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G = C24.311C23order 128 = 27

151st non-split extension by C24 of C23 acting via C23/C2=C22

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

Aliases: C24.311C23, C23.421C24, C22.2142+ 1+4, C22.1632- 1+4, C22⋊C4.74D4, C23.44(C2×D4), C2.48(D46D4), C2.64(D45D4), C23.4Q818C2, C23.7Q860C2, C23.11D437C2, (C2×C42).536C22, (C23×C4).385C22, (C22×C4).531C23, C22.284(C22×D4), C24.C2273C2, C23.10D4.15C2, C23.23D4.31C2, (C22×D4).157C22, (C22×Q8).124C22, C23.78C2314C2, C23.63C2374C2, C23.67C2354C2, C2.37(C22.45C24), C2.C42.169C22, C2.29(C22.26C24), C2.28(C22.50C24), C2.38(C22.36C24), C2.64(C23.36C23), (C4×C4⋊C4)⋊80C2, (C2×C4).69(C2×D4), (C4×C22⋊C4)⋊80C2, (C2×C422C2)⋊8C2, (C2×C4).140(C4○D4), (C2×C4⋊C4).284C22, (C2×C4.4D4).23C2, C22.298(C2×C4○D4), (C2×C22⋊C4).166C22, SmallGroup(128,1253)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C24.311C23
C1C2C22C23C22×C4C23×C4C4×C22⋊C4 — C24.311C23
C1C23 — C24.311C23
C1C23 — C24.311C23
C1C23 — C24.311C23

Generators and relations for C24.311C23
 G = < a,b,c,d,e,f,g | a2=b2=c2=f2=1, d2=c, 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: 500 in 259 conjugacy classes, 100 normal (82 characteristic)
C1, C2 [×7], C2 [×3], C4 [×19], C22 [×7], C22 [×17], C2×C4 [×12], C2×C4 [×37], D4 [×4], Q8 [×4], C23, C23 [×2], C23 [×13], C42 [×7], C22⋊C4 [×4], C22⋊C4 [×19], C4⋊C4 [×13], C22×C4 [×13], C22×C4 [×5], C2×D4 [×5], C2×Q8 [×5], C24 [×2], C2.C42 [×10], C2×C42 [×4], C2×C22⋊C4 [×11], C2×C4⋊C4 [×7], C4.4D4 [×4], C422C2 [×4], C23×C4, C22×D4, C22×Q8, C4×C22⋊C4, C4×C4⋊C4, C23.7Q8, C23.23D4, C23.63C23, C24.C22 [×3], C23.67C23, C23.10D4, C23.78C23, C23.11D4, C23.4Q8, C2×C4.4D4, C2×C422C2, C24.311C23
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- 1+4, C23.36C23, C22.26C24, C22.36C24, D45D4, D46D4, C22.45C24, C22.50C24, C24.311C23

Smallest permutation representation of C24.311C23
On 64 points
Generators in S64
(1 25)(2 26)(3 27)(4 28)(5 54)(6 55)(7 56)(8 53)(9 48)(10 45)(11 46)(12 47)(13 52)(14 49)(15 50)(16 51)(17 39)(18 40)(19 37)(20 38)(21 42)(22 43)(23 44)(24 41)(29 33)(30 34)(31 35)(32 36)(57 64)(58 61)(59 62)(60 63)
(1 61)(2 62)(3 63)(4 64)(5 31)(6 32)(7 29)(8 30)(9 23)(10 24)(11 21)(12 22)(13 17)(14 18)(15 19)(16 20)(25 58)(26 59)(27 60)(28 57)(33 56)(34 53)(35 54)(36 55)(37 50)(38 51)(39 52)(40 49)(41 45)(42 46)(43 47)(44 48)
(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 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 21 25 42)(2 43 26 22)(3 23 27 44)(4 41 28 24)(5 20 54 38)(6 39 55 17)(7 18 56 40)(8 37 53 19)(9 60 48 63)(10 64 45 57)(11 58 46 61)(12 62 47 59)(13 32 52 36)(14 33 49 29)(15 30 50 34)(16 35 51 31)
(1 51)(2 39)(3 49)(4 37)(5 9)(6 24)(7 11)(8 22)(10 32)(12 30)(13 59)(14 27)(15 57)(16 25)(17 26)(18 60)(19 28)(20 58)(21 29)(23 31)(33 42)(34 47)(35 44)(36 45)(38 61)(40 63)(41 55)(43 53)(46 56)(48 54)(50 64)(52 62)
(1 30 58 53)(2 35 59 5)(3 32 60 55)(4 33 57 7)(6 27 36 63)(8 25 34 61)(9 39 44 13)(10 18 41 49)(11 37 42 15)(12 20 43 51)(14 45 40 24)(16 47 38 22)(17 23 52 48)(19 21 50 46)(26 31 62 54)(28 29 64 56)

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

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

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

38 conjugacy classes

class 1 2A···2G2H2I2J4A···4H4I···4X4Y4Z4AA
order12···22224···44···4444
size11···14482···24···4888

38 irreducible representations

dim111111111111112244
type++++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2C2C2D4C4○D42+ 1+42- 1+4
kernelC24.311C23C4×C22⋊C4C4×C4⋊C4C23.7Q8C23.23D4C23.63C23C24.C22C23.67C23C23.10D4C23.78C23C23.11D4C23.4Q8C2×C4.4D4C2×C422C2C22⋊C4C2×C4C22C22
# reps1111113111111141611

Matrix representation of C24.311C23 in GL6(𝔽5)

400000
040000
004000
000400
000010
000001
,
100000
010000
004000
000400
000040
000004
,
100000
010000
001000
000100
000040
000004
,
400000
310000
003100
002200
000024
000003
,
320000
020000
003100
000200
000043
000001
,
100000
010000
001200
000400
000010
000044
,
320000
020000
004300
000100
000020
000002

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

C24.311C23 in GAP, Magma, Sage, TeX

C_2^4._{311}C_2^3
% in TeX

G:=Group("C2^4.311C2^3");
// GroupNames label

G:=SmallGroup(128,1253);
// by ID

G=gap.SmallGroup(128,1253);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,120,758,723,675,192]);
// Polycyclic

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