 0 95 77 56 13
G:=sub<GL(4,GF(113))| [106,0,0,0,0,106,0,0,0,0,106,0,0,0,0,106],[1,0,0,30,0,112,0,0,0,0,1,98,0,0,0,112],[1,0,0,30,0,1,0,100,0,0,112,0,0,0,0,112],[112,0,0,0,0,112,0,0,0,0,112,0,0,0,0,112],[0,30,0,95,0,100,1,77,1,98,0,56,0,111,0,13] >;

C7×C23.C8 in GAP, Magma, Sage, TeX

C_7\times C_2^3.C_8
% in TeX

G:=Group("C7xC2^3.C8");
// GroupNames label

G:=SmallGroup(448,153);
// by ID

G=gap.SmallGroup(448,153);
# by ID

G:=PCGroup([7,-2,-2,-7,-2,-2,-2,-2,392,421,7059,4911,102,124]);
// Polycyclic

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

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