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## G = C23.612C24order 128 = 27

### 329th central stem extension by C23 of C24

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C23 — C23.612C24
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C23.67C23 — C23.612C24
 Lower central C1 — C23 — C23.612C24
 Upper central C1 — C23 — C23.612C24
 Jennings C1 — C23 — C23.612C24

Generators and relations for C23.612C24
G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=g2=a, f2=b, ab=ba, ac=ca, ede-1=ad=da, geg-1=ae=ea, af=fa, ag=ga, bc=cb, fdf-1=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef-1=ce=ec, cf=fc, cg=gc, gdg-1=abd, fg=gf >

Subgroups: 612 in 278 conjugacy classes, 96 normal (22 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×16], C22 [×3], C22 [×4], C22 [×24], C2×C4 [×8], C2×C4 [×36], D4 [×12], Q8 [×8], C23, C23 [×2], C23 [×20], C42 [×4], C22⋊C4 [×4], C22⋊C4 [×21], C4⋊C4 [×3], C22×C4 [×2], C22×C4 [×10], C22×C4 [×3], C2×D4 [×13], C2×Q8 [×10], C24, C24 [×2], C2.C42 [×2], C2.C42 [×8], C2×C42 [×2], C2×C22⋊C4 [×2], C2×C22⋊C4 [×12], C2×C4⋊C4, C2×C4⋊C4 [×2], C4.4D4 [×8], C23×C4, C22×D4, C22×D4 [×2], C22×Q8 [×2], C23.8Q8, C23.23D4 [×2], C24.C22 [×2], C23.67C23 [×2], C232D4, C23⋊Q8 [×2], C23.10D4 [×2], C23.11D4, C2×C4.4D4 [×2], C23.612C24
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C22×D4, C2×C4○D4 [×2], 2+ 1+4 [×3], 2- 1+4, C22.31C24, C22.36C24 [×2], D45D4 [×2], C22.53C24, C24⋊C22, C23.612C24

Smallest permutation representation of C23.612C24
On 64 points
Generators in S64
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)(25 27)(26 28)(29 31)(30 32)(33 35)(34 36)(37 39)(38 40)(41 43)(42 44)(45 47)(46 48)(49 51)(50 52)(53 55)(54 56)(57 59)(58 60)(61 63)(62 64)
(1 41)(2 42)(3 43)(4 44)(5 20)(6 17)(7 18)(8 19)(9 57)(10 58)(11 59)(12 60)(13 25)(14 26)(15 27)(16 28)(21 34)(22 35)(23 36)(24 33)(29 45)(30 46)(31 47)(32 48)(37 56)(38 53)(39 54)(40 55)(49 61)(50 62)(51 63)(52 64)
(1 11)(2 12)(3 9)(4 10)(5 50)(6 51)(7 52)(8 49)(13 31)(14 32)(15 29)(16 30)(17 63)(18 64)(19 61)(20 62)(21 40)(22 37)(23 38)(24 39)(25 47)(26 48)(27 45)(28 46)(33 54)(34 55)(35 56)(36 53)(41 59)(42 60)(43 57)(44 58)
(1 4)(2 3)(5 63)(6 62)(7 61)(8 64)(9 12)(10 11)(13 28)(14 27)(15 26)(16 25)(17 50)(18 49)(19 52)(20 51)(21 39)(22 38)(23 37)(24 40)(29 48)(30 47)(31 46)(32 45)(33 55)(34 54)(35 53)(36 56)(41 44)(42 43)(57 60)(58 59)
(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 49 41 61)(2 5 42 20)(3 51 43 63)(4 7 44 18)(6 57 17 9)(8 59 19 11)(10 52 58 64)(12 50 60 62)(13 34 25 21)(14 56 26 37)(15 36 27 23)(16 54 28 39)(22 32 35 48)(24 30 33 46)(29 53 45 38)(31 55 47 40)
(1 27 3 25)(2 26 4 28)(5 37 7 39)(6 40 8 38)(9 47 11 45)(10 46 12 48)(13 41 15 43)(14 44 16 42)(17 55 19 53)(18 54 20 56)(21 49 23 51)(22 52 24 50)(29 57 31 59)(30 60 32 58)(33 62 35 64)(34 61 36 63)

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

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

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

32 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 4A ··· 4P 4Q 4R 4S 4T order 1 2 ··· 2 2 2 2 2 4 ··· 4 4 4 4 4 size 1 1 ··· 1 4 4 8 8 4 ··· 4 8 8 8 8

32 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 4 4 type + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 D4 C4○D4 2+ 1+4 2- 1+4 kernel C23.612C24 C23.8Q8 C23.23D4 C24.C22 C23.67C23 C23⋊2D4 C23⋊Q8 C23.10D4 C23.11D4 C2×C4.4D4 C22⋊C4 C2×C4 C22 C22 # reps 1 1 2 2 2 1 2 2 1 2 4 8 3 1

Matrix representation of C23.612C24 in GL6(𝔽5)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 4 0 0 0 0 0 0 4 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 4 0 0 0 0 0 0 4
,
 1 0 0 0 0 0 2 4 0 0 0 0 0 0 4 4 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 4 4 0 0 0 0 2 1 0 0 0 0 0 0 1 0 0 0 0 0 0 4
,
 3 2 0 0 0 0 0 2 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 1 4 0 0 0 0 0 4 0 0 0 0 0 0 3 0 0 0 0 0 4 2 0 0 0 0 0 0 1 0 0 0 0 0 0 1

G:=sub<GL(6,GF(5))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,2,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,4,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,2,0,0,0,0,4,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4],[3,0,0,0,0,0,2,2,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[1,0,0,0,0,0,4,4,0,0,0,0,0,0,3,4,0,0,0,0,0,2,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C23.612C24 in GAP, Magma, Sage, TeX

C_2^3._{612}C_2^4
% in TeX

G:=Group("C2^3.612C2^4");
// GroupNames label

G:=SmallGroup(128,1444);
// by ID

G=gap.SmallGroup(128,1444);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,253,120,758,723,184,1571,346,80]);
// Polycyclic

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

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