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## G = C7×C23.C23order 448 = 26·7

### Direct product of C7 and C23.C23

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C7×C23.C23
 Chief series C1 — C2 — C22 — C23 — C22×C14 — C7×C22⋊C4 — C7×C23⋊C4 — C7×C23.C23
 Lower central C1 — C2 — C22 — C7×C23.C23
 Upper central C1 — C28 — C22×C28 — C7×C23.C23

Generators and relations for C7×C23.C23
G = < a,b,c,d,e,f,g | a7=b2=c2=d2=f2=1, e2=c, g2=d, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, ebe-1=bd=db, bf=fb, bg=gb, fcf=cd=dc, ce=ec, cg=gc, de=ed, df=fd, dg=gd, fef=bde, eg=ge, fg=gf >

Subgroups: 274 in 158 conjugacy classes, 78 normal (26 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C7, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C14, C14, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C28, C28, C28, C2×C14, C2×C14, C2×C14, C23⋊C4, C42⋊C2, C2×C4○D4, C2×C28, C2×C28, C2×C28, C7×D4, C7×Q8, C22×C14, C22×C14, C23.C23, C4×C28, C7×C22⋊C4, C7×C4⋊C4, C22×C28, C22×C28, D4×C14, D4×C14, Q8×C14, C7×C4○D4, C7×C23⋊C4, C7×C42⋊C2, C14×C4○D4, C7×C23.C23
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, C23.C23, C7×C22⋊C4, C22×C28, D4×C14, C14×C22⋊C4, C7×C23.C23

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

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

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

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

154 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 4E 4F ··· 4O 7A ··· 7F 14A ··· 14F 14G ··· 14X 14Y ··· 14AJ 28A ··· 28L 28M ··· 28AD 28AE ··· 28CL order 1 2 2 2 2 2 2 4 4 4 4 4 4 ··· 4 7 ··· 7 14 ··· 14 14 ··· 14 14 ··· 14 28 ··· 28 28 ··· 28 28 ··· 28 size 1 1 2 2 2 4 4 1 1 2 2 2 4 ··· 4 1 ··· 1 1 ··· 1 2 ··· 2 4 ··· 4 1 ··· 1 2 ··· 2 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 C4 C4 C4 C7 C14 C14 C14 C28 C28 C28 D4 C7×D4 C23.C23 C7×C23.C23 kernel C7×C23.C23 C7×C23⋊C4 C7×C42⋊C2 C14×C4○D4 C22×C28 D4×C14 Q8×C14 C23.C23 C23⋊C4 C42⋊C2 C2×C4○D4 C22×C4 C2×D4 C2×Q8 C2×C28 C2×C4 C7 C1 # reps 1 4 2 1 4 2 2 6 24 12 6 24 12 12 4 24 2 12

Matrix representation of C7×C23.C23 in GL4(𝔽29) generated by

 20 0 0 0 0 20 0 0 0 0 20 0 0 0 0 20
,
 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0
,
 1 0 0 0 0 1 0 0 0 0 28 0 0 0 0 28
,
 28 0 0 0 0 28 0 0 0 0 28 0 0 0 0 28
,
 1 0 0 0 0 28 0 0 0 0 0 28 0 0 1 0
,
 0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0
,
 17 0 0 0 0 17 0 0 0 0 17 0 0 0 0 17
G:=sub<GL(4,GF(29))| [20,0,0,0,0,20,0,0,0,0,20,0,0,0,0,20],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,28,0,0,0,0,28],[28,0,0,0,0,28,0,0,0,0,28,0,0,0,0,28],[1,0,0,0,0,28,0,0,0,0,0,1,0,0,28,0],[0,0,1,0,0,0,0,1,1,0,0,0,0,1,0,0],[17,0,0,0,0,17,0,0,0,0,17,0,0,0,0,17] >;

C7×C23.C23 in GAP, Magma, Sage, TeX

C_7\times C_2^3.C_2^3
% in TeX

G:=Group("C7xC2^3.C2^3");
// GroupNames label

G:=SmallGroup(448,818);
// by ID

G=gap.SmallGroup(448,818);
# by ID

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

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

׿
×
𝔽