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G = C24.293C23order 128 = 27

133rd non-split extension by C24 of C23 acting via C23/C2=C22

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

Aliases: C24.293C23, C23.376C24, C22.1802+ 1+4, C22⋊C4.133D4, C23.179(C2×D4), C2.57(D45D4), C23.37(C4○D4), (C2×C42).35C22, C23.8Q856C2, C23.23D450C2, C23.10D434C2, (C23×C4).364C22, (C22×C4).822C23, C22.256(C22×D4), C24.C2256C2, C24.3C2246C2, (C22×D4).141C22, C23.81C2320C2, C23.63C2354C2, C2.48(C22.19C24), C2.C42.132C22, C2.15(C22.34C24), C2.11(C22.49C24), C2.28(C22.47C24), C2.42(C23.36C23), (C4×C22⋊C4)⋊70C2, (C2×C4).902(C2×D4), (C2×C4⋊D4).30C2, (C2×C42⋊C2)⋊25C2, (C2×C4).371(C4○D4), (C2×C4⋊C4).253C22, C22.253(C2×C4○D4), (C2×C22⋊C4).459C22, SmallGroup(128,1208)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C24.293C23
C1C2C22C23C22×C4C23×C4C4×C22⋊C4 — C24.293C23
C1C23 — C24.293C23
C1C23 — C24.293C23
C1C23 — C24.293C23

Generators and relations for C24.293C23
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=f2=1, e2=b, 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: 596 in 291 conjugacy classes, 100 normal (82 characteristic)
C1, C2 [×7], C2 [×5], C4 [×17], C22 [×7], C22 [×27], C2×C4 [×10], C2×C4 [×43], D4 [×12], C23, C23 [×4], C23 [×19], C42 [×5], C22⋊C4 [×4], C22⋊C4 [×19], C4⋊C4 [×10], C22×C4 [×12], C22×C4 [×11], C2×D4 [×15], C24 [×3], C2.C42 [×8], C2×C42 [×3], C2×C22⋊C4 [×13], C2×C4⋊C4 [×6], C42⋊C2 [×4], C4⋊D4 [×4], C23×C4 [×2], C22×D4 [×3], C4×C22⋊C4, C23.8Q8, C23.23D4 [×2], C23.63C23, C24.C22 [×3], C24.3C22, C23.10D4 [×3], C23.81C23, C2×C42⋊C2, C2×C4⋊D4, C24.293C23
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×8], C24, C22×D4, C2×C4○D4 [×4], 2+ 1+4 [×2], C22.19C24, C23.36C23, C22.34C24, D45D4 [×2], C22.47C24, C22.49C24, C24.293C23

Smallest permutation representation of C24.293C23
On 64 points
Generators in S64
(1 39)(2 40)(3 37)(4 38)(5 17)(6 18)(7 19)(8 20)(9 41)(10 42)(11 43)(12 44)(13 45)(14 46)(15 47)(16 48)(21 49)(22 50)(23 51)(24 52)(25 53)(26 54)(27 55)(28 56)(29 57)(30 58)(31 59)(32 60)(33 62)(34 63)(35 64)(36 61)
(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 51)(2 52)(3 49)(4 50)(5 36)(6 33)(7 34)(8 35)(9 53)(10 54)(11 55)(12 56)(13 57)(14 58)(15 59)(16 60)(17 61)(18 62)(19 63)(20 64)(21 37)(22 38)(23 39)(24 40)(25 41)(26 42)(27 43)(28 44)(29 45)(30 46)(31 47)(32 48)
(1 19)(2 8)(3 17)(4 6)(5 37)(7 39)(9 15)(10 48)(11 13)(12 46)(14 44)(16 42)(18 38)(20 40)(21 36)(22 62)(23 34)(24 64)(25 31)(26 60)(27 29)(28 58)(30 56)(32 54)(33 50)(35 52)(41 47)(43 45)(49 61)(51 63)(53 59)(55 57)
(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 15)(2 60)(3 13)(4 58)(5 41)(6 26)(7 43)(8 28)(9 17)(10 62)(11 19)(12 64)(14 50)(16 52)(18 54)(20 56)(21 29)(22 46)(23 31)(24 48)(25 36)(27 34)(30 38)(32 40)(33 42)(35 44)(37 45)(39 47)(49 57)(51 59)(53 61)(55 63)
(1 41 37 11)(2 42 38 12)(3 43 39 9)(4 44 40 10)(5 45 19 15)(6 46 20 16)(7 47 17 13)(8 48 18 14)(21 55 51 25)(22 56 52 26)(23 53 49 27)(24 54 50 28)(29 63 59 36)(30 64 60 33)(31 61 57 34)(32 62 58 35)

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

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

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

38 conjugacy classes

class 1 2A···2G2H2I2J2K2L4A···4H4I···4V4W4X4Y
order12···2222224···44···4444
size11···1444482···24···4888

38 irreducible representations

dim111111111112224
type+++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2D4C4○D4C4○D42+ 1+4
kernelC24.293C23C4×C22⋊C4C23.8Q8C23.23D4C23.63C23C24.C22C24.3C22C23.10D4C23.81C23C2×C42⋊C2C2×C4⋊D4C22⋊C4C2×C4C23C22
# reps1112131311141242

Matrix representation of C24.293C23 in GL6(𝔽5)

100000
010000
004000
000400
000010
000001
,
400000
040000
001000
000100
000040
000004
,
100000
010000
001000
000100
000040
000004
,
030000
200000
001200
000400
000003
000020
,
200000
020000
003100
002200
000001
000040
,
010000
100000
004000
000400
000001
000010
,
300000
030000
004300
001100
000030
000003

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],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[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],[0,2,0,0,0,0,3,0,0,0,0,0,0,0,1,0,0,0,0,0,2,4,0,0,0,0,0,0,0,2,0,0,0,0,3,0],[2,0,0,0,0,0,0,2,0,0,0,0,0,0,3,2,0,0,0,0,1,2,0,0,0,0,0,0,0,4,0,0,0,0,1,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[3,0,0,0,0,0,0,3,0,0,0,0,0,0,4,1,0,0,0,0,3,1,0,0,0,0,0,0,3,0,0,0,0,0,0,3] >;

C24.293C23 in GAP, Magma, Sage, TeX

C_2^4._{293}C_2^3
% in TeX

G:=Group("C2^4.293C2^3");
// GroupNames label

G:=SmallGroup(128,1208);
// by ID

G=gap.SmallGroup(128,1208);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,253,232,758,723,675,192]);
// Polycyclic

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