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## G = C42.697C23order 128 = 27

### 112nd non-split extension by C42 of C23 acting via C23/C22=C2

p-group, metabelian, nilpotent (class 2), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2 — C42.697C23
 Chief series C1 — C2 — C4 — C2×C4 — C42 — C2×C42 — C4×C4○D4 — C42.697C23
 Lower central C1 — C2 — C42.697C23
 Upper central C1 — C2×C4 — C42.697C23
 Jennings C1 — C2 — C2 — C2×C4 — C42.697C23

Generators and relations for C42.697C23
G = < a,b,c,d,e | a4=b4=d2=e2=1, c2=b, ab=ba, ac=ca, dad=a-1, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=a2c, ede=a2d >

Subgroups: 276 in 216 conjugacy classes, 174 normal (18 characteristic)
C1, C2 [×3], C2 [×6], C4 [×2], C4 [×8], C4 [×7], C22, C22 [×6], C22 [×6], C8 [×8], C2×C4 [×3], C2×C4 [×21], C2×C4 [×9], D4 [×12], Q8 [×4], C23 [×3], C42, C42 [×9], C22⋊C4 [×6], C4⋊C4 [×6], C2×C8 [×8], C2×C8 [×6], C22×C4 [×9], C2×D4 [×3], C2×Q8, C4○D4 [×8], C4×C8 [×6], C22⋊C8 [×6], C4⋊C8, C4⋊C8 [×9], C2×C42 [×3], C42⋊C2 [×3], C4×D4 [×6], C4×Q8 [×2], C22×C8 [×6], C2×C4○D4, C2×C4⋊C8 [×3], C42.12C4 [×3], C8×D4 [×6], C8×Q8 [×2], C4×C4○D4, C42.697C23
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C8 [×8], C2×C4 [×28], C23 [×15], C2×C8 [×28], C22×C4 [×14], C24, C22×C8 [×14], C23×C4, 2+ 1+4, 2- 1+4, C23.33C23, C23×C8, Q8○M4(2), C42.697C23

Smallest permutation representation of C42.697C23
On 64 points
Generators in S64
(1 10 55 63)(2 11 56 64)(3 12 49 57)(4 13 50 58)(5 14 51 59)(6 15 52 60)(7 16 53 61)(8 9 54 62)(17 46 31 35)(18 47 32 36)(19 48 25 37)(20 41 26 38)(21 42 27 39)(22 43 28 40)(23 44 29 33)(24 45 30 34)
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 32)(33 35 37 39)(34 36 38 40)(41 43 45 47)(42 44 46 48)(49 51 53 55)(50 52 54 56)(57 59 61 63)(58 60 62 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)
(1 10)(2 11)(3 12)(4 13)(5 14)(6 15)(7 16)(8 9)(17 46)(18 47)(19 48)(20 41)(21 42)(22 43)(23 44)(24 45)(25 37)(26 38)(27 39)(28 40)(29 33)(30 34)(31 35)(32 36)(49 57)(50 58)(51 59)(52 60)(53 61)(54 62)(55 63)(56 64)
(1 46)(2 36)(3 48)(4 38)(5 42)(6 40)(7 44)(8 34)(9 24)(10 31)(11 18)(12 25)(13 20)(14 27)(15 22)(16 29)(17 63)(19 57)(21 59)(23 61)(26 58)(28 60)(30 62)(32 64)(33 53)(35 55)(37 49)(39 51)(41 50)(43 52)(45 54)(47 56)

G:=sub<Sym(64)| (1,10,55,63)(2,11,56,64)(3,12,49,57)(4,13,50,58)(5,14,51,59)(6,15,52,60)(7,16,53,61)(8,9,54,62)(17,46,31,35)(18,47,32,36)(19,48,25,37)(20,41,26,38)(21,42,27,39)(22,43,28,40)(23,44,29,33)(24,45,30,34), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64), (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), (1,10)(2,11)(3,12)(4,13)(5,14)(6,15)(7,16)(8,9)(17,46)(18,47)(19,48)(20,41)(21,42)(22,43)(23,44)(24,45)(25,37)(26,38)(27,39)(28,40)(29,33)(30,34)(31,35)(32,36)(49,57)(50,58)(51,59)(52,60)(53,61)(54,62)(55,63)(56,64), (1,46)(2,36)(3,48)(4,38)(5,42)(6,40)(7,44)(8,34)(9,24)(10,31)(11,18)(12,25)(13,20)(14,27)(15,22)(16,29)(17,63)(19,57)(21,59)(23,61)(26,58)(28,60)(30,62)(32,64)(33,53)(35,55)(37,49)(39,51)(41,50)(43,52)(45,54)(47,56)>;

G:=Group( (1,10,55,63)(2,11,56,64)(3,12,49,57)(4,13,50,58)(5,14,51,59)(6,15,52,60)(7,16,53,61)(8,9,54,62)(17,46,31,35)(18,47,32,36)(19,48,25,37)(20,41,26,38)(21,42,27,39)(22,43,28,40)(23,44,29,33)(24,45,30,34), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64), (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), (1,10)(2,11)(3,12)(4,13)(5,14)(6,15)(7,16)(8,9)(17,46)(18,47)(19,48)(20,41)(21,42)(22,43)(23,44)(24,45)(25,37)(26,38)(27,39)(28,40)(29,33)(30,34)(31,35)(32,36)(49,57)(50,58)(51,59)(52,60)(53,61)(54,62)(55,63)(56,64), (1,46)(2,36)(3,48)(4,38)(5,42)(6,40)(7,44)(8,34)(9,24)(10,31)(11,18)(12,25)(13,20)(14,27)(15,22)(16,29)(17,63)(19,57)(21,59)(23,61)(26,58)(28,60)(30,62)(32,64)(33,53)(35,55)(37,49)(39,51)(41,50)(43,52)(45,54)(47,56) );

G=PermutationGroup([(1,10,55,63),(2,11,56,64),(3,12,49,57),(4,13,50,58),(5,14,51,59),(6,15,52,60),(7,16,53,61),(8,9,54,62),(17,46,31,35),(18,47,32,36),(19,48,25,37),(20,41,26,38),(21,42,27,39),(22,43,28,40),(23,44,29,33),(24,45,30,34)], [(1,3,5,7),(2,4,6,8),(9,11,13,15),(10,12,14,16),(17,19,21,23),(18,20,22,24),(25,27,29,31),(26,28,30,32),(33,35,37,39),(34,36,38,40),(41,43,45,47),(42,44,46,48),(49,51,53,55),(50,52,54,56),(57,59,61,63),(58,60,62,64)], [(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)], [(1,10),(2,11),(3,12),(4,13),(5,14),(6,15),(7,16),(8,9),(17,46),(18,47),(19,48),(20,41),(21,42),(22,43),(23,44),(24,45),(25,37),(26,38),(27,39),(28,40),(29,33),(30,34),(31,35),(32,36),(49,57),(50,58),(51,59),(52,60),(53,61),(54,62),(55,63),(56,64)], [(1,46),(2,36),(3,48),(4,38),(5,42),(6,40),(7,44),(8,34),(9,24),(10,31),(11,18),(12,25),(13,20),(14,27),(15,22),(16,29),(17,63),(19,57),(21,59),(23,61),(26,58),(28,60),(30,62),(32,64),(33,53),(35,55),(37,49),(39,51),(41,50),(43,52),(45,54),(47,56)])

68 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 4A 4B 4C 4D 4E ··· 4Z 8A ··· 8AF order 1 2 2 2 2 ··· 2 4 4 4 4 4 ··· 4 8 ··· 8 size 1 1 1 1 2 ··· 2 1 1 1 1 2 ··· 2 2 ··· 2

68 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 4 4 4 type + + + + + + + - image C1 C2 C2 C2 C2 C2 C4 C4 C4 C4 C8 2+ 1+4 2- 1+4 Q8○M4(2) kernel C42.697C23 C2×C4⋊C8 C42.12C4 C8×D4 C8×Q8 C4×C4○D4 C42⋊C2 C4×D4 C4×Q8 C2×C4○D4 C4○D4 C4 C4 C2 # reps 1 3 3 6 2 1 6 6 2 2 32 1 1 2

Matrix representation of C42.697C23 in GL5(𝔽17)

 16 0 0 0 0 0 0 0 16 0 0 0 0 0 16 0 1 0 0 0 0 0 1 0 0
,
 13 0 0 0 0 0 16 0 0 0 0 0 16 0 0 0 0 0 16 0 0 0 0 0 16
,
 9 0 0 0 0 0 7 1 0 0 0 1 10 0 0 0 0 0 7 1 0 0 0 1 10
,
 1 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 0
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 16 0 0 0 16 0 0 0 1 0 0 0

G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,16,0,0,0,0,0,16,0,0],[13,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16],[9,0,0,0,0,0,7,1,0,0,0,1,10,0,0,0,0,0,7,1,0,0,0,1,10],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,1,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,16,0,0,0,16,0,0,0,1,0,0,0] >;

C42.697C23 in GAP, Magma, Sage, TeX

C_4^2._{697}C_2^3
% in TeX

G:=Group("C4^2.697C2^3");
// GroupNames label

G:=SmallGroup(128,1720);
// by ID

G=gap.SmallGroup(128,1720);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,219,675,80,124]);
// Polycyclic

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

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