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## G = C56.9C23order 448 = 26·7

### 2nd non-split extension by C56 of C23 acting via C23/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C28 — C56.9C23
 Chief series C1 — C7 — C14 — C28 — D28 — C2×D28 — C2×C4○D28 — C56.9C23
 Lower central C7 — C14 — C28 — C56.9C23
 Upper central C1 — C4 — C22×C4 — C2×M4(2)

Generators and relations for C56.9C23
G = < a,b,c,d | a56=b2=1, c2=d2=a28, bab=a27, ac=ca, dad-1=a29, bc=cb, bd=db, cd=dc >

Subgroups: 1380 in 262 conjugacy classes, 107 normal (21 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, D7, C14, C14, C2×C8, M4(2), D8, SD16, Q16, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, Dic7, C28, C28, D14, C2×C14, C2×C14, C2×C14, C2×M4(2), C4○D8, C8⋊C22, C8.C22, C2×C4○D4, C56, Dic14, Dic14, C4×D7, D28, D28, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C22×D7, C22×C14, D8⋊C22, C56⋊C2, D56, Dic28, C2×C56, C7×M4(2), C2×Dic14, C2×C4×D7, C2×D28, C4○D28, C4○D28, C2×C7⋊D4, C22×C28, D567C2, C8⋊D14, C8.D14, C14×M4(2), C2×C4○D28, C56.9C23
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C24, D14, C22×D4, D28, C22×D7, D8⋊C22, C2×D28, C23×D7, C22×D28, C56.9C23

Smallest permutation representation of C56.9C23
On 112 points
Generators in S112
(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 28)(3 55)(4 26)(5 53)(6 24)(7 51)(8 22)(9 49)(10 20)(11 47)(12 18)(13 45)(14 16)(15 43)(17 41)(19 39)(21 37)(23 35)(25 33)(27 31)(30 56)(32 54)(34 52)(36 50)(38 48)(40 46)(42 44)(57 107)(58 78)(59 105)(60 76)(61 103)(62 74)(63 101)(64 72)(65 99)(66 70)(67 97)(69 95)(71 93)(73 91)(75 89)(77 87)(79 85)(80 112)(81 83)(82 110)(84 108)(86 106)(88 104)(90 102)(92 100)(94 98)(109 111)
(1 68 29 96)(2 69 30 97)(3 70 31 98)(4 71 32 99)(5 72 33 100)(6 73 34 101)(7 74 35 102)(8 75 36 103)(9 76 37 104)(10 77 38 105)(11 78 39 106)(12 79 40 107)(13 80 41 108)(14 81 42 109)(15 82 43 110)(16 83 44 111)(17 84 45 112)(18 85 46 57)(19 86 47 58)(20 87 48 59)(21 88 49 60)(22 89 50 61)(23 90 51 62)(24 91 52 63)(25 92 53 64)(26 93 54 65)(27 94 55 66)(28 95 56 67)
(1 68 29 96)(2 97 30 69)(3 70 31 98)(4 99 32 71)(5 72 33 100)(6 101 34 73)(7 74 35 102)(8 103 36 75)(9 76 37 104)(10 105 38 77)(11 78 39 106)(12 107 40 79)(13 80 41 108)(14 109 42 81)(15 82 43 110)(16 111 44 83)(17 84 45 112)(18 57 46 85)(19 86 47 58)(20 59 48 87)(21 88 49 60)(22 61 50 89)(23 90 51 62)(24 63 52 91)(25 92 53 64)(26 65 54 93)(27 94 55 66)(28 67 56 95)

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

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

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

82 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 4A 4B 4C 4D 4E 4F 4G 4H 4I 7A 7B 7C 8A 8B 8C 8D 14A ··· 14I 14J ··· 14O 28A ··· 28L 28M ··· 28R 56A ··· 56X order 1 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 7 7 7 8 8 8 8 14 ··· 14 14 ··· 14 28 ··· 28 28 ··· 28 56 ··· 56 size 1 1 2 2 2 28 28 28 28 1 1 2 2 2 28 28 28 28 2 2 2 4 4 4 4 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4 4 ··· 4

82 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 D4 D4 D7 D14 D14 D14 D28 D28 D8⋊C22 C56.9C23 kernel C56.9C23 D56⋊7C2 C8⋊D14 C8.D14 C14×M4(2) C2×C4○D28 C2×C28 C22×C14 C2×M4(2) C2×C8 M4(2) C22×C4 C2×C4 C23 C7 C1 # reps 1 4 4 4 1 2 3 1 3 6 12 3 18 6 2 12

Matrix representation of C56.9C23 in GL6(𝔽113)

 0 1 0 0 0 0 112 24 0 0 0 0 0 0 0 0 98 0 0 0 0 15 0 63 0 0 98 104 0 0 0 0 15 9 0 98
,
 112 0 0 0 0 0 89 1 0 0 0 0 0 0 1 0 0 0 0 0 72 112 0 0 0 0 0 91 112 111 0 0 112 22 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 15 0 0 0 0 0 0 15 0 0 0 0 0 0 15 0 0 0 0 0 0 15
,
 112 0 0 0 0 0 0 112 0 0 0 0 0 0 15 0 0 0 0 0 0 15 0 0 0 0 0 0 98 0 0 0 0 9 0 98

G:=sub<GL(6,GF(113))| [0,112,0,0,0,0,1,24,0,0,0,0,0,0,0,0,98,15,0,0,0,15,104,9,0,0,98,0,0,0,0,0,0,63,0,98],[112,89,0,0,0,0,0,1,0,0,0,0,0,0,1,72,0,112,0,0,0,112,91,22,0,0,0,0,112,0,0,0,0,0,111,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,15,0,0,0,0,0,0,15,0,0,0,0,0,0,15,0,0,0,0,0,0,15],[112,0,0,0,0,0,0,112,0,0,0,0,0,0,15,0,0,0,0,0,0,15,0,9,0,0,0,0,98,0,0,0,0,0,0,98] >;

C56.9C23 in GAP, Magma, Sage, TeX

C_{56}._9C_2^3
% in TeX

G:=Group("C56.9C2^3");
// GroupNames label

G:=SmallGroup(448,1201);
// by ID

G=gap.SmallGroup(448,1201);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,184,675,570,80,1684,102,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^56=b^2=1,c^2=d^2=a^28,b*a*b=a^27,a*c=c*a,d*a*d^-1=a^29,b*c=c*b,b*d=d*b,c*d=d*c>;
// generators/relations

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