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G = C42.388C23order 128 = 27

249th non-split extension by C42 of C23 acting via C23/C2=C22

p-group, metabelian, nilpotent (class 3), monomial

Aliases: C42.388C23, (C4×D8)⋊41C2, C82D46C2, C87D436C2, C84D419C2, C4⋊C4.248D4, D8⋊C414C2, C8.11(C4○D4), C2.25(D4○D8), C22⋊C4.88D4, C8.5Q818C2, C23.85(C2×D4), C4⋊C4.115C23, (C4×C8).224C22, (C2×C8).567C23, (C2×C4).374C24, (C2×D8).64C22, (C4×D4).95C22, C82M4(2)⋊18C2, C4.Q8.26C22, (C2×D4).129C23, C41D4.65C22, C4⋊D4.36C22, C8⋊C4.131C22, C2.D8.185C22, (C22×C8).303C22, C22.634(C22×D4), C42.C2.20C22, D4⋊C4.207C22, C22.34C243C2, (C22×C4).1054C23, C42.29C2221C2, C42⋊C2.331C22, (C2×M4(2)).284C22, C2.71(C22.26C24), C4.59(C2×C4○D4), (C2×C4).146(C2×D4), SmallGroup(128,1908)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.388C23
C1C2C4C2×C4C42C4×C8C82M4(2) — C42.388C23
C1C2C2×C4 — C42.388C23
C1C22C42⋊C2 — C42.388C23
C1C2C2C2×C4 — C42.388C23

Generators and relations for C42.388C23
 G = < a,b,c,d,e | a4=b4=c2=1, d2=b2, e2=cbc=b-1, ab=ba, ac=ca, dad-1=ab2, ae=ea, bd=db, be=eb, dcd-1=a2c, ece-1=b-1c, de=ed >

Subgroups: 428 in 193 conjugacy classes, 88 normal (26 characteristic)
C1, C2, C2 [×2], C2 [×5], C4 [×2], C4 [×9], C22, C22 [×15], C8 [×4], C8 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×10], D4 [×16], C23, C23 [×4], C42 [×2], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4 [×6], C4⋊C4 [×4], C2×C8 [×4], C2×C8 [×2], M4(2) [×2], D8 [×8], C22×C4, C22×C4 [×4], C2×D4 [×4], C2×D4 [×8], C4×C8 [×2], C8⋊C4 [×2], D4⋊C4 [×8], C4.Q8 [×2], C2.D8 [×2], C42⋊C2, C4×D4 [×4], C4⋊D4 [×4], C4⋊D4 [×4], C22.D4 [×4], C42.C2 [×2], C41D4 [×2], C22×C8, C2×M4(2), C2×D8 [×4], C82M4(2), C4×D8 [×2], D8⋊C4 [×2], C87D4 [×2], C82D4 [×2], C42.29C22 [×2], C84D4, C8.5Q8, C22.34C24 [×2], C42.388C23
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C22×D4, C2×C4○D4 [×2], C22.26C24, D4○D8 [×2], C42.388C23

Smallest permutation representation of C42.388C23
On 64 points
Generators in S64
(1 35 27 10)(2 36 28 11)(3 37 29 12)(4 38 30 13)(5 39 31 14)(6 40 32 15)(7 33 25 16)(8 34 26 9)(17 58 45 54)(18 59 46 55)(19 60 47 56)(20 61 48 49)(21 62 41 50)(22 63 42 51)(23 64 43 52)(24 57 44 53)
(1 7 5 3)(2 8 6 4)(9 15 13 11)(10 16 14 12)(17 23 21 19)(18 24 22 20)(25 31 29 27)(26 32 30 28)(33 39 37 35)(34 40 38 36)(41 47 45 43)(42 48 46 44)(49 55 53 51)(50 56 54 52)(57 63 61 59)(58 64 62 60)
(1 42)(2 41)(3 48)(4 47)(5 46)(6 45)(7 44)(8 43)(9 64)(10 63)(11 62)(12 61)(13 60)(14 59)(15 58)(16 57)(17 32)(18 31)(19 30)(20 29)(21 28)(22 27)(23 26)(24 25)(33 53)(34 52)(35 51)(36 50)(37 49)(38 56)(39 55)(40 54)
(1 51 5 55)(2 52 6 56)(3 53 7 49)(4 54 8 50)(9 45 13 41)(10 46 14 42)(11 47 15 43)(12 48 16 44)(17 38 21 34)(18 39 22 35)(19 40 23 36)(20 33 24 37)(25 61 29 57)(26 62 30 58)(27 63 31 59)(28 64 32 60)
(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)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F2G2H4A···4F4G4H4I4J4K4L4M8A8B8C8D8E···8J
order1222222224···4444444488888···8
size1111488882···2444888822224···4

32 irreducible representations

dim11111111112224
type+++++++++++++
imageC1C2C2C2C2C2C2C2C2C2D4D4C4○D4D4○D8
kernelC42.388C23C82M4(2)C4×D8D8⋊C4C87D4C82D4C42.29C22C84D4C8.5Q8C22.34C24C22⋊C4C4⋊C4C8C2
# reps11222221122284

Matrix representation of C42.388C23 in GL6(𝔽17)

400000
040000
000010
000001
0016000
0001600
,
100000
010000
0011500
0011600
0000115
0000116
,
040000
1300000
0049100
00013107
007049
00710013
,
010000
100000
007049
0007413
0049100
00413010
,
100000
010000
000600
0014600
000006
0000146

G:=sub<GL(6,GF(17))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,15,16,0,0,0,0,0,0,1,1,0,0,0,0,15,16],[0,13,0,0,0,0,4,0,0,0,0,0,0,0,4,0,7,7,0,0,9,13,0,10,0,0,10,10,4,0,0,0,0,7,9,13],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,7,0,4,4,0,0,0,7,9,13,0,0,4,4,10,0,0,0,9,13,0,10],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,14,0,0,0,0,6,6,0,0,0,0,0,0,0,14,0,0,0,0,6,6] >;

C42.388C23 in GAP, Magma, Sage, TeX

C_4^2._{388}C_2^3
% in TeX

G:=Group("C4^2.388C2^3");
// GroupNames label

G:=SmallGroup(128,1908);
// by ID

G=gap.SmallGroup(128,1908);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,456,758,520,1018,80,4037,124]);
// Polycyclic

G:=Group<a,b,c,d,e|a^4=b^4=c^2=1,d^2=b^2,e^2=c*b*c=b^-1,a*b=b*a,a*c=c*a,d*a*d^-1=a*b^2,a*e=e*a,b*d=d*b,b*e=e*b,d*c*d^-1=a^2*c,e*c*e^-1=b^-1*c,d*e=e*d>;
// generators/relations

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