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G = C24.285C23order 128 = 27

125th non-split extension by C24 of C23 acting via C23/C2=C22

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

Aliases: C24.285C23, C23.365C24, C22.1712+ 1+4, C22.1262- 1+4, C2.17(D4×Q8), C4⋊C4.331D4, C22⋊C414Q8, C23.15(C2×Q8), C2.48(D45D4), C23⋊Q8.4C2, C2.16(D43Q8), C23.Q8.4C2, C22.82(C22×Q8), (C22×C4).818C23, (C23×C4).358C22, (C2×C42).508C22, C22.245(C22×D4), C23.8Q8.15C2, (C22×Q8).111C22, C23.65C2359C2, C23.78C2311C2, C23.67C2345C2, C23.63C2344C2, C2.37(C22.19C24), C24.C22.14C2, C2.C42.122C22, C2.18(C23.37C23), C2.15(C22.50C24), C2.22(C22.36C24), (C2×C4×Q8)⋊19C2, (C2×C4).35(C2×Q8), (C2×C4).340(C2×D4), (C4×C22⋊C4).43C2, (C2×C22⋊Q8).28C2, (C2×C4).115(C4○D4), (C2×C4⋊C4).245C22, C22.242(C2×C4○D4), (C2×C22⋊C4).497C22, SmallGroup(128,1197)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C24.285C23
C1C2C22C23C22×C4C23×C4C4×C22⋊C4 — C24.285C23
C1C23 — C24.285C23
C1C23 — C24.285C23
C1C23 — C24.285C23

Generators and relations for C24.285C23
 G = < a,b,c,d,e,f,g | a2=b2=c2=f2=1, d2=e2=a, g2=ba=ab, ac=ca, ede-1=gdg-1=ad=da, ae=ea, af=fa, ag=ga, bc=cb, fdf=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef=ce=ec, cf=fc, cg=gc, eg=ge, fg=gf >

Subgroups: 452 in 254 conjugacy classes, 108 normal (82 characteristic)
C1, C2 [×7], C2 [×2], C4 [×22], C22 [×7], C22 [×10], C2×C4 [×16], C2×C4 [×38], Q8 [×12], C23, C23 [×2], C23 [×6], C42 [×7], C22⋊C4 [×4], C22⋊C4 [×9], C4⋊C4 [×4], C4⋊C4 [×16], C22×C4 [×14], C22×C4 [×5], C2×Q8 [×11], C24, C2.C42 [×12], C2×C42 [×4], C2×C22⋊C4 [×6], C2×C4⋊C4 [×10], C4×Q8 [×4], C22⋊Q8 [×4], C23×C4, C22×Q8 [×2], C4×C22⋊C4, C23.8Q8, C23.63C23 [×3], C24.C22 [×2], C23.65C23, C23.67C23, C23⋊Q8, C23.78C23 [×2], C23.Q8, C2×C4×Q8, C2×C22⋊Q8, C24.285C23
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×4], C23 [×15], C2×D4 [×6], C2×Q8 [×6], C4○D4 [×6], C24, C22×D4, C22×Q8, C2×C4○D4 [×3], 2+ 1+4, 2- 1+4, C22.19C24, C23.37C23, C22.36C24, D45D4, D4×Q8, D43Q8, C22.50C24, C24.285C23

Smallest permutation representation of C24.285C23
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 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 63 3 61)(2 62 4 64)(5 58 7 60)(6 57 8 59)(9 19 11 17)(10 18 12 20)(13 23 15 21)(14 22 16 24)(25 36 27 34)(26 35 28 33)(29 40 31 38)(30 39 32 37)(41 51 43 49)(42 50 44 52)(45 55 47 53)(46 54 48 56)
(2 42)(4 44)(5 62)(6 51)(7 64)(8 49)(10 58)(12 60)(14 26)(16 28)(17 63)(18 52)(19 61)(20 50)(21 40)(22 56)(23 38)(24 54)(30 46)(32 48)(33 39)(34 55)(35 37)(36 53)
(1 15 43 25)(2 14 44 28)(3 13 41 27)(4 16 42 26)(5 56 18 39)(6 55 19 38)(7 54 20 37)(8 53 17 40)(9 31 59 45)(10 30 60 48)(11 29 57 47)(12 32 58 46)(21 49 36 63)(22 52 33 62)(23 51 34 61)(24 50 35 64)

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

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

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

38 conjugacy classes

class 1 2A···2G2H2I4A···4H4I···4X4Y4Z4AA4AB
order12···2224···44···44444
size11···1442···24···48888

38 irreducible representations

dim11111111111122244
type++++++++++++-++-
imageC1C2C2C2C2C2C2C2C2C2C2C2Q8D4C4○D42+ 1+42- 1+4
kernelC24.285C23C4×C22⋊C4C23.8Q8C23.63C23C24.C22C23.65C23C23.67C23C23⋊Q8C23.78C23C23.Q8C2×C4×Q8C2×C22⋊Q8C22⋊C4C4⋊C4C2×C4C22C22
# reps111321112111441211

Matrix representation of C24.285C23 in GL6(𝔽5)

100000
010000
001000
000100
000040
000004
,
400000
040000
001000
000100
000010
000001
,
400000
040000
004000
000400
000010
000001
,
010000
100000
004000
000400
000020
000003
,
010000
100000
000100
001000
000001
000040
,
100000
040000
001000
000400
000010
000001
,
200000
020000
004000
000400
000001
000040

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

C24.285C23 in GAP, Magma, Sage, TeX

C_2^4._{285}C_2^3
% in TeX

G:=Group("C2^4.285C2^3");
// GroupNames label

G:=SmallGroup(128,1197);
// by ID

G=gap.SmallGroup(128,1197);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,120,758,723,675,192]);
// Polycyclic

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

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