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## G = C7×C8.C22order 224 = 25·7

### Direct product of C7 and C8.C22

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C7×C8.C22
 Chief series C1 — C2 — C4 — C28 — C7×D4 — C7×SD16 — C7×C8.C22
 Lower central C1 — C2 — C4 — C7×C8.C22
 Upper central C1 — C14 — C2×C28 — C7×C8.C22

Generators and relations for C7×C8.C22
G = < a,b,c,d | a7=b8=c2=d2=1, ab=ba, ac=ca, ad=da, cbc=b3, dbd=b5, dcd=b4c >

Subgroups: 84 in 60 conjugacy classes, 40 normal (24 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, Q8, C14, C14, M4(2), SD16, Q16, C2×Q8, C4○D4, C28, C28, C2×C14, C2×C14, C8.C22, C56, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C7×Q8, C7×Q8, C7×M4(2), C7×SD16, C7×Q16, Q8×C14, C7×C4○D4, C7×C8.C22
Quotients: C1, C2, C22, C7, D4, C23, C14, C2×D4, C2×C14, C8.C22, C7×D4, C22×C14, D4×C14, C7×C8.C22

Smallest permutation representation of C7×C8.C22
On 112 points
Generators in S112
(1 87 42 28 93 79 34)(2 88 43 29 94 80 35)(3 81 44 30 95 73 36)(4 82 45 31 96 74 37)(5 83 46 32 89 75 38)(6 84 47 25 90 76 39)(7 85 48 26 91 77 40)(8 86 41 27 92 78 33)(9 111 21 52 69 103 57)(10 112 22 53 70 104 58)(11 105 23 54 71 97 59)(12 106 24 55 72 98 60)(13 107 17 56 65 99 61)(14 108 18 49 66 100 62)(15 109 19 50 67 101 63)(16 110 20 51 68 102 64)
(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 4)(3 7)(6 8)(10 12)(11 15)(14 16)(18 20)(19 23)(22 24)(25 27)(26 30)(29 31)(33 39)(35 37)(36 40)(41 47)(43 45)(44 48)(49 51)(50 54)(53 55)(58 60)(59 63)(62 64)(66 68)(67 71)(70 72)(73 77)(74 80)(76 78)(81 85)(82 88)(84 86)(90 92)(91 95)(94 96)(97 101)(98 104)(100 102)(105 109)(106 112)(108 110)
(1 97)(2 102)(3 99)(4 104)(5 101)(6 98)(7 103)(8 100)(9 48)(10 45)(11 42)(12 47)(13 44)(14 41)(15 46)(16 43)(17 95)(18 92)(19 89)(20 94)(21 91)(22 96)(23 93)(24 90)(25 106)(26 111)(27 108)(28 105)(29 110)(30 107)(31 112)(32 109)(33 66)(34 71)(35 68)(36 65)(37 70)(38 67)(39 72)(40 69)(49 78)(50 75)(51 80)(52 77)(53 74)(54 79)(55 76)(56 73)(57 85)(58 82)(59 87)(60 84)(61 81)(62 86)(63 83)(64 88)

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

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

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

C7×C8.C22 is a maximal subgroup of   D28.39D4  M4(2).15D14  M4(2).16D14  D28.40D4  D56⋊C22  C56.C23  D28.44D4

77 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 7A ··· 7F 8A 8B 14A ··· 14F 14G ··· 14L 14M ··· 14R 28A ··· 28L 28M ··· 28AD 56A ··· 56L order 1 2 2 2 4 4 4 4 4 7 ··· 7 8 8 14 ··· 14 14 ··· 14 14 ··· 14 28 ··· 28 28 ··· 28 56 ··· 56 size 1 1 2 4 2 2 4 4 4 1 ··· 1 4 4 1 ··· 1 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4 4 ··· 4

77 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 4 4 type + + + + + + + + - image C1 C2 C2 C2 C2 C2 C7 C14 C14 C14 C14 C14 D4 D4 C7×D4 C7×D4 C8.C22 C7×C8.C22 kernel C7×C8.C22 C7×M4(2) C7×SD16 C7×Q16 Q8×C14 C7×C4○D4 C8.C22 M4(2) SD16 Q16 C2×Q8 C4○D4 C28 C2×C14 C4 C22 C7 C1 # reps 1 1 2 2 1 1 6 6 12 12 6 6 1 1 6 6 1 6

Matrix representation of C7×C8.C22 in GL4(𝔽113) generated by

 16 0 0 0 0 16 0 0 0 0 16 0 0 0 0 16
,
 110 101 110 9 3 0 12 104 92 0 104 101 21 21 12 12
,
 1 1 1 0 0 112 0 0 0 0 112 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 111 112 112 112 0 1 0 0
G:=sub<GL(4,GF(113))| [16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[110,3,92,21,101,0,0,21,110,12,104,12,9,104,101,12],[1,0,0,0,1,112,0,0,1,0,112,0,0,0,0,1],[1,0,111,0,0,0,112,1,0,0,112,0,0,1,112,0] >;

C7×C8.C22 in GAP, Magma, Sage, TeX

C_7\times C_8.C_2^2
% in TeX

G:=Group("C7xC8.C2^2");
// GroupNames label

G:=SmallGroup(224,172);
// by ID

G=gap.SmallGroup(224,172);
# by ID

G:=PCGroup([6,-2,-2,-2,-7,-2,-2,697,679,2090,5044,2530,88]);
// Polycyclic

G:=Group<a,b,c,d|a^7=b^8=c^2=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^3,d*b*d=b^5,d*c*d=b^4*c>;
// generators/relations

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