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G = C24.249C23order 128 = 27

89th non-split extension by C24 of C23 acting via C23/C2=C22

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

Aliases: C24.249C23, C23.314C24, C22.1292+ 1+4, C4⋊C433D4, (C22×C4)⋊23D4, C232D48C2, C2.7(Q86D4), C23.153(C2×D4), C2.19(D45D4), C4.162(C4⋊D4), C23.20(C4○D4), (C22×C4).49C23, C23.7Q834C2, C23.10D410C2, (C2×C42).463C22, (C23×C4).332C22, C22.194(C22×D4), C24.C2231C2, C24.3C2228C2, (C22×D4).502C22, C23.65C2334C2, C2.21(C22.19C24), C2.11(C22.29C24), C2.C42.78C22, C2.6(C22.49C24), C2.12(C22.47C24), (C2×C4×D4)⋊21C2, (C2×C4⋊D4)⋊5C2, (C2×C4).679(C2×D4), C2.18(C2×C4⋊D4), (C2×C4).94(C4○D4), (C2×C42⋊C2)⋊20C2, (C2×C4⋊C4).205C22, C22.193(C2×C4○D4), (C2×C22⋊C4).109C22, SmallGroup(128,1146)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C24.249C23
C1C2C22C23C22×C4C23×C4C2×C4×D4 — C24.249C23
C1C23 — C24.249C23
C1C23 — C24.249C23
C1C23 — C24.249C23

Generators and relations for C24.249C23
 G = < a,b,c,d,e,f,g | a2=b2=c2=e2=f2=1, d2=b, g2=a, ab=ba, ac=ca, ede=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: 740 in 352 conjugacy classes, 112 normal (42 characteristic)
C1, C2 [×7], C2 [×6], C4 [×4], C4 [×14], C22 [×7], C22 [×34], C2×C4 [×14], C2×C4 [×38], D4 [×28], C23, C23 [×4], C23 [×26], C42 [×6], C22⋊C4 [×24], C4⋊C4 [×4], C4⋊C4 [×8], C22×C4 [×5], C22×C4 [×10], C22×C4 [×10], C2×D4 [×30], C24 [×2], C24 [×2], C2.C42 [×4], C2×C42, C2×C42 [×2], C2×C22⋊C4 [×2], C2×C22⋊C4 [×12], C2×C4⋊C4 [×4], C2×C4⋊C4 [×2], C42⋊C2 [×4], C4×D4 [×4], C4⋊D4 [×8], C23×C4 [×2], C22×D4 [×2], C22×D4 [×4], C23.7Q8, C24.C22 [×2], C23.65C23, C24.3C22, C24.3C22 [×2], C232D4 [×2], C23.10D4 [×2], C2×C42⋊C2, C2×C4×D4, C2×C4⋊D4 [×2], C24.249C23
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C4○D4 [×6], C24, C4⋊D4 [×4], C22×D4 [×2], C2×C4○D4 [×3], 2+ 1+4 [×2], C2×C4⋊D4, C22.19C24, C22.29C24, D45D4, Q86D4, C22.47C24, C22.49C24, C24.249C23

Smallest permutation representation of C24.249C23
On 64 points
Generators in S64
(1 23)(2 24)(3 21)(4 22)(5 16)(6 13)(7 14)(8 15)(9 27)(10 28)(11 25)(12 26)(17 64)(18 61)(19 62)(20 63)(29 39)(30 40)(31 37)(32 38)(33 50)(34 51)(35 52)(36 49)(41 55)(42 56)(43 53)(44 54)(45 59)(46 60)(47 57)(48 58)
(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 25)(2 26)(3 27)(4 28)(5 38)(6 39)(7 40)(8 37)(9 21)(10 22)(11 23)(12 24)(13 29)(14 30)(15 31)(16 32)(17 51)(18 52)(19 49)(20 50)(33 63)(34 64)(35 61)(36 62)(41 47)(42 48)(43 45)(44 46)(53 59)(54 60)(55 57)(56 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 41)(2 56)(3 43)(4 54)(5 51)(6 35)(7 49)(8 33)(9 59)(10 46)(11 57)(12 48)(13 52)(14 36)(15 50)(16 34)(17 38)(18 29)(19 40)(20 31)(21 53)(22 44)(23 55)(24 42)(25 47)(26 58)(27 45)(28 60)(30 62)(32 64)(37 63)(39 61)
(1 2)(3 4)(5 15)(6 14)(7 13)(8 16)(9 10)(11 12)(17 50)(18 49)(19 52)(20 51)(21 22)(23 24)(25 26)(27 28)(29 40)(30 39)(31 38)(32 37)(33 64)(34 63)(35 62)(36 61)(41 58)(42 57)(43 60)(44 59)(45 54)(46 53)(47 56)(48 55)
(1 13 23 6)(2 7 24 14)(3 15 21 8)(4 5 22 16)(9 37 27 31)(10 32 28 38)(11 39 25 29)(12 30 26 40)(17 46 64 60)(18 57 61 47)(19 48 62 58)(20 59 63 45)(33 43 50 53)(34 54 51 44)(35 41 52 55)(36 56 49 42)

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

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

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

38 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M4A···4H4I···4V4W4X
order12···22222224···44···444
size11···14444882···24···488

38 irreducible representations

dim111111111122224
type+++++++++++++
imageC1C2C2C2C2C2C2C2C2C2D4D4C4○D4C4○D42+ 1+4
kernelC24.249C23C23.7Q8C24.C22C23.65C23C24.3C22C232D4C23.10D4C2×C42⋊C2C2×C4×D4C2×C4⋊D4C4⋊C4C22×C4C2×C4C23C22
# reps112132211244842

Matrix representation of C24.249C23 in GL6(𝔽5)

400000
040000
001000
000100
000010
000001
,
100000
010000
004000
000400
000010
000001
,
100000
010000
001000
000100
000040
000004
,
100000
040000
000400
001000
000040
000004
,
020000
300000
001000
000100
000001
000010
,
100000
010000
000100
001000
000040
000001
,
010000
400000
001000
000100
000040
000004

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

C24.249C23 in GAP, Magma, Sage, TeX

C_2^4._{249}C_2^3
% in TeX

G:=Group("C2^4.249C2^3");
// GroupNames label

G:=SmallGroup(128,1146);
// by ID

G=gap.SmallGroup(128,1146);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,758,723,675,80]);
// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=e^2=f^2=1,d^2=b,g^2=a,a*b=b*a,a*c=c*a,e*d*e=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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